Conductance of junctions of multiple interacting quantum wires andlong Aharonov-Bohm-Kondo ringsbyZheng ShiB. Sc., Peking University, 2010M. Sc., The University Of British Columbia, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Physics)The University Of British Columbia(Vancouver)July 2017c© Zheng Shi, 2017AbstractIn this thesis, we calculate the linear dc conductance of two types of multi-terminal interactingsystems: junctions of interacting quantum wires attached to Tomonaga-Luttinger liquid (TLL) leads,and closed and open long Aharonov-Bohm-Kondo (ABK) rings. In both cases, we obtain correctionsto the non-interacting Landauer formula, arising from interactions in the TLL leads and the quantumdot (QD) in the Kondo regime respectively.In junctions of interacting quantum wires, if the wires are attached to Fermi liquid (FL) leads, theconductance is formally given by the Landauer formula with renormalized single-particle S-matrixelements. If, however, the wires are attached to TLL leads, i.e. the interaction does not vanish evenin the leads, the conductance has an additional contribution dependent on the interaction strengthin the leads. We calculate this additional contribution both at the first order in interaction andin the random phase approximation (RPA), and heuristically relate the FL conductance to the TLLconductance through a “contact resistance” between an FL lead and a TLL wire.In long ABK rings, where the interaction is due to spin-flip scattering at a QD in the Kondoregime, the linear dc conductance consists of two parts: a disconnected part of the Landauer form,and a connected part that can be approximately eliminated at low temperatures. For a closed longABK ring, where the electric current is conserved in the ring, the high-temperature conductancehas qualitatively different behaviors for temperatures greater than and lower than the characteristicenergy scale vF/L, where vF is the Fermi velocity and L is the ring circumference. Meanwhile, foran open long ABK ring where electrons may leak into the side leads coupled to ring arms, as longas the ring arms have both small transmission and small reflection, the ring behaves as a two-pathAharonov-Bohm (AB) interferometer, and we predict the observation of a pi/2 phase shift due toscattering off the Kondo singlet formed at low energies around the impurity spin.iiLay SummaryIn nanostructures such as quantum wires and quantum dots, electrons are spatially confined alongthe wire or inside the dot, and become much more likely to meet each other in their motion. In-teractions between electrons are therefore especially important in nanostructures; combined withquantum mechanical laws, they can lead to many interesting properties that promise applicationsin quantum circuits and quantum computing. In this thesis, we theoretically study the effects ofinteractions between electrons on the electric current response to external voltages, focusing on twotypes of nanostructures: a junction connecting multiple quantum wires with interacting electrons,and also a nano-ring with an embedded quantum dot and a threading magnetic field. Our quantita-tive predictions are potentially applicable to quantum circuit design.iiiPrefaceThis thesis is based on notes I wrote during my PhD studies. The concept and scope of the re-search were developed collectively by my supervisor Prof. Ian Affleck, Dr. Yashar Komijani (forChapter 3) and me. I performed all analytical and numerical calculations in consultation with mysupervisor and (for Chapter 3) Dr. Yashar Komijani.Section 1.2 and Sections 2.3 to 2.9 have been published in the following paper: Zheng Shi andIan Affleck, Physics Review B 94, 035106 (2016).Section 1.3, Sections 3.2 to 3.8, Appendix B and most of Appendix C have been published inthe following paper: Zheng Shi and Yashar Komijani, Physics Review B 95, 075147 (2017).Appendix A has been published in the following paper: Zheng Shi, Journal of Statistical Me-chanics: Theory and Experiment (2016) 063106.I verify that all hyperlinks in the bibliography are active as of July 10, 2017.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Landauer formula for multi-terminal mesoscopic systems . . . . . . . . . . . . . . 21.2 Junctions of multiple interacting quantum wires . . . . . . . . . . . . . . . . . . . 41.3 Linear dc conductance through ABK rings . . . . . . . . . . . . . . . . . . . . . . 51.4 This thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Conductance of junctions of multiple interacting quantum wires with TLL leads . . 102.1 Bosonic approach to interacting quantum wires with TLL leads . . . . . . . . . . . 112.1.1 Uniform wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.2 TLL leads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Bosonic approach to junctions of two wires and Y-junctions . . . . . . . . . . . . . 162.2.1 2-lead junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.2 Y-junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Fermionic formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4 Wilsonian approach to S-matrix renormalization . . . . . . . . . . . . . . . . . . . 27v2.5 First-order perturbation theory conductance . . . . . . . . . . . . . . . . . . . . . 302.5.1 Zeroth order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.5.2 First order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.6 First-order Callan-Symanzik (CS) formulation of renormalization group (RG) . . . 392.7 RPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.7.1 Details of the RPA conductance . . . . . . . . . . . . . . . . . . . . . . . 442.7.2 Real space integral Eq. (2.118) . . . . . . . . . . . . . . . . . . . . . . . . 482.8 2-lead junctions and Y-junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.9 Conclusion and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 Conductance of long ABK rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.1 The spin-1/2 single-channel Kondo effect . . . . . . . . . . . . . . . . . . . . . . 543.2 Anderson model and Kondo model . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2.1 Screening and non-screening channels . . . . . . . . . . . . . . . . . . . . 583.2.2 Kondo model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.3 dc conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.3.1 Kubo formula in terms of screening and non-screening channels . . . . . . 633.3.2 Disconnected part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.3.3 Connected part and its low-temperature elimination . . . . . . . . . . . . . 683.4 Perturbation theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.4.1 Weak coupling perturbation theory . . . . . . . . . . . . . . . . . . . . . . 723.4.2 FL perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.5 Comparison with early results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.5.1 Short ABK ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.5.2 Finite quantum wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.6 Closed long ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.6.1 Kondo temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.6.2 High-temperature conductance . . . . . . . . . . . . . . . . . . . . . . . . 853.6.3 FL conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.7 Open long ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.7.1 Wave function on a single lossy arm . . . . . . . . . . . . . . . . . . . . . 913.7.2 Background S-matrix and coupling site wave functions . . . . . . . . . . . 933.7.3 Kondo temperature and conductance . . . . . . . . . . . . . . . . . . . . . 963.8 Conclusion and open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109viAppendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122A S-matrix RG equation and fixed points for 2-lead junctions and Y-junctions . . . . . 122A.1 First order in interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122A.1.1 2-lead junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122A.1.2 Y-junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123A.2 RPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128A.2.1 2-lead junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128A.2.2 Y-junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129B Details of the disconnected contribution . . . . . . . . . . . . . . . . . . . . . . . . . 134B.1 Properties of the S-matrix and the wave functions . . . . . . . . . . . . . . . . . . 135B.2 Background transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136B.3 Terms linear in T-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139B.4 Terms quadratic in T-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141C Details of weak-coupling and FL perturbation theory . . . . . . . . . . . . . . . . . . 144C.1 Weak-coupling perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . 144C.1.1 T-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144C.1.2 Connected contribution to the conductance . . . . . . . . . . . . . . . . . 146C.2 FL perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149C.2.1 T-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149C.2.2 Connected contribution to the conductance . . . . . . . . . . . . . . . . . 151viiList of TablesTable 3.1 Different regimes of energy scales discussed in this chapter. T , TK and EV arerespectively the temperature, the Kondo temperature, and the energy scale overwhich V 2k varies significantly. We also assume Econn ∼ EV , where Econn is theenergy scale over which S and Γ vary significantly. For the low-temperatureconductance in the small Kondo cloud regime, see discussion in Section 3.8. . . 105viiiList of FiguresFigure 2.1 Sketch of a quantum wire connected to two leads. The leads are modeled withLuttinger parameters KL and KR that are different from the Luttinger parameterof the wire, KW . For FL leads KL = KR = 1. . . . . . . . . . . . . . . . . . . . 13Figure 2.2 a) The sketch of a 2-lead junction, with possibly different Luttinger parametersand speeds of sound in the two wires. The wires extend from x = 0 to ∞, andthe junction is at x = 0. b) The RG fixed points of this model: D (smoothlyconnected wires) and N (decoupled wires). c) The RG flow diagram from thebosonic approach. RG flows are towards N when Ke < 1 (repulsive interactions)and towards D when Ke > 1 (attractive interactions); here Ke is the harmonicaverage of K1 and K2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Figure 2.3 a) Typical RG fixed points of the Y-junction, as predicted by the bosonic ap-proach (with the exception of the M fixed point): N (decoupled wires), χ+(chiral transmission which breaks time-reversal symmetry: 1 to 2, 2 to 3 and 3to 1), M (electrons incident from any wire can be reflected or transmitted), D(Andreev reflection) and the asymmetric A3 (1 and 2 are smoothly connectedand 3 is decoupled). b) The RG flow diagram for a Z3 symmetric Y-junctionat K < 1. The horizontal axis measures the coupling strength between any twowires, so the N fixed point occupies the entire vertical axis. The vertical axismeasures time-reversal symmetry breaking (e.g. due to an AB flux at the junc-tion), which means the time-reversal symmetric M should be on the horizontalaxis, but χ± should not. RG flows are always towards N when K < 1, althoughit is found from the fermionic approach that M is more stable than χ±. c) Sameas b) but with 1 < K < 3. RG flows are towards χ± whenever time-reversalsymmetry is broken; in the time-reversal symmetric case, however, the RG flowis from N to M. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24ixFigure 2.4 Sketch of a typical junction system with the number of quantum wires N = 3.The shaded junction area to the left is modeled by quadratic hopping terms be-tween ends of the wires, which are all aligned at the origin x= 0. The electron-electron interaction strengths inside the wires are also plotted for comparison.The interaction strength in wire j= 1 is uniform, and that in wire j= 2 also goesto a constant nonzero value as x→ ∞; these two wires are said to be attachedto TLL leads. In contrast, the wire j = 3 has a vanishing interaction strength faraway from the junction, and is connected to an FL lead (the FL lead itself is toowide to be shown in full). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Figure 2.5 Diagrammatic representation of the electron-electron interaction. . . . . . . . . 28Figure 2.6 Diagrams contributing to the linear dc conductance at the first order in inter-action. The second line shows the self-energy dressed bubble diagrams, whilevertex correction diagrams are in the third line. . . . . . . . . . . . . . . . . . 32Figure 2.7 Dressing of the first order vertex correction diagrams by the first order self-energy diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Figure 2.8 The RPA diagrammatics: (a) effective interaction in the RPA represented by thickwavy lines; (b) dressed propagator in the RPA, to O(δD/D) in RG, representedby thick straight lines; and (c) diagrams contributing to the Kubo conductancein the RPA. The dressed propagator in (b) is calculated to O(δD/D) only, be-cause higher order terms in δD/D do not contribute to the renormalization ofthe S-matrix [Eq. (2.106)]— see Section 2.6 for an explanation in the first ordercontext. (a) and (c) do not involve truncation at O(δD/D) because any renor-malization of the interaction [Eq. (2.107)] and the conductance [Eqs. (2.105)and (2.109)] can be attributed to the renormalization of the S-matrix. Note that(c) features a thin interaction line (rather than a thick one) to avoid double-counting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 2.9 Schematic representation of the relation between the conductance of a junc-tion with FL leads and that of a junction with TLL leads, through the “contactresistance”Eq. (2.30). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Figure 3.1 Sketch of a generic system which allows the application of our formalism. HereN = 5 and M = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59xFigure 3.2 Disconnected (self-energy) and connected (vertex correction) contributions tothe density-density correlation function, which is directly related to the conduc-tance through the Kubo formula Eq. (3.31). The dashed lines represent externallegs at times t¯ and 0, the solid lines represent fully dressed Ψ fermion propa-gators, and the hatched circle represents all connected 4-point vertices of thescreening channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Figure 3.3 Diagrammatics of weak-coupling perturbation theory. a) The vertices corre-sponding to the Kondo coupling and the potential scattering in Eq. (3.25a). b)Diagrams contributing to the T-matrix of the screening channel ψ electrons upto O(J2) ∼ O(K2). We have traced over the impurity spin so that the doubledashed lines (impurity spin propagators) form loops, and arranged the internaltime variables from left to right in increasing order. c) Connected diagramscontributing to the linear dc conductance up to O(J2). . . . . . . . . . . . . . 73Figure 3.4 Diagrammatics of FL perturbation theory. a) The two vertices given by theleading irrelevant operator, Eq. (3.87). b) Diagrams contributing to the T-matrixof ψ˜ electrons up to O(1/T 2K). The propagators are those of the phase-shiftedscreening channel operators ψ˜ . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Figure 3.5 The short ABK ring studied in Refs. [66, 75]. . . . . . . . . . . . . . . . . . . 80Figure 3.6 The finite quantum wire geometry studied in Ref. [116]. . . . . . . . . . . . . 81Figure 3.7 Geometry of the long ABK ring with short upper arms and a pinched referencearm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Figure 3.8 Kondo temperature TK for the closed long ABK ring, calculated by numericalintegration of the weak coupling RG equation Eq. (3.27), plotted against theAB phase ϕ . TK (ϕ) is an even function of ϕ and has a period of 2pi , so only0 ≤ ϕ ≤ pi is shown. System parameters are: dre f = 60, θ = pi/2, |r˜| = 0,tL = tR, D0 = 10. The curves with TK ∆ (small Kondo cloud regime) havea large bare Kondo coupling(t2L+ t2R)j0/pi = 0.15, whereas the curves withTK ∆ (large Kondo cloud regime) have a much smaller bare Kondo coupling(t2L+ t2R)j0/pi = 0.02. In the small cloud regime TK is almost independent of ϕand kF , as the curves are flat and overlapping with each other. In the large cloudregime, however, TK highly sensitive to both ϕ and kF . . . . . . . . . . . . . . 85xiFigure 3.9 Kondo-type correction to the conductance δG at T TK for the closed longABK ring with a particle-hole symmetric QD, calculated by RG improved per-turbation theory Eq. (3.106), plotted against the AB phase ϕ . Again only0 ≤ ϕ ≤ pi is shown. System parameters are: dre f = 60, θ = pi/2, |r˜| = 0,tL = tR,(t2L+ t2R)j0/pi = 0.02 at D0 = 10 (i.e. the system is in the large cloudregime). T/∆= 0.0955 in panel a) and T/∆= 19.1 in panel b). For T ∆ theconductance shows considerable kF dependence, while for T ∆ such depen-dence essentially vanishes and curves at different kF overlap. Also, for T ∆the first harmonic cosϕ drops out as predicted by Eq. (3.109), and δG(ϕ) hasa period of pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Figure 3.10 Geometry of the open long ABK ring. Side leads are appended to the QD armsand the reference arms, which are all of comparable lengths. . . . . . . . . . . 91Figure 3.11 A single lossy arm attached to side leads. . . . . . . . . . . . . . . . . . . . . 92Figure 3.12 Normalization factor V 2k from Eq. (3.19) as a function of k for different ABphases ϕ in the open long ABK ring, obtained by solving the full tight-bindingmodel. We focus on a small slice of momentum |k−pi/3| < 0.05. Two valuesof tx are considered: tx = 0 corresponding to the closed ring without electronleakage, and tx = 0.3t corresponding to strong leakage along and small trans-mission across the arms. System parameters are: dL = dR = dre f /2 = 100,tL,RJL = tL,RJQ = tL,RJR = t, and symmetric QD coupling tL = tR. For comparison wehave also plotted the analytic prediction Eq. (3.134) for tx = 0.3t, which agreesquantitatively with the full tight-binding solution. While V 2k for the closed ringis extremely sensitive to kF and ϕ , the sensitivity is strongly suppressed byelectron leakage, and curves for different ϕ overlap when tx = 0.3t. Since V 2kcontrols the renormalization of the Kondo coupling, the Kondo temperature ofthe open long ABK ring is not sensitive to mesoscopic details in the small trans-mission limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Figure 3.13 Low-temperature and high-temperature conductances G as functions of ABphase ϕ in the open long ABK ring with a particle-hole asymmetric QD, cal-culated with Eqs. (3.141a) and (3.145). We assume TK t so that the ther-mal averaging in the high temperature case is trivial. System parameters are:tx = 0.3t, kF = pi/3, dL = dR = dre f /2= 100, tL,RJL = tL,RJQ = tL,RJR = t, and particle-hole symmetry breaking phase shift δP = 0.1. A phase shift of approximatelypi/2 is clearly visible as the temperature is lowered and Kondo correlations be-come important. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101xiiFigure C.1 The three connected diagrams at O(T 2/T 2K)contributing to the conductance.ZS, ZS’ and BCS label only the topology of the diagrams and not necessarilythe physics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152xiiiList of AbbreviationsRG renormalization groupFL Fermi liquidTLL Tomonaga-Luttinger liquidQD quantum dotAB Aharonov-BohmABK Aharonov-Bohm-KondoRPA random phase approximationCS Callan-SymanzikxivAcknowledgmentsFirst of all, I am profoundly grateful to my supervisor, Professor Ian Affleck. He has been veryconsiderate and supportive throughout my time at Vancouver, and has influenced me as a physicistmore than anyone else does. I have benefited greatly from his deep insights and his extensiveknowledge of condensed matter physics, and perhaps learned as much from his attitude towardwork and life in general. I have been very fortunate to have the opportunity to work with him.It is a pleasure to thank Dr. Yashar Komijani for his contribution to the work presented inChapter 3, as well as many very enjoyable discussions. I would also like to thank my supervisorycommittee– Prof. Mona Berciu, Prof. Joshua Folk, and Prof. Ariel Zhitnitsky– whose questionsand suggestions were immensely helpful in improving this thesis.I want to thank the University of British Columbia. The Point Grey campus has been a wonderfulplace to live; also, the Department of Physics and Astronomy gave me a chance to meet many newfriends, and granted me teaching experience that I find invaluable in shaping my character.Finally, I cannot thank my parents enough for their support and encouragement. They are thebest parents in the world and I dedicate my thesis to them.xvChapter 1IntroductionQuantum wires and quantum dot (QD)s have received increasing attention from condensed mattertheorists and experimentalists alike[53]. One realization of a quantum wire has electrons in a two-dimensional electron gas confined in a lateral direction, so that any motion of these electrons is effec-tively one-dimensional, taking place in the direction perpendicular to the confinement. A QD, whichcomprises a small droplet of electrons, can also be realized by confining a two-dimensional electrongas in both lateral directions. Such spatial confinement at mesoscopic length scales not only ex-poses interference effects, as demonstrated by the Aharonov-Bohm (AB) effect in one-dimensionalrings, but also enhances interaction and correlation. In particular, an interacting quantum wire canbehave as a paradigmic Tomonaga-Luttinger liquid (TLL), characterized by bosonic elementary ex-citations formed by electron-hole pairs, together with the power law decay of various correlationfunctions with interaction-dependent exponents[44]. In another example, tunneling through a QDfeatures the Coulomb blockade phenomenon: unless the plunger gate voltage falls outside certainnarrow intervals (the Coulomb peaks), the number of electrons in the QD is always an integer, andit takes a finite energy to add/remove an electron to/from the QD due to the Coulomb repulsion,which strongly suppresses tunneling. Meanwhile, when an odd number of electrons are on the QD,tunneling will be anomalously enhanced at low temperatures due to the Kondo effect[53].Both quantum wires and QDs have a broad range of potential applications; in particular, theyare conceived to be the building blocks of quantum computers. In quantum-circuit-based quantumcomputers, qubits can be realized by the spin states of the QDs[73], which are coupled to eachother and manipulated externally through quantum wires; on the other hand, in topological quantumcomputers[31], the Majorana zero modes forming topological qubits are thought to live at the endsof quantum wires, and their braiding operations are achievable through quantum wire networks[8,52].In this thesis, we study two types of multi-terminal devices consisting of quantum wires andQDs, namely junctions of interacting quantum wires[111] and long Aharonov-Bohm-Kondo (ABK)1rings[112], with a focus on the influence of interactions on their transport properties. The former arean essential ingredient of any practical quantum circuit and may find utilization as nano-switches,while the latter prove useful in the quest for mesoscopic manifestations of Kondo physics. Beforeelaborating on our motivations, it is helpful to first revisit the non-interacting problem.1.1 Landauer formula for multi-terminal mesoscopic systemsIn this section, we illustrate a number of basic concepts by outlining the generic setup for a multi-terminal junction, and review the Landauer formula for the case without interactions[53, 54].Consider N quantum wires meeting at a junction; each wire extends from x= 0 to x→ ∞, withthe x→ ∞ part representing a reservoir (or a “lead”). We assume that each wire supports only asingle channel and a single spin orientation; this is not a crucial assumption as we may view eachchannel/spin in a realistic setup effectively as a separate “wire”. To define the linear dc conductance,we apply a weak bias voltage Vj to the reservoir connected to wire j, j = 1, 2, ..., N. The dc current(in the +x direction) probed in wire j′ is then given byI j′ =N∑j=1G j′ jVj; (1.1)the conductance tensor G j j′ obeys∑ jG j j′ =∑ j′G j j′ = 0, because the total current flowing out of thejunction is zero, and a uniform voltage applied to all leads results in no current. We emphasize thathere the conductance tensor is defined as the linear response coefficients to external bias voltages; itis possible that experimentally measured voltage drops also contain contributions from the electricpolarization[55, 65, 91, 104, 125].In the case where the junction, the wires and the reservoirs are all free from various interactions,the conductance tensor is directly related to the scattering S-matrix via the Landauer formula, whosederivation is sketched below.The purely quadratic Hamiltonian consists of a kinetic term, and a potential scattering termconfined in the junction region. The potential scattering vanishes inside the wires x j > 0, so thatthe scattering state incident from wire j′ at energy E has the following asymptotic wave function onwire j at x j > 0:Φ j′,E ( j,x j) =1√2piv j (E){δ j j′ exp [−ik j (E)x j]+S j j′ (E)exp [ik j (E)x j]}, (1.2)where k j (E) is the wave vector at energy E in wire j, and v j = (∂k j/∂E)−1 is the correspondinggroup velocity. (Hereafter we let h¯= 1.) The N×N scattering S-matrix S (E) has diagonal elementsrepresenting reflection amplitudes and off-diagonal elements representing transmission amplitudes.S (E) is a unitary matrix,2∑j′′S j j′′ (E)S∗j′ j′′ (E) = δ j j′ , (1.3)due to current conservation and orthogonality between different scattering states.The central assumption of the Landauer formula is that electrons leaving a reservoir (“source”)retain their shifted Fermi distribution in that reservoir, until they are scattered into the reservoirs(“sinks”) where they are immediately absorbed and reach local thermal equilibrium. If we let theequilibrium chemical potential be µ , in the presence of dc bias voltages Vj, the electrochemicalpotential in reservoir j becomes µ + eVj where e is the electron charge. Consequently, taking theexpectation value of the electric current operator Iˆ j = evˆ j in the scattering state Eq. (1.2) wherevˆ j is the velocity operator, we find the following contribution from electrons in the scattering stateincident from wire j′ to the current in wire j:I j, j′ =−e∫dEv j (E)12piv j (E)[δ j j′−∣∣S j j′ (E)∣∣2] f (E−µ− eVj′) ; (1.4)here f (ε) = 1/(eβε +1)is the Fermi function, with β = 1/(kBT ). The δ j j′ part comes from theincident component in Eq. (1.2) and the∣∣S j j′∣∣2 part comes from the scattered component. We haveneglected cross terms between the two components, which are associated with Friedel oscillations;contributions from these cross terms are suppressed by O(T/(vF jkF j)) relative to the terms that wehave retained, where vF j and kF j are respectively the Fermi velocity and the Fermi wave vector inwire j. Summing over j′ and expanding in powers ofVj, we find that the zeroth order term vanishesdue to the unitarity of S (E), i.e. there is no current in the absence of any bias voltage. We thusobtain the total current of the first order in V :I j =− e22pi∑j′∫dE[δ j j′−∣∣S j j′ (E)∣∣2][− f ′ (E−µ)]Vj′ . (1.5)Comparing with Eq. (1.1) we find the linear dc conductance tensor[53, 54]G j j′ =− e22pi∫dE[− f ′ (E−µ)][δ j j′− ∣∣S j j′ (E)∣∣2] . (1.6)As expected, it follows from the unitarity of S (E) that G j j′ satisfies ∑ jG j j′ = ∑ j′G j j′ = 0.The derivation above relies on the existence of noninteracting scattering states characterized bythe scattering S-matrix, and their thermal equilibrium with quasiparticles in the Fermi liquid (FL)reservoirs. As we will see in this thesis, interactions can change the Landauer picture significantlywhether they are in the reservoirs, in the wires or at the junction itself.31.2 Junctions of multiple interacting quantum wiresA class of powerful theoretical approaches to junctions of interacting quantum wires models thequantum wires as conformally invariant bulk TLL[40, 41, 51, 60, 61, 93, 130]. In the spirit ofboundary conformal field theory, at low energies the junction with its boundary operators shouldeventually renormalize to conformally invariant boundary conditions; it is these conformally invari-ant boundary conditions that control the low-energy physics of any realistic junction, regardless oftheir microscopic details. Possible fixed points of the renormalization group (RG) flow are thenpostulated, and their various properties, such as zero-temperature conductance and operator scalingdimensions, are explored. Details of the RG flow, however, are largely open to conjecture exceptin the vicinity of these fixed points. These approaches are often consolidated with the techniqueof bosonization, as the elementary excitations of TLLs are bosonic in nature, and various boundaryconditions imposed by the junction are often conveniently expressed in bosonic field variables.Alternate formalisms have been independently developed in the language of fermions. Con-sider first the junction system without bulk electron-electron interaction in the quantum wires; wefurther ignore local interactions so that this system becomes completely non-interacting. (For afree-fermion system, unless localized discrete states exist, local interactions are irrelevant in theRG sense and do not affect leading order physics.) A single-particle S-matrix determines thescattering basis, a new set of single-particle states which diagonalizes the non-interacting sys-tem. The bulk electron-electron interaction is then reintroduced and handled by perturbation the-ory, whose infrared divergence is resummed in an RG procedure. The S-matrix elements arenow scale-dependent coupling constants. In the simplest approximation scheme, both the renor-malized single-particle self-energy and the renormalized two-particle vertex depend only on therenormalized S-matrix at every point in the RG flow; the S-matrix elements (or the transmissionprobabilities) are thus treated as the only running coupling constants of the theory, and by solv-ing their RG equations we gain information about the conductance. This scheme is first adoptedby Refs. [28, 70, 77, 132] to the first order in interaction; various generalizations include resonanttunneling with an energy-dependent S-matrix[87, 99], second order perturbation theory in interac-tion in a setup with a tunnel electrode[16], random phase approximation (RPA) in interaction[11–15], superconducting junctions[27, 29, 30, 126], and refermionization of fixed points proposed bybosonic theory[45], to list a few. In particular, it has been found that the RPA in the Tomonaga-Luttinger model with a linear dispersion reproduces various scaling dimensions of the conductanceknown from bosonic methods. (The term RPA has been used interchangeably with “ladder ap-proximation” in Refs. [11, 13, 15].) An improved approximation, known as the functional RGmethod[18, 19, 36, 78–81], explicitly studies the flows of single-particle self-energy and the two-particle vertex. Despite its basis on perturbation theory in interaction, the functional RG showsexcellent agreement with analytic results at Luttinger parameter K = 1/2, and with numerical den-sity matrix renormalization group data for fairly strong interactions. The merit of these formalisms4are that the crossover behavior between different fixed points can, in principle, be found to anyorder in interaction. Nevertheless, when the interaction becomes sufficiently strong in a junctionof three wires (a “Y-junction”), the fixed points and the RG flow predicted by the RPA fermionicapproach and the bosonic approach begin to differ qualitatively. Also a careful analysis reveals that,in the RPA approach, the β function of the S-matrix beyond one-loop order contains non-universalterms[11, 13] which depend on the precise cutoff scheme of the theory, and may potentially changeits predictions.To our knowledge, many aspects of the junction problem have not been explored in the fermionicformalism. One such example is the well-known distinction between a semi-infinite TLL wire anda finite TLL wire connected to a FL reservoir[10, 42, 55, 56, 61, 65, 68, 69, 76, 90, 91, 97, 100, 105,106, 125]. There have been controversies on the nature of the conductance measured in a realisticexperimental setup[55, 125]. However, if we consider the linear dc response to an externally appliedbias voltage[65, 91], then it has been predicted that the corresponding linear response coefficients(which we have been referring to as the “conductances”) with and without the FL reservoirs aregenerally different[104]. (These coefficients are well-defined and can be studied numerically.) Forinstance, the conductance of a finite TLL wire attached to FL leads on both sides is e2/h, irrespectiveof the interaction strength; on the other hand, the conductance of an infinite spinless TLL wire isKe2/h, where K is the Luttinger parameter. The Landauer formula based on a perfectly transmittingS-matrix alone cannot recover the Ke2/h result. In existing literature employing the fermionicformalism, the case of FL leads has been well studied, but the effects of TLL leads on the conductanceare not discussed.The reasons are twofold for our interest in the effects of TLL leads on the conductance from thefermionic perspective. At the fixed points well understood in the bosonic approach, such as the per-fect transmission fixed point in the two-lead junction and the chiral fixed points in the Y-junction, theagreement of these results in both approaches is a necessary validation of the fermionic approach.On the other hand, for the fixed points eluding the bosonic treatment, such as the maximally openfixed point of the Y-junction (known as the “M fixed point”[13, 15, 19, 93, 99]), these results can bedirectly compared to numerics[104] where available.1.3 Linear dc conductance through ABK ringsAnother important ingredient in mesoscopic devices is the QD[26, 47, 48, 102, 114, 127]. Theconductance through a QD can be measured by embedding it between two reservoirs. When the QDhas a non-zero spin in its ground state, as is the case when there is an odd number of electrons,the conductance may undergo a strong enhancement as the temperature is reduced well below thecharging energy. This is due to spin-flip cotunneling processes; for a spin-1/2 dot in e.g. the spin-down state, such a process can be a spin-up electron hopping onto the QD from one reservoir, and aspin-down electron hopping off the QD at the same time to the other reservoir. An effective model5thus features a localized impurity spin coupled with a Fermi sea of conduction electrons, in analogywith the famous Kondo problem[49, 67], which usually refers to the low-temperature resistivityminimum in a system of magnetic impurities embedded in a non-magnetic metallic host.In the Kondo model, perturbation theory in the coupling constant is plagued by infrared di-vergence, but after much theoretical endeavor[9, 89, 129] it has been recognized that the modelhas a relatively simple low energy behavior. For the single-channel spin-1/2 model, at temper-atures well below the Kondo temperature TK , the impurity spin is “screened” by the conductionelectrons, forming a local singlet state. The spatial extent of this singlet state, commonly termedthe “Kondo screening cloud”, is expected to be LK = vF/TK where vF is the Fermi velocity. Theremaining conduction electrons are well described by a FL theory at zero temperature, and acquirea phase shift (pi/2 in the presence of particle-hole symmetry) upon being elastically scattered bythe Kondo singlet. Moreover, at finite temperatures, scattering by the Kondo impurity can haveboth elastic and inelastic contributions[21, 134], and it has been suggested that the inelastic scat-tering can be the origin of decoherence in mesoscopic structure, as measured for example by weaklocalization[83, 98, 134]. The possibility of imitating the impurity spin by a QD has triggered re-newed experimental and theoretical interest in mesoscopic manifestations of Kondo physics, such asthe observation of the length scale LK , the pi/2 phase shift, and also decoherence effects of inelasticKondo scattering.Many mesoscopic configurations have been proposed in order to observe LK . These includeQD-terminated finite quantum wires[95], and also various geometries with an embedded QD, in-cluding finite quantum wires[25, 94, 116, 117], small metallic grains/larger QDs[24, 63, 64, 71,72, 119, 124], and in particular, closed long AB rings with[118, 131] and without[4, 115] externalelectrodes. (A closed ring conserves the electric current and there is no leakage current.) Anothermotivation for quantum rings is that they may be used to answer the question of whether or notthe inelastic scattering from the Kondo QD can cause decoherence by suppressing the amplitude ofAB oscillations. A common feature of all these configurations is that they introduce at least oneadditional mesoscopic length scale L. When the bare Kondo coupling strength is adjusted so thatLK crosses the scale L, the dependence of observables on other control parameters changes qualita-tively. In the closed long AB ring with an embedded QD [also known as the ABK ring], for instance,L is the circumference of the ring: it is known that both LK itself and the conductance through thering can have drastically different AB phase dependences for LK L and LK . L[118, 131]. In the“large Kondo cloud” regime LK L, corresponding to a relatively small bare Kondo coupling, theKondo cloud “leaks” out of the ring and the size of the cloud becomes strongly influenced by thering size and other mesoscopic details of the system. For a given bare Kondo coupling, LK can beextremely sensitive to the AB phase at certain values of Fermi energy, varying by many orders ofmagnitude. This sensitivity is completely lost in the opposite “small Kondo cloud” regime LK . L,where the bare Kondo coupling is relatively large.6The conductance calculation of ABK rings, however, involves an additional layer of complication[66]that went neglected in a number of early works. In mesoscopic Kondo problems with FL electrodes,it is usually convenient to work with the scattering states and rotate to the basis of the so-calledscreening and non-screening channels: the screening channel ψ is coupled to the QD and there-fore has a nonzero T-matrix, while the non-screening channel φ is described by a decoupled non-interacting theory[22, 46, 75, 103, 131]. A careful evaluation by Kubo formula at finite temperaturesreveals that, unlike a QD directly coupled to external leads, the interaction effects on the linear dcconductance of short ABK rings are generally not fully encoded by the screening channel T-matrixin the single-particle sector, or equivalently the two-point function. Instead, there exists a contri-bution from connected four-point diagrams, corresponding to two-particle scattering processes inthe screening channel, which cannot be interpreted as resulting from a single-particle scatteringamplitude[66]. This is not in contradiction with the famous Meir-Wingreen formula[57, 82] due tothe violation of the proportionate coupling condition[33]. For the short ABK ring, the four-pointcontribution becomes comparable to the two-point contribution well above the Kondo tempera-ture T TK , but can be approximately eliminated at temperatures low compared to the bandwidthand the on-site repulsion of the QD, T min{t,U}, by applying the bias voltage and probingthe current in a particular fashion. (This does not mean the four-point contribution is negligible forT min{t,U}, however.) One naturally wonders how this result generalizes to the closed long ringat high and low temperatures, and how it possibly modifies early predictions on conductance[131],which is again expected to display qualitatively different behaviors for LK L and LK . L.On the other hand, efforts to measure the pi/2 phase shift are mainly concentrated on two-pathAB interferometer devices[17, 58, 59, 108, 123, 133]. In these devices, electrons from the sourcelead propagate through two possible paths (QD path and reference path) to the drain lead; the twopaths enclose a tunable AB phase ϕ , and a QD tuned into the Kondo regime is embedded in theQD path. Most importantly, the complex transmission amplitudes through the two paths tdeiϕ andtre f should be independent of each other, and the total coherent transmission amplitude at zerotemperature tsd = tre f + tdeiϕ is the sum of the individual amplitudes (the “two-slit condition”),meaning multiple traversals of the ring are negligible. Using a multi-particle scattering formalism,and assuming that only single-particle scattering processes are coherent, Ref. [23] calculates theconductance of such an interferometer with an embedded Kondo QD in terms of the single-particleT-matrix through the QD, and concludes that the AB oscillations are suppressed by inelastic multi-particle scattering processes due to the Kondo QD.The two-path interferometer can in principle be realized through open AB rings, where in con-trast to closed rings, the propagating electrons may leak into side leads attached to the ring. For anon-interacting QD, Ref. [7] presents the criteria for an open long ring to yield the intrinsic trans-mission phase through the QD: all lossy arms with side leads should have a small transmission anda small reflection. A small transmission suppresses multiple traversals of the ring and guarantees7the validity of the two-slit assumption, while a small reflection prevents electrons from “rattling”(tunneling back and forth) across the QD. However, when the QD is in the Kondo regime, as withthe previously discussed closed AB rings, the transmission probability through the QD[6] and eventhe Kondo temperature[118] may be sensitive to the AB phase and other details of the geometry,hampering the detection of the intrinsic phase shift across the QD. In addition, since the screeningchannels in the open ABK ring and in the simple embedded QD geometry are usually not the same,it is not obvious that the single-particle sector T-matrices coincide in the two geometries. Theseissues are not addressed in Ref. [23], which simply assumes that the two-slit condition is obeyed bythe coherent processes, and that the T-matrix of the open ABK ring is identical to that of the QD em-bedded between source and drain leads. To our knowledge, it has been a mystery whether in certainparameter regimes the open long ABK ring realizes the two-path interferometer with a Kondo QD,where the Kondo temperature and the transmission probability through the QD are independent ofthe details of other parts of the ring, and the T-matrix of the ring truthfully reflects the T-matrix ofthe QD.The aforementioned problems in closed and open ABK rings prompt a unified treatment of lineardc conductance in different mesoscopic geometries containing an interacting QD. Much work hasbeen done on generic mesoscopic geometries[33, 37, 62], but in the formalism to be presented in thisthesis, we aim to take the connected contribution into account expressly, and refrain from makingassumptions about the geometry in question (such as parity symmetry).1.4 This thesisIn this thesis, we discuss how the linear dc transport through a multi-terminal system is affectedby interactions, namely interacting quantum wires and reservoirs[111], and an interacting QD in theKondo regime[112].In the first part of the thesis, we adopt the RPA fermionic approach to study the conductancetensor for a generic multi-lead junction in the presence of TLL leads. Our theory makes extensiveuse of the scattering basis transformation of the non-interacting part of the system; as a result itis explicitly formulated on the basis of the single-particle S-matrix (much like that in Ref. [70]),and is formally independent of the number of wires. This stands in contrast to previous RPA treat-ments of junctions attached to FL leads, whose formulation heavily depends on the parametrizationof the conductance tensors, different for two-lead junctions[11] and Y-junctions[13, 15]. We derivea Landauer-type conductance formula, appropriate for the renormalized S-matrix, and recover theadditional contribution from the TLL leads to the conductance, absent in the naive Landauer formal-ism. Our theory is applied to the two-lead junction and Y-junction problems, where in addition toverifying existing results on the fixed points and the phase diagrams, the conductance of the M fixedpoint attached to TLL leads is calculated.In the second part of this thesis, we study a QD represented by an Anderson impurity, which8is embedded in a junction connecting an arbitrary number of FL leads. The junction is regardedas a black box characterized only by its scattering S-matrix and its coupling with the QD, and allleads (including source, drain and possibly side leads) are treated on equal footing. In parallel withRef. [66] we find that the linear dc conductance is given by the sum of a “disconnected” part and a“connected” part. The disconnected part has the appearance of a linear response Landauer formula,where the “transmission amplitude” is linear in the T-matrix of the screening channel in the single-particle sector, and indeed reduces at zero temperature to a non-interacting transmission amplitudeappropriate for the local FL theory. The connected part is again a Fermi surface property, can beeliminated by proper application of bias voltages, and is calculated perturbatively at weak couplingT TK , as well as at strong coupling T TK provided the local FL theory applies. Our formalismis subsequently applied to long ABK rings. In the case of closed rings, we show that for T TK ,the high-temperature conductance exhibits qualitatively different behaviors as a function of the ABphase for T vF/L and T vF/L. In the case of open rings, when the small transmission con-dition is met, we find the mesoscopic fluctuations are suppressed, and the two-path interferometerbehavior is indeed recovered at low temperatures. If in addition the small reflection condition issatisfied, the Kondo temperature of the QD and the complex transmission amplitude through theQD are both unaffected by the details of the ring. We then find the conductance at T TK andT TK , and in particular rigorously calculate the normalized visibility[23] of the AB oscillations inthe FL regime. We show that while the deviation of normalized visibility from unity is indeed pro-portional to inelastic scattering as predicted by Ref. [66], the constant of proportionality dependson non-universal particle-hole symmetry breaking potential scattering. Our findings also suggestthat the pi/2 phase shift across the QD is measurable in our two-path interferometer when the cri-teria of small transmission and small reflection are fulfilled. While we focus on long ABK ringsin this thesis, our general formalism is applicable to a Kondo impurity embedded in an arbitrarynon-interacting multi-terminal mesoscopic structure.9Chapter 2Conductance of junctions of multipleinteracting quantum wires with TLLleadsThis chapter is devoted to junctions of multiple interacting quantum wires[111]. We first review thebosonic approach to quantum wires in Section 2.1, and outline its application to 2-lead junctions andY-junctions in Section 2.2. Switching to the fermionic approach, we then elaborate on our modelfor a generic multi-lead junction in Section 2.3. Viewing the single-particle S-matrix elements asrunning coupling constants, we derive an S-matrix RG equation a` la Wilson in Section 2.4. Thisderivation does not directly shed light on the conductance, however, which prompts us to calculatethe linear dc conductance of the junction to the first order in interaction in Section 2.5. Section 2.6 isbased on perturbative RG, again to the first order in interaction. We recover the S-matrix RG equationfrom a Callan-Symanzik (CS) approach, using the Kubo conductance calculated in Section 2.5. Thisestablishes a modified Landauer formula involving the renormalized S-matrix in the case of FL leads.An additional contribution to the conductance is shown to arise from TLL leads. In Section 2.7, theconductance is found in the RPA to arbitrary order in interaction; we derive an S-matrix RG equationin the RPA, and again find the conductance in terms of the renormalized S-matrix both with andwithout TLL leads. Section 2.8 applies our results to the fixed points of 2-lead junctions and Y-junctions at the first order and in the RPA. In particular, we find the conductance at the M fixedpoint of a Z3 symmetric Y-junction attached to TLL leads. We discuss some open questions andsummarize our findings in Section 2.9. Finally, a review of various RG fixed points in the fermionicapproach is given in Appendix A.102.1 Bosonic approach to interacting quantum wires with TLL leadsAs a first example of how TLL leads may modify the Landauer formula in quasi one-dimensionalmesoscopic structures, we examine a simple model of an interacting quantum wire of spinless elec-trons connected to TLL leads from the bosonization perspective[42, 61, 76, 100, 105, 106, 125].2.1.1 Uniform wireLet us begin by considering a uniform quantum wire with spinless electrons. Retaining the de-grees of freedom in the vicinity of the Fermi wave vector k ≈ ±kF only, and approximating theirdispersions by linear ones, we have the HamiltonianH = H0+Hg2 +Hg4 , (2.1)whereH0 =∫ ∞−∞dx(ivF)[ψ†L (x)∂xψL (x)−ψ†R (x)∂xψR (x)], (2.2)Hg2 = g2∫ ∞−∞dxρL (x)ρR (x) , (2.3)Hg4 =g42∫ ∞−∞dx[ρ2L (x)+ρ2R (x)]. (2.4)H0 is a kinetic term describing right-moving and left-moving electrons ψR and ψL with velocities±vF and dispersions ε±,k = vF (±k− kF) respectively. The full electron annihilation operator is sim-ply ψ (x) = eikFxψR (x)+e−ikFxψL (x). ρR/L = :ψ†R/LψR/L : are the normal-ordered chiral density op-erators (i.e. with their ground state expectation value subtracted); Hg2 then accounts for the interac-tion between electrons of opposite chiralities, and Hg4 accounts for the interaction between electronsof the same chirality[122]. We ignore Umklapp processes [e4ikFxψ†R (x)ψ†R (x+a)ψL (x)ψL (x+a)+h.c.where a is a small distance] which do not play a significant role away from commensurate fillings,as well as backscattering processes (ψ†RψLψ†LψR+h.c.) which can be absorbed into Hg2 for spinlessfermions[42, 44].The above model can be solved exactly by the bosonization technique[44], which takes advan-tage of the fact that the elementary excitations are density fluctuations. We introduce bosonic fieldsφ and θ such that∂xφ (x) =−pi [ρR (x)+ρL (x)] , (2.5)∂xθ (x) = pi [ρR (x)−ρL (x)] ; (2.6)11thus ∂xφ is proportional to the zero-momentum piece of the density fluctuations (the other pieceis ψ†RψL+h.c. of momentum ±2kF ), and ∂xθ is proportional to the zero-momentum piece of thecurrent operator. φ commutes with itself and so does θ , but they obey the commutation relation[φ (x) ,θ (y)] = ipi2sgn(y− x) ; (2.7)thus the canonical momentum conjugate to φ isΠ(x) =1pi∂xθ (x) . (2.8)The original fermion operators are solitons in the boson language:ψR/L (x) =UR/L1√2piαexp [∓iφ (x)+ iθ (x)] , (2.9)where the short-distance cutoff α mimics a finite bandwidth which scales as 1/α . UR/L are Kleinfactors encoding the Fermi statistics of ψR/L; they commute with both bosons and anticommutewith each other. Their presence ensures that the application of ψR/L on a state reduces the fermionnumber of that state by 1.It turns out that both the kinetic part of the Hamiltonian H0 and the interaction parts Hg2 andHg4 are quadratic in bosons. Therefore, Hg2 and Hg4 does not change the linear spectrum other thanrenormalizing the speed of sound from vF :H0 =vF2pi∫dx[(piΠ(x))2+(∂xφ (x))2], (2.10)H = H0+Hg2 +Hg4 =u2pi∫dx[K (piΠ(x))2+1K(∂xφ (x))2], (2.11)whereu= vF√(1+g42pivF)2−(g22pivF)2(2.12)is the renormalized speed of sound,K =√1+(g4−g2)/(2pivF)1+(g4+g2)/(2pivF)(2.13)is known as the Luttinger parameter. For repulsive interactions g2 > 0, K < 1, while for attractiveinteractions g2 < 0, K > 1; free electrons have K = 1. Also notice that, when g2 = 0, g4 alone doesnot change the Luttinger parameter from 1, which means a minimal model of interacting electronsrequires the g2 term but not the g4 term.12Figure 2.1: Sketch of a quantum wire connected to two leads. The leads are modeled withLuttinger parameters KL and KR that are different from the Luttinger parameter of thewire, KW . For FL leads KL = KR = 1.2.1.2 TLL leadsTo investigate the effect of TLL leads on conductance, it is convenient to introduce a slightly moregeneral model, with position-dependent speed of sound and Luttinger parameter:H =12pi∫dx[u(x)K (x)(piΠ(x))2+u(x)K (x)(∂xφ (x))2], (2.14)whereu(x) =uL, x< xLuW , xL < x< xRuR, x> xR, (2.15)K (x) =KL, x< xLKW , xL < x< xRKR, x> xR. (2.16)This model may result from e.g. a position-dependent g2 interaction, which is stronger inside thequantum wire segment xL < x< xR than in the reservoirs[42, 76, 105, 106, 125]; see Fig. 2.1. Notethat the infinitely long wire limit |xR− xL| → ∞ would not be well-defined if the asymptotic behav-ior of K (x) as x→±∞ were not specified, because as we shall see momentarily, this asymptoticbehavior is required to determine the Green’s function which in turn determines the conductance.Assuming the wire is subjected to an electric field which exists between reservoirs only and is13turned on slowly from t→−∞,E =−∂A∂ t= E0 (x)e−i(ω+i0)t , (2.17)where E0 (x) = 0 for x< xL and x> xR, the Hamiltonian is modified according topiΠ(x)→ piΠ(x)− eA(x, t) ; (2.18)Thus the current is, in the linear response regime[44],I (x, t) =−〈δHδA(x, t)〉= eu(x)K (x)〈Π(x, t)〉− e2piu(x)K (x)A(x, t)=−e2u(x)K (x)∫dt ′∫dx′u(x′)K(x′)A(x′, t ′)GΠΠ(x,x′, t− t ′)− e2piu(x)K (x)A(x, t) ,(2.19)where the equilibrium retarded current-current correlation function isGΠΠ(x,x′, t− t ′)≡ (−i)H (t− t ′)〈[Π(x, t) ,Π(x′, t ′)]〉 . (2.20)Here we denote the Heaviside unit-step function by H (t). It is useful to study similarly definedobjects GΠφ and Gφφ ; for instance,GφΠ(x,x′, t− t ′)≡ (−i)H (t− t ′)〈[Π(x, t) ,φ (x′, t ′)]〉 . (2.21)From the quadratic Hamiltonian Eq. (2.14) we can work out the equations of motion for GΠφ andGφφ :∂t ′GΠφ(x,x′, t− t ′)= δ (x− x′)δ (t− t ′)+piu(x′)K (x′)GΠΠ (x,x′, t− t ′) , (2.22)∂tGφφ(x,x′, t− t ′)= piu(x)K (x)GΠφ (x,x′, t− t ′) . (2.23)With these two equations, we can remove the diamagnetic term and cast the current into the follow-ing form:14I (x, t) =− e2pi2∫dt ′∫dx′A(x′, t ′)∂t ′∂tGφφ(x,x′, t− t ′)=e2pi2∫dx′E0(x′)iωGφφ(x,x′,ω)e−iωt . (2.24)In the second line we have performed Fourier transform Gφφ (x,x′, t− t ′)=∫ dω2pi e−iω(t−t ′)Gφφ (x,x′,ω).For our particular choice of u(x) and K (x), it is not difficult to solve for Gφφ . Writing down theequation of motion for GΠφ ,∂tGΠφ(x,x′, t− t ′)=−δ (x− x′)δ (t− t ′)+∂x [ 1pi u(x)K (x)∂xGφφ (x,x′, t− t ′)], (2.25)we may eliminate GΠφ to find a closed equation for Gφφ ,ω2Gφφ(x,x′,ω)= piu(x)K (x)δ(x− x′)−u(x′)K (x′)∂x′ [ u(x′)K (x′)∂x′Gφφ (x,x′,ω)]. (2.26)Gφφ should be continuous everywhere, [u(x′)/K (x′)]∂x′Gφφ should be continuous everywhere ex-cept when x′ = x, and as x′ →±∞, Gφφ should describe purely outgoing wave. Choosing x < xL(because in the dc limit it is unimportant where we choose to measure the current), the aforemen-tioned boundary conditions lead to the following solution:Gφφ(x,x′,ω)=piKWKLiωe−iωuLx (KW +KR)ei ωuW x′− (KW −KR)e2iωuWxRe−iωuWx′(KW +KL)(KW +KR)− (KW −KL)(KW −KR)e2iωuW(xR−xL) . (2.27)Therefore, in the dc limit ω → 0,I (x, t) =e22pi2KLKRKL+KR∫dx′E0(x′), (2.28)i.e. the linear dc conductance isG=e22pi2KLKRKL+KR. (2.29)Remarkably, G is independent of properties of the quantum wire, but does depend on the in-teraction strength in the reservoirs. In particular, for Fermi liquid reservoirs interactions becomenegligible (KL = KR = 1), and G reduces to the Landauer prediction e2/(2pi); on the other hand,when an interacting quantum wire acts as its own reservoir (KL = KR = KW ), G = KW e2/(2pi) de-15viates from the Landauer prediction.It is also worth mentioning that, in the limit of tunneling through a junction of two quantumwires with Luttinger parameters KL and KR (i.e. xL → xR so that the central segment becomesinfinitely short), there exists a heuristic understanding of Eq. (2.29): a series connection between aninfinite wire with Luttinger parameter KL and a “contact resistance” Gc[32]. If we assume loss ofcoherence, we have G−1c +[KLe2/(2pi)]−1= G−1. Solving for Gc, we findG−1c =(e22pi)−1 12(1KR− 1KL). (2.30)Note that here Gc is defined for the right wire relative to the left wire: the sign of Gc changes if it isdefined for the left wire relative to the right wire.2.2 Bosonic approach to junctions of two wires and Y-junctionsWe now review the RG flow diagrams of junctions of two wires and Y-junctions[40, 41, 51, 60, 61,93, 130]. The RG fixed points of these models control their low-energy behavior, and correspondto conformally invariant boundary conditions. The simplest way to identify the RG fixed points isthe method of delayed evaluation of boundary condition[51, 93]. While bosonizing the electronoperators at the junction according to Eq. (2.9), we keep both types of bosonic variables φ and θwithout imposing any specific conformally invariant boundary condition on them, thus doubling thedegrees of freedom in the system. The boundary conditions are reintroduced and the extraneousdegrees of freedom eliminated only when we compute the scaling dimensions of various tunnelingprocesses at the junction.2.2.1 2-lead junctionsWe illustrate the method through 2-lead junctions. Each wire extends from x = 0 to ∞, and thejunction is located at x= 0, as shown in Fig. 2.2 panel a).Introducing boson fields (φ j,θ j) for wire j, we further define rescaled fields(φ˜ j, θ˜ j)such thatφ˜ j =φ j√K j, θ˜ j = θ j√K j, (2.31)where K j is the Luttinger parameter for wire j; this transformation preserves the commutation16Figure 2.2: a) The sketch of a 2-lead junction, with possibly different Luttinger parametersand speeds of sound in the two wires. The wires extend from x= 0 to ∞, and the junctionis at x = 0. b) The RG fixed points of this model: D (smoothly connected wires) and N(decoupled wires). c) The RG flow diagram from the bosonic approach. RG flows aretowards N when Ke < 1 (repulsive interactions) and towards D when Ke > 1 (attractiveinteractions); here Ke is the harmonic average of K1 and K2.relation. The imaginary time action of the bulk wires may then be written as a non-interacting form,S0 = ∑j=1,212pi∫ ∞0dx˜∫ β0dτ[(∂τ φ˜ j)2+(∂x˜φ˜ j)2] (2.32a)= ∑j=1,212pi∫ ∞0dx˜∫ β0dτ[(∂τ θ˜ j)2+(∂x˜θ˜ j)2] , (2.32b)where we have also rescaled x= u jx˜ for wire j.17For future convenience we define the following linear combinations of θ˜ :Θ0 =√K1θ˜1+√K2θ˜2√K1+K2, (2.33a)Θ1 =√K2θ˜1−√K1θ˜2√K1+K2. (2.33b)The bulk action S0 is also of a non-interacting form in terms of Θ0 and Θ1. We can similarly defineconjugate variablesΦ0 andΦ1. The right- and left-moving chiral fields for these linear combinationsareϕ0R/L =Θ0∓Φ0, (2.34a)ϕ1R/L =Θ1∓Φ1. (2.34b)The right-movers depend only on τ− ix˜ and the left-movers depend only on τ+ ix˜. The LL and RRcorrelation functions are unaffected by the boundary at x= 0:〈ϕ jL (τ, x˜)ϕ j′L (0,0)〉= δ j j′ lnατ+ ix˜+α sgnτ, (2.35a)〈ϕ jR (τ, x˜)ϕ j′R (0,0)〉= δ j j′ lnατ− ix˜+α sgnτ ; (2.35b)the LR and RL correlation functions, however, are usually nonzero due to the junction.Since the total current is conserved at the junction, from Eq. (2.6) we have the boundary condi-tion∂x (u1K1θ1+u2K2θ2)|x=0 =√K1+K2 ∂x˜Θ0|x˜=0 = 0; (2.36)Thus ∂τΦ0|x˜=0 = ∂x˜Θ0|x˜=0 = 0, and Φ0 is constant (“pinned”) at the boundary. This correspondsto the following boundary condition on ϕ0R/L:ϕ0R (x˜= 0) = ϕ0L (x˜= 0)+ const. (2.37a)We need an additional boundary condition for the remaining degrees of freedom, in this case ϕ1R/L.A natural ansatz isϕ1R (x˜= 0) =R11ϕ1L (x˜= 0)+ const., (2.37b)where R is an orthogonal matrix; this is because all ϕ fields are real, and the correlation functions18〈ϕRϕR〉 and 〈ϕLϕL〉 should have the same constant prefactor. For a junction of two wires,R is 1×1and the only possibilities areR11 =±1.We may view the right-movers as analytic continuation of the left-movers. For x˜< 0, based onthe above boundary conditions, we have the following “unfolding” transformation:ϕ0R (x˜) = ϕ0L (−x˜)+ const., (2.38a)ϕ1R (x˜) =R11ϕ1L (−x˜)+ const. (2.38b)It then follows that, for instance,〈ϕ1R (τ, x˜)ϕ1L (0,0)〉=R11 ln ατ− ix˜+α sgnτ .Let us calculate the scaling dimensions of some tunneling processes. Defining Ke as the har-monic average of K1 and K2,Ke =2K1K2K1+K2, (2.39)we can write down the bosonized form of several typical processes using Eq. (2.9),ψ†2Rψ1L∣∣∣x=0∝ exp [i(φ2−θ2+φ1+θ1)] = exp[i(√K1+K2Φ0+√2KeΘ1)], (2.40a)ψ†1Rψ1L∣∣∣x=0∝ exp(2iφ1) = exp[i(2K1Φ0√K1+K2+√2KeΦ1)], (2.40b)ψ†2Rψ1R∣∣∣x=0∝ exp [i(φ2−θ2−φ1+θ1)] = exp[i((K2−K1)Φ0√K1+K2−√2KeΦ1+√2KeΘ1)],(2.40c)where all boson fields are evaluated at x˜ = 0. We have discarded the Klein factors which do notcontribute to the power law decay of the correlation functions. Neglecting the constant Φ0, thecorrelation functions are easily evaluated:〈Tτ exp(i√2Ke[Θ1 (τ,0)−Θ1 (0,0)])〉∝ exp(2Ke〈TτΘ1 (τ,0)Θ1 (0,0)〉)=(ατ) 1+R11Ke ,(2.41)meaning ψ†2Rψ1L∣∣∣x=0has scaling dimension (1+R11)/(2Ke). Similarly, ψ†1Rψ1L∣∣∣x=0has dimen-19sion (1−R11)Ke/2, and ψ†2Rψ1R∣∣∣x=0has dimension (1+R11)/(2Ke)+(1−R11)Ke/2.We examine the two conformally invariant boundary conditions separately:1) WhenR11 = 1, ψ†2Rψ1L∣∣∣x=0and ψ†2Rψ1R∣∣∣x=0have dimension 1/Ke, whereas ψ†1Rψ1L∣∣∣x=0hasdimension zero for any Ke, i.e. becomes the identity operator. The latter result implies that ψ1R canbe identified with ψ1L up to a constant phase, indicative of an open boundary condition at x= 0 forboth wires. Thus, at this fixed point, the two wires are completely decoupled. Alternatively, note thatΦ0 andΦ1 are both pinned at x˜= 0, so φ1 and φ2 are also pinned; Eq. (2.6) then leads to zero currentin each wire. This fixed point is known as the N (Neumann) fixed point in literature. It is stablewhen Ke < 1, and unstable when Ke > 1. The dimension of leading irrelevant operators is 1/Ke;second-order perturbation theory thus implies that, when Ke < 1, the conductance is proportional toT 2/Ke−2 at low temperatures.2) WhenR11 =−1, ψ†1Rψ1L∣∣∣x=0and ψ†2Rψ1R∣∣∣x=0have dimension Ke, and ψ†2Rψ1L∣∣∣x=0becomesidentity. The latter result implies that ψ2R can be identified with ψ1L up to a constant phase, whichmeans the two wires are smoothly connected without backscattering. This fixed point is known asthe D (Dirichlet) fixed point. For K1 = K2 it further reduces to the case of a single infinite wire.The D fixed point is stable when Ke > 1, and unstable when Ke < 1. The dimension of leadingirrelevant operators is Ke, so when Ke > 1, the low-temperature correction to the zero-temperatureconductance should scale as T 2Ke−2[51].These two RG fixed points are shown in Fig. 2.2 panel b), and the RG flow diagram is shown inpanel c).Finally, we calculate the linear dc conductance, again using the Kubo formula. In the dc limitwhere the current is measured or how the external electric field is applied should be unimportant.Measuring the current at position x on wire j and applying a uniform electric field from 0 to L onwire j′, we have the Kubo formulaG j j′ =e2pi2u jK ju j′K j′ limω→01iωL[∫dτeiωτ∫ L0dy〈Tτ∂yθ j′ (τ,y)∂xθ j (0,x)〉− (ω → 0)] . (2.42)Taking G12 as an example,G12 =e2pi2√K1K2K1+K2limω→01iω Lu2[∫dτeiωτ∫ Lu20dy˜×〈Tτ∂y˜ (√K2Θ0−√K1Θ1)(τ, y˜)∂x˜ (√K1Θ0+√K2Θ1)(0, x˜)〉− (ω → 0)] ; (2.43)the 〈Θ0Θ1〉 and 〈Θ1Θ0〉 cross terms vanish asΘ0 andΘ1 are decoupled, and the 〈Θ0Θ0〉 contributionmust also vanish since it corresponds to the conductance of a semi-infinite wire. The remaining〈Θ1Θ1〉 contribution is straightforward to evaluate:20G12 =− e2pi2K1K2K1+K2limω→014iω Lu2[∫dτeiωτ∫ Lu20dy˜∂y˜∂x˜ 〈Tτ (ϕ1R+ϕ1L)(τ, y˜)(ϕ1R+ϕ1L)(0, x˜)〉− (ω → 0)]=− e2pi2K1K2K1+K2limω→014iω Lu2{∫dτeiωτ∫ Lu20dy˜∂y˜∂x˜[lnα2τ2+(y˜− x˜)2 +R11 lnα2τ2+(y˜+ x˜)2]− (ω → 0)}=− e2pi2K1K2K1+K2limω→014iω Lu2{(2pii)[e−ω∣∣∣ Lu2−x˜∣∣∣ sgn(Lu2− x˜)+ e−ω x˜−R11e−ω(Lu2+x˜)+R11e−ω x˜]− (ω → 0)}=1−R112Kee22pi. (2.44)In other words, the conductance at the N fixed point is 0, and the conductance at the D fixed pointis Kee2/(2pi). This is a special case of Eq. (2.29).2.2.2 Y-junctionsWe proceed to apply the method above to Y-junctions[51]. In this case, the “center-of-mass” fieldΘ0 should beΘ0 =√K1θ˜1+√K2θ˜2+√K3θ˜3√K1+K2+K3, (2.45)while a convenient set of definitions of the remaining two Θ fields isΘ1 =√K2θ˜1−√K1θ˜2√K1+K2, (2.46)Θ2 =√K1K3θ˜1+√K2K3θ˜2− (K1+K2) θ˜3√K1+K2+K3√K1+K2. (2.47)(Other linear combinations are possible, as long as they are orthogonal to each other and to Θ0.)Again Φ fields can be similarly defined, and their chiral fields are given as before. Φ0 is pinned atthe junction x˜ = 0 by current conservation, and ϕ0R/L obey an open boundary condition. We groupthe remaining fields as ~Θ ≡ (Θ1,Θ2) and ~Φ ≡ (Φ1,Φ2), and express the boundary condition forthese fields as ~ϕR =R~ϕL. R is now a 2×2 rotation matrix which can be parametrized asR =(cosξ sinξ−sinξ cosξ). (2.48)It is tedious but straightforward to enumerate all single-particle processes at the junction and expresstheir scaling dimensions in terms ofR.It turns out that there are at least four different types of fixed points, namely the N fixed pointwith three decoupled wires where ξ = 0 and R = 1, the A j fixed point with wire j ( j = 1,2,3)21decoupled from the other two wires which are smoothly connected, and also two types of fixedpoints with novel physics: D (Dirichlet) fixed point and χ± (chiral) fixed points.At the D fixed point, ξ = pi and R = −1. None of the single-particle tunneling processesbecomes identity, but some two-particle processes do, such as ψ†2Rψ1Lψ†2Lψ1R which describes anelectron pair tunneling from wire 1 to wire 2. This evinces Andreev reflection physics at the D fixedpoint. We can also look at the linear dc conductanceGDj j′ = 2e22pi(−K jδ j j′+K jK j′K1+K2+K3), (2.49)which indicates G11 = −(4/3)e2/(2pi) in the case of non-interacting wires K1 = K2 = K3 = 1.(In this case the D fixed point would only be possible for an interacting junction.) |G11| exceedse2/(2pi), and the enhancement can be attributed to Andreev reflection. The D fixed point cannotpossibly be stable unless the interactions in all three wires are very strongly attractive.At the χ± fixed points, tanξ = ±√(K1+K2+K3)/(K1K2K3). At χ+, ψ†2Rψ1L, ψ†3Rψ2L andψ†1Rψ3L simultaneously become identity; heuristically, this means incident electrons are scatteredfrom wire j into wire j+ 1 (here we identify j+ 3 ≡ j). Similarly, at χ−, ψ†1Rψ2L, ψ†2Rψ3L andψ†3Rψ1L simultaneously become identity. The linear dc conductance is given byGχ±j j′ =−2e22piK j (K1+K2+K3)δ j j′+(±K1K2K3ε j j′−K jK j′)K1+K2+K3+K1K2K3, (2.50)where the anti-symmetric tensor ε j j′ is defined as follows: ε12 = ε23 = ε31 = 1, ε21 = ε32 = ε13 =−1,ε j j = 0.We now focus on the Z3 symmetric Y-junction of three identical quantum wires, with Luttingerparameter K j = K. The RG flow diagram is greatly simplified under this assumption[93].1) At K < 1, the only stable fixed point is the N fixed point. Neither χ± nor D is stable.2) At K = 1, the system is non-interacting in the bulk and all single-particle processes aremarginal in the RG sense. There is a manifold of fixed points, which contains both N and χ±,although not D.3) At 1<K < 3, N becomes unstable, and the RG flows are eventually toward χ± which becomestable. However, while N preserves time-reversal symmetry, χ± explicitly breaks it (since Gχ±j j′ 6=Gχ±j′ j when j 6= j′). Therefore the most economic assumption is that there exists an intermediate fixedpoint (termed M for mysterious) that preserves time-reversal symmetry. M is stable against time-reversal symmetric perturbations, such as tunneling between any two wires; however, it is unstableagainst time-reversal symmetry breaking perturbations, such as an AB phase at the junction that isseen in either tunneling cycle (1→ 2→ 3→ 1 or 1→ 3→ 2→ 1). It is not difficult to see from anon-interacting model that M also belongs to the fixed manifold when K = 1. D remains unstable.4) At K = 3, the system again has non-interacting quasiparticles but they are not the bare elec-22trons. There is a manifold of fixed points, which contains χ± but not N, M or D.5) At K > 3, N remains unstable, χ± becomes unstable once more, and D becomes stable.(There are in fact two types of D fixed points which have the same conductance tensor; however,only one type can be reached by an RG flow that preserves time-reversal symmetry. This type of Dfixed point is stable when K > 9.)We are particularly interested in the realistic case of K close to 1, when the interactions are nottoo strong. In this case the D fixed point is never stable, and important fixed points are N, χ±,M and also the asymmetric A j if Z3 asymmetry is allowed. The RG flow diagrams for K < 1 and1 < K < 3 are depicted in Fig. 2.3, along with schematic representations of N, χ+, M, A3 and D.Unfortunately, important as the M fixed point may be in the presence of time-reversal symmetryand Z3 symmetry, we are unaware of its corresponding conformally invariant boundary conditionin terms of bosonic variables. In the remainder of this chapter, we turn to the fermionic approachto junctions of quantum wires, which appears to be a more convenient description of the M fixedpoint.2.3 Fermionic formulationIn this section, we establish the model Hamiltonian for a generic junction of interacting quantumwires, to be studied by the fermionic approach in the remainder of the chapter.The system consists of N quantum wires of interacting spinless electrons, numbered j = 1, 2,..., N, meeting at a junction which we choose as the origin x= 0. We align the wires so that they areparallel to the +x axis; see Fig. 2.4.In the continuum limit of the model, on each quantum wire we retain right- and left-movers innarrow bands of wave vectors around the Fermi points ±kF j:ψ j (x)≈ eikF jxψ jR (x)+e−ikF jxψ jL (x)=∫ D−DdE√2pivF j[ψ jR (E)ei(EvF j+kF j)x+ψ jL (E)e−i(EvF j+kF j)x],(2.51)where vF j is the Fermi velocity in wire j, the dispersion relation is linearized as E = E j (k) = vF jk,and D vF jkF j is the high-energy cutoff. Left-movers ψ jL are incident on the junction, scattered,and turned into right-movers ψ j′R; ψ jL and ψ j′R are not independent degrees of freedom, but relatedby the single-particle S-matrix of the junction S j j′ [see also Eq. (2.57)].The Hamiltonian consists of three parts:H =N∑j=1(H j0,wire+Hjint)+H0,B. (2.52)H j0,wire is the non-interacting part of the Hamiltonian for wire j, quadratic in electron operators,23Figure 2.3: a) Typical RG fixed points of the Y-junction, as predicted by the bosonic approach(with the exception of the M fixed point): N (decoupled wires), χ+ (chiral transmissionwhich breaks time-reversal symmetry: 1 to 2, 2 to 3 and 3 to 1), M (electrons incidentfrom any wire can be reflected or transmitted), D (Andreev reflection) and the asymmetricA3 (1 and 2 are smoothly connected and 3 is decoupled). b) The RG flow diagram for aZ3 symmetric Y-junction at K < 1. The horizontal axis measures the coupling strengthbetween any two wires, so the N fixed point occupies the entire vertical axis. The verticalaxis measures time-reversal symmetry breaking (e.g. due to an AB flux at the junction),which means the time-reversal symmetric M should be on the horizontal axis, but χ±should not. RG flows are always towards N when K < 1, although it is found from thefermionic approach that M is more stable than χ±. c) Same as b) but with 1<K < 3. RGflows are towards χ± whenever time-reversal symmetry is broken; in the time-reversalsymmetric case, however, the RG flow is from N to M.24Figure 2.4: Sketch of a typical junction system with the number of quantum wires N = 3. Theshaded junction area to the left is modeled by quadratic hopping terms between ends ofthe wires, which are all aligned at the origin x = 0. The electron-electron interactionstrengths inside the wires are also plotted for comparison. The interaction strength inwire j = 1 is uniform, and that in wire j = 2 also goes to a constant nonzero value asx→ ∞; these two wires are said to be attached to TLL leads. In contrast, the wire j = 3has a vanishing interaction strength far away from the junction, and is connected to anFL lead (the FL lead itself is too wide to be shown in full).H j0,wire ≈ ivF j∫ ∞0dx[ψ†jL∂xψ jL−ψ†jR∂xψ jR](x)≈∫ D−DdE E(ψ†jR (E)ψ jR (E)−ψ†jL (E)ψ jL (E)),(2.53)while the quartic term H jint to be specified below describes the electron-electron interaction in wirej. The boundary term H0,B is quadratic, and is responsible for electron transfer between wires acrossthe junction. For simplicity we again assume that each wire only supports one single channel, andignore quartic interactions between wires, at the junction and between the junction and the wires.To model the electron-electron interaction, we assume it is short-ranged and the system is awayfrom half-filling, so that the Umklapp processes are unimportant. We further ignore processes wheretwo chiral densities of the same chirality interact with one another, ψ†RψRψ†RψR or ψ†LψLψ†LψL; asdiscussed in Section 2.1 these g4 processes renormalize the Fermi velocity but do not change theLuttinger parameter by themselves. For spinless fermions, this leaves us with only processes in-volving two chiral densities of different chiralities (g2 processes) ψ†RψRψ†LψL. The electron-electroninteraction is then represented by a spatially variant g2 term:25H jint =∫ ∞0dxg j2 (x)ψ†jR (x)ψ jR (x)ψ†jL (x)ψ jL (x) . (2.54)g j2 (x→ ∞) is a constant. A finite g j2 (∞) 6= 0 corresponds to a TLL lead attached to wire j, whileif g j2 (∞) = 0 the junction is considered to be connected to an FL lead. We define a dimensionlessinteraction strengthα j (x) = g j2 (x)/(2pivF j) . (2.55)Along the lines of Refs. [77, 132], viewing the electron-electron interaction as a perturbation, wecan first diagonalize the quadratic part of the Hamiltonian. The resultant eigenstates, which form theso-called scattering basis, can be related to the S-matrix in the low-energy theory. Note that sucha scattering basis transformation is independent of the actual eigenstates of the fully interactingsystem; we are therefore always able to proceed with this transformation, regardless of whetherthe interaction is present at x → ∞. For non-resonant scattering, which we assume throughoutthis chapter, the S-matrix elements S j j′ (E) ≡ S j j′ are independent of the electron energy E at lowenergies, and the single-particle scattering state incident from wire j′ with energy E ′ readsφ †j′(E ′) |0〉=∑j∫ ∞0dx1√2pivF j(δ j j′e−i E′vF j xψ†jL (x)+S j j′ei E′vF jxψ†jR (x))|0〉+ · · · , (2.56)where |0〉 corresponds to the filled Fermi sea, and the omitted terms represent the part of the wavefunction from the junction area. Under our assumption of non-resonant scattering, these omittedterms do not contribute to the renormalization of the interaction[132]. Inverting Eq. (2.56) we mayexpress the original electrons ψ in terms of the scattering basis operators φ ,ψ jR (E) =N∑j′=1∫dE ′∫ ∞0dx(1√2pivF jei EvF j x)∗(1√2pivF jS j j′ei E′vF jx)φ j′(E ′)=∑j′∫ D−DdE ′2pi−iE−E ′− i0S j j′φ j′(E ′). (2.57a)Similarlyψ jL (E) =∫ D−DdE ′2piiE−E ′+ i0φ j(E ′). (2.57b)The coefficients of this transformation would be δ -functions if the Fourier transform were per-formed on the real axis; however, they pick up a principal value part here because our system isdefined on the positive x-axis.26Now recast the Hamiltonian in the scattering basis. By definition, the quadratic part of theHamiltonian is diagonal:N∑j=1H j0,wire+H0,B =∑j∫dE Eφ †j (E)φ j (E) . (2.58)We insert the scattering basis transformation into the interaction Eq. (2.54). Allowing the energiesto run freely from −∞ to ∞ and calculating the energy integrals using the method of residues[34],we findH jint =∫ ∞0dxg j2 (x) ∑l1l2l3l4∫ dE1dE2dE3dE4(2pi)2 v2F jφ †l1 (E1)φl2 (E2)φ†l3 (E3)φl4 (E4)ei(−E1+E2+E3−E4) xvF j S∗jl1S jl2δ jl3δ jl4(2.59)This is a plausible manipulation, seeing that the scattering basis transformation should not introduceadditional singularities at the band edge. NowH jint =∫ ∞0dxg j2 (x) ∑l1l2l3l4∫ dE1dE2dE3dE4(2pi)2 v2F jρ jl1l2l3l4 (E1,E2,E3,E4;x)φ†l1 (E1)φl2 (E2)φ†l3 (E3)φl4 (E4) ,(2.60)where we introduce the functionρ jl1l2l3l4 (E1,E2,E3,E4;x)≡12[ei(−E1+E2+E3−E4) xvF j S∗jl1S jl2δ jl3δ jl4 + ei(−E3+E4+E1−E2) xvF j S∗jl3S jl4δ jl1δ jl2].(2.61)Note that we have symmetrized the function ρ so that ρ jl1l2l3l4 (E1,E2,E3,E4;x)= ρjl3l4l1l2 (E3,E4,E1,E2;x).This interaction is diagrammatically represented by the symmetric vertex in Fig. 2.5. We may wellopt not to symmetrize ρ; however, the two created electrons E1l1 and E3l3 (or the two annihilatedelectrons E2l2 and E4l4) would be inequivalent in that case, and the diagrammatic bookkeepingwould be more difficult.2.4 Wilsonian approach to S-matrix renormalizationWe now review the derivation of the S-matrix RG equation using the Wilsonian scaling approach inRef. [132].Starting from Eq. (2.60), we reduce the energy cutoff D to D− δD (δD D), and inte-grate out the so-called “fast modes” with energies in one of the two slices (−D+δD,−D) and(D−δD,D). This procedure generates corrections of O(αδD/D) to the quadratic part of theHamiltonian [Eq. (2.58)] as well as the quartic part [Eq. (2.60)]. We assume that the corrections27Figure 2.5: Diagrammatic representation of the electron-electron interaction.to the quartic part are unimportant; the rationale is that the quartic part originates entirely from thebulk, so it should renormalize independently of the junction. In fact, since the quartic part is freeof Umklapp processes, it should be exactly marginal in the RG sense[109]. Meanwhile, the renor-malized quadratic part becomes off-diagonal and must be diagonalized with a new scattering basis,which is in turn associated with a running (i.e. cutoff-dependent) S-matrix.The quadratic correction generated by Eq. (2.60) readsδH j0 =−2∫ ∞0dxg j2 (x) ∑l1l2l3∫δDdE1∫ D−DdE2dE3(2pi)2 v2F jρ jl1l2l3l1 (E1,E2,E3,E1;x) f (E1)φ†l3 (E3)φl2 (E2) .(2.62)The E2E3 contraction is equivalent to the E1E4 contraction; hence the factor of 2. The E1E2 andE3E4 contractions are discarded because, once we sum over l taking into account the S-matrixunitarity ∑l1∣∣S jl1∣∣2 = 1, we find the resulting “tadpole” diagrams only harmlessly shift the chemicalpotential[109].We let φ ′ be the renormalized scattering basis after integrating out fast modes. φ ′ is related to φby another S-matrix, Sδj j′ , which only weakly deviates from the N×N identity matrix:φ j (E) =∫ dE ′2pi[iE−E ′+ i0φ′j(E ′)+−iE−E ′− i0∑j′Sδj j′(E ′;E)φ ′j′(E ′)](2.63)The inverse transformation is obtained by calculating anti-commutators:(φ ′j′(E ′))†=∑j∫ dE2pi[iE−E ′+ i0δ j j′+−iE−E ′− i0Sδj j′(E ′;E)]φ †j (E) (2.64)By definition φ ′ diagonalizes the renormalized quadratic Hamiltonian,28[∑jH j0,wire+H0,B+∑jδH j0 ,(φ ′j′(E ′))†]= E ′(φ ′j′(E ′))† (2.65)Substituting Eq. (2.64) into the above, we find to O(δD/D)i[δ j j′−Sδj j′(E ′;E)]= δD∫ ∞0dx{f (D)[α j (x)vF jS∗j jS j j′e−i 2D−E−E′vF j x+α j′ (x)vF j′S j′ j′S∗j′ jei 2D−E−E′vF j′x]+ f (−D)[α j (x)vF jS∗j jS j j′ei 2D+E+E′vF jx+α j′ (x)vF j′S j′ j′S∗j′ je−i 2D+E+E′vF j′ x]}. (2.66)We now perform the x integral in a simple model. Let us assume that the junction is connectedthrough wire n to a TLL or FL lead at x= Ln; in other words, when x≥ Ln, αn (x) = αn (∞) becomesa constant independent of x and dαn (x)/dx = 0. We further assume that the interaction inside thewire is also uniform, i.e.α j (x) = α j (0)+ [α j (∞)−α j (0)]H (x−L j) (2.67)where H (x) is the Heaviside unit-step function. Nowδ j j′−Sδj j′(E ′;E)= δD f (D)S∗j jS j j′ (α j (0)−α j (∞))e−i2D−E−E′vF jL j −α j (0)2D−E−E ′+S j′ j′S∗j′ j(α j′ (0)−α j′ (∞))ei 2D−E−E′vF j′L j′ −α j′ (0)2D−E−E ′+ f (−D)S∗j jS j j′ (α j (0)−α j (∞))ei2D+E+E′vF jL j −α j (0)−2D−E−E ′+S j′ j′S∗j′ j(α j′ (0)−α j′ (∞))e−i 2D+E+E′vF j′ L j′ −α j′ (0)−2D−E−E ′ (2.68)If D& |E|, |E ′|,±2D−E−E ′ can be approximated as±2D, thus giving rise to a scaling contributionO(δD/D). If D & vFn/Ln, exp(±i2DLn/vFn) oscillates rapidly with D and is negligible; on theother hand, when D. vFn/Ln, exp(±i2DLn/vFn)≈ 1. Finally, if D& T , the factors f (D)≈ 0 andf (−D)≈ 1 are approximately independent of D. Therefore, to O(δD/D), Eq. (2.68) predicts thatSδj j′(E;E ′)= δ j j′− δD2D(α j (D)S∗j jS j j′−α j′ (D)S j′ j′S∗j′ j)(2.69)independent of |E|, |E ′| and T as long as D & max{|E| , |E ′| ,T}. Here we have defined a cutoff-dependent interaction strength29αn (D)≡{αn (0) , D& vFn/Lnαn (∞) , D. vFn/Ln. (2.70)This means the renormalization will stop at the energy scale of the incident/scattered electron or thetemperature, whichever is higher. In addition, the energy scale associated with the inverse length ofwire n, vFn/Ln, determines whether the renormalization due to that wire is controlled by interactionstrength in the wire αn (0) or that in the lead αn (∞): the effective interaction strength crosses overfrom αn (0) to αn (∞) as the D is reduced below vFn/Ln.The renormalized S-matrix S+ δS relates φ ′ to the original fermions ψ . Inserting Eq. (2.63)into Eq. (2.57) we find that δS and Sδ obey the simple matrix relation δS= SSδ −S, and accordingto Eq. (2.69), δS j j′ is given byδS j j′ =−δD2D(∑nαn (D)S jnS∗nnSn j′−α j (D)S j jδ j j′). (2.71)We are now in a position to write down the RG equation for the S-matrix valid to O(α). Restoringthe explicit cutoff dependence, we have− dS j j′ (D)d lnD=−12∑nαn (D)[S jn (D)S∗nn (D)Sn j′ (D)−δ j′nSnn (D)δn j](2.72)where the RG flow is cut off at the temperature T . In the special case of αn independent of D, this isthe equation given in Refs. [70, 77]. It can be readily checked that Eq. (2.72) preserves the unitarityof the S-matrix.We pause to remark that, as the cutoff is reduced below the inverse length of one of the wires,renormalization due to that wire is governed only by the lead to which that wire is attached. This isreasonable because a junction of finite-length TLL wires attached to FL leads should, at low energies,renormalize into a junction connected directly to FL leads[12, 42, 93].2.5 First-order perturbation theory conductanceIntuitively, once the renormalization flow of the S-matrix is stopped by a physical infrared cutoff,the renormalized S-matrix should represent the non-interacting part of the low-energy theory of thejunction, and can be taken as an input to the Landauer formalism. However, such an argumentdoes not address the role of the low-energy residual interaction, which turns out to be especiallyimportant in the case of TLL leads. Also, in principle, the Landauer formalism is well-founded onlyin the absence of inelastic scattering. We are therefore motivated to study the conductance in the CSformulation, which fully exposes possible deviations from the Landauer predictions.In the CS formulation of RG, we start from a field theory with a running cutoff D, and calculatelow-energy physical observables (in our case the linear dc conductance tensor G j j′) as a function ofthe running coupling constants of the theory [in our case the S-matrix elements S j j′ (D)]. This is30once again accomplished by perturbation theory in interaction. We require that when D is greaterthan the energy scales at which the system is probed, namely the finite temperature T , G j j′ shouldbe independent of D. Therefore, by allowing the cutoff to run from D to D−δD, where δD D,we can find the RG equation satisfied by the coupling constants S j j′ (D), and subsequently find theconductance. In this section and the next, we first compute the linear dc conductance in Kuboformalism to the first order in interaction, then apply the CS formulation to our perturbation theoryresults.The current operator at coordinate x in wire j is first written in terms of the fermion fields:Iˆ j (x) = evF j(ψ†jRψ jR−ψ†jLψ jL)(x) . (2.73)Note that Iˆ j is not changed by the interaction; it is proportional to the commutator of the elec-tron density with the Hamiltonian, but the interaction commutes with the electron density. UsingEq. (2.57) we find the imaginary time correlation functionΩ j j′ (x,x′;τ− τ ′)≡−〈Tτ I j (x,τ) I j′ (x′,τ ′)〉to beΩ j j′(x,x′;τ− τ ′)=− e2(2pi)2 ∑j1 j2 j′1 j′2∫dε1dε2dε ′1dε′2[ei ε2−ε1vF j xS∗j j1S j j2− e−i ε2−ε1vF j xδ j j1δ j j2]×[eiε ′2−ε ′1vF j′x′S∗j′ j′1S j′ j′2− e−i ε′2−ε ′1vF j′x′δ j′ j′1δ j′ j′2]〈Tτφ †j1 (ε1,τ)φ j2 (ε2,τ)φ†j′1(ε ′1,τ′)φ j′2 (ε ′2,τ ′)〉H .(2.74)The imaginary time-ordered expectation value should be evaluated in the Heisenberg picture. Thelinear dc conductance G j j′ is then given by the retarded current-current correlation function Ω,G j j′(x,x′)= limω→0limηω→0+1iω[Ω j j′(x,x′;ω+)−Ω j j′ (x,x′;0)] , (2.75)where again ω+ ≡ ω + iηω . The coordinate dependence should vanish in the ω → 0 limit, sincewhere exactly we apply the bias or measure the current is inconsequential in a dc experiment[42, 93].Eq. (2.75) is now calculated in perturbation theory. Switching to the interaction picture, weperform a Wick decomposition of the time-ordered product, go to the frequency space and sumover the Matsubara frequencies. The retarded correlation function is then obtained by analyticcontinuation iωn → ω+ ≡ ω + iηω where the ηω → 0+ limit is taken. The energy integrals arecalculated afterwards, followed by real space integrals [which appear in Eq. (2.60)] in the end.Feynman diagrams involved to the first order are shown in Fig. 2.6, with the final results given byEqs. (2.88) and (2.98).31Figure 2.6: Diagrams contributing to the linear dc conductance at the first order in interaction.The second line shows the self-energy dressed bubble diagrams, while vertex correctiondiagrams are in the third line.2.5.1 Zeroth orderAt the zeroth order, there is only a single bubble diagram for the current-current correlation function[35,38]. Wick’s theorem gives〈Tτφ †j1 (ε1,τ)φ j2 (ε2,τ)φ†j′1(ε ′1,τ′)φ j′2 (ε ′2,τ ′)〉=−δ j2 j′1δ(ε2− ε ′1)G j2(ε2,τ− τ ′)δ j1 j′2δ(ε1− ε ′2)G j1(ε1,τ ′− τ). (2.76)Here G is the free scattering basis Matsubara Green’s function G j (E, iωn) = 1/(iωn−E), ωn =(2n+1)pi/β . We have dropped the H subscript in Eq. (2.74) when switching to the interactionpicture. Going to the frequency space, we have the standard Matsubara sum1β ∑iωnG j2 (E2, iωn)G j1 (E1, iωn− ipm) =f (E2)− f (E1)−ipm+E2−E1 , (2.77)where pm = 2mpi/β is a bosonic frequency and f (ε) = 1/(eβε +1)is the Fermi distribution attemperature β = 1/T . Performing analytic continuation ipm → ω+ ≡ ω + iηω (ηω → 0+) thenyields the zeroth order retarded correlation function,32Ω(0)j j′(x,x′;ω+)=e2(2pi)2 ∑j1 j2∫dε1dε2f (ε2)− f (ε1)−ω++ ε2− ε1[δ j j′ei(ε2−ε1)(xvF j− x′vF j′)+δ j j′ei(ε1−ε2)(xvF j− x′vF j′)− ∣∣S j′ j∣∣2 ei(ε1−ε2)(xvF j+ x′vF j′)− ∣∣S j j′∣∣2 ei(ε2−ε1)(xvF j+ x′vF j′)]. (2.78)We have done the j1 and j2 sums using unitarity of the S-matrix. Employing contour techniques, weintegrate over ε1 on (−∞,∞) for the term proportional to f (ε2), and integrate over ε2 on (−∞,∞)for the term proportional to f (ε1):Ω(0)j j′(x,x′;ω+)=e2(2pi)2∫dε2 (2pii) f (ε2)[δ j j′eiω+∣∣∣∣ xvF j− x′vF j′∣∣∣∣−0− ∣∣S j j′∣∣2 eiω+(xvF j+ x′vF j′)]− e2(2pi)2∫dε1 (2pii) f (ε1)[δ j j′eiω+∣∣∣∣ xvF j− x′vF j′∣∣∣∣−0− ∣∣S j j′∣∣2 eiω+(xvF j+ x′vF j′)]. (2.79)We note that the∣∣S j′ j∣∣2 term vanishes because the associated singularities are on the wrong sideof the contour. Now combine the f (ε2) and f (ε1) terms and restore the cutoff D, recalling thatε2− ε1 = ω+. This givesΩ(0)j j′(x,x′;ω+)= ie22pi∫dε2[f (ε2)− f(ε2−ω+)][δ j j′eiω+∣∣∣∣ xvF j− x′vF j′∣∣∣∣−0− ∣∣S j j′∣∣2 eiω+(xvF j+ x′vF j′)].(2.80)Substituting into Eq. (3.31), taking the ηω → 0+ limit and then the ω → 0 limit, we obtainG(0)j j′ =−e22pi(δ j j′−∣∣S j j′∣∣2) , (2.81)which is the usual linearized Landauer formula.2.5.2 First orderHigher order diagrams can be classified into two basic types, namely self-energy diagrams andvertex corrections. They are discussed separately in the following.33Self-energy diagramsAt the first order, as shown in Fig. 2.6, the bubble diagram is dressed by two types of self-energies:contraction of E1 with E2 or E3 with E4 in Eq. (2.60) (the “tadpole”), and contraction of E1 with E4or E2 with E3.For the self-energy diagrams and the dressed conductance bubbles we need two more types ofMatsubara frequency sums. The first one is1β ∑iωnG j (ε, iωn) = f (ε) . (2.82)The second one is1β ∑iωnG j′1(ε ′1, iωn)G j1 (ε1, iωn)G j2 (ε2, iωn+ ipm)= f (ε2)1(ε2− ipm)− ε ′11(ε2− ipm)− ε1 −∫ dε˜2piif (ε˜)× 1ε˜+ ipm− ε2[1ε˜+ i0− ε ′11ε˜+ i0− ε1 −1ε˜− i0− ε ′11ε˜− i0− ε1]. (2.83)To compute this sum, we consider the following contour integral,∮ dz2piif (z)1z− ε ′11z− ε11z+ ipm− ε2 , (2.84)where the integration contour is wrapped around the branch cut on the real axis[74], so that polesinside the contour are z= iωn (n running over all integers) and also z= ε2− ipm. The f (ε2) term inEq. (2.83) comes from z= ε2− ipm, and the f (ε˜) term comes from the branch cut z= 0.We ignore the tadpole-type self-energy diagrams, again on the grounds that they only modifythe chemical potential. The other type of self-energy diagrams turn out to dress the S-matrix. Oneinstance of these diagrams reads34Ω(1),SE,non-tadpole,1j j′(x,x′;τ− τ ′)=− e2(2pi)2 ∑j1 j2 j′1 j′2∫dε1dε2dε ′1dε′2×[ei (ε2−ε1)vF jxS∗j j1S j j2− e−i (ε2−ε1)vF j xδ j j1δ j j2][ei (ε ′2−ε ′1)vF j′x′S∗j′ j′1S j′ j′2− e−i (ε′2−ε ′1)vF j′x′δ j′ j′1δ j′ j′2]× (−)∫ β0dτ1∑n∫dygn2 (y) ∑l1l2l3l4∫ dE1dE2dE3dE4(2pi)2 v2Fnρnl1l2l3l4 (E1,E2,E3,E4;y)×δ j2 j′1δ(ε2− ε ′1)G j2(ε2,τ− τ ′)δ j′2l1δ(ε ′2−E1)G j′2(ε ′2,τ′− τ1)×δl2l3δ (E2−E3)Gl2 (E2,0)δl4 j1δ (E4− ε1)G j1 (ε1,τ1− τ) . (2.85)Going to the frequency space, performing Matsubara sums and analytic continuation, we findΩ(1),SE,non-tadpole,1j j′(x,x′;ω+)=− e2(2pi)2 ∑j1 j2 j′2∫dε1dε2dε ′2×[ei (ε2−ε1)vF jxS∗j j1S j j2− e−i (ε2−ε1)vF j xδ j j1δ j j2][ei (ε ′2−ε2)vF j′x′S∗j′ j2S j′ j′2− e−i (ε′2−ε2)vF j′x′δ j′ j2δ j′ j′2]× (−)∑n∫dygn2 (y)∫ dE2(2pi)2 v2Fnf (E2)ρnj′2nn j1(ε ′2,E2,E2,ε1;y)×[f (ε2)1(ε2−ω+)− ε ′21(ε2−ω+)− ε1 −∫ dε˜2piif (ε˜)1ε˜+ω+− ε2×(1ε˜+ i0− ε ′21ε˜+ i0− ε1 −1ε˜− i0− ε ′21ε˜− i0− ε1)]. (2.86)Carrying out the ε1 and ε ′2 integrations, and also the ε2 integration in the f (ε˜) term, this becomes inthe x, x′→ ∞ limitΩ(1),SE,non-tadpole,1j j′(x,x′;ω+)=− e22pi∫dε2[f (ε2)− f(ε2−ω+)]× (−)∑n∫dyαn (y)∫ dE2vFnf (E2)12eiω+(xvF j+ x′vF j′)×S j j′[S∗jnSnnS∗n j′e2i(E2−(ε2−ω+)) yvFn +δ jnS∗nnδ j′ne−2i(E2−(ε2−ω+)) yvFn]. (2.87)This is just one of the four terms which contribute to the dressing of the S-matrix in Eq. (3.31).Another identical term comes from contracting E1 with E4 (completely equivalent to contracting E2with E3 which we have done). The remaining two terms have all their electron propagators reverted,so that their contributions to the conductance are the complex conjugate of the first two terms. In35the end, contributions from first-order self-energy diagrams can be integrated into a Landauer-typeformula:G(0)j j′ +G(1),SEj j′ =−e22pi(δ j j′−∫dε [− f (ε)]∣∣∣Sd(1)j j′ (ε)∣∣∣2) (2.88)where the first order “dressed S-matrix” Sd(1) is given bySd(1)j j′ (ε) = S j j′− i∑n∫ ∞0dyαn (y)∫ dε ′vFnf(ε ′)×S jnS∗nnSn j′ exp(2i(ε− ε ′) yvFn)+δ jnSnnδ j′n exp(−2i(ε− ε ′) yvFn); (2.89)αn (y) has been defined in Eq. (2.55). For a non-interacting system Sd(1)j j′ (ε) = S j j′ ; this is in agree-ment with our intuitive expectation.For the simple model Eq. (2.67), integrating over y:Sd(1)j j′ (ε) = S j j′ (ε)−∑n∫dε ′f (ε ′)2(ε ′− ε)(S jnS∗nnSn j′[(αn (∞)−αn (0))e2i(ε−ε′) LnvFn +αn (0)]−δ jnSnnδ j′n[(αn (∞)−αn (0))e−2i(ε−ε′) LnvFn +αn (0)]). (2.90)The ε ′ integral is infrared divergent, which prompts an RG resummation of leading logarithms. Wewill determine the renormalization of the S-matrix using Eq. (2.90) and discuss its implications inSection 2.6.Vertex correction diagramsThere are two types of first order vertex correction diagrams, the “cracked egg” diagram and thering diagram. Neither type of vertex corrections to the conductance requires Matsubara sums otherthan Eq. (2.77).By summing over all dummy wire indices, we can show that the “cracked egg” contribution tothe dc conductance is proportional to δ j j′ . On the other hand, due to current conservation and theabsence of equilibrium currents, the full dc conductance G j j′ obeys∑jG j j′ =∑j′G j j′ = 0; (2.91)this must also be true at O(α). Since Eq. (2.91) is already satisfied by the self-energy contributionEq. (2.88), and also by the ring diagram contribution Eq. (2.98) as we shall see below, it must beseparately satisfied by the “cracked egg” diagrams as well. But ∑ j δ j j′ = ∑ j′ δ j j′ = 1, and we infer36that the “cracked egg” diagrams must be identically zero.An example of the ring diagram isΩ(1),VC,ring,1j j′(x,x′;τ− τ ′)=− e2(2pi)2 ∑j1 j2 j′1 j′2∫dε1dε2dε ′1dε′2×[ei (ε2−ε1)vF jxS∗j j1S j j2− e−i (ε2−ε1)vF j xδ j j1δ j j2][ei (ε ′2−ε ′1)vF j′x′S∗j′ j′1S j′ j′2− e−i (ε′2−ε ′1)vF j′x′δ j′ j′1δ j′ j′2]× (−)∫ β0dτ1∑n∫dygn2 (y) ∑l1l2l3l4∫ dE1dE2dE3dE4(2pi)2 v2Fnρnl1l2l3l4 (E1,E2,E3,E4;y)×δ j2l1δ (ε2−E1)G j2 (ε2,τ− τ1)δl4 j′1δ(ε ′1−E4)G j′1(ε ′1,τ1− τ ′)×δ j′2l3δ(ε ′2−E3)G j′2(ε ′2,τ′− τ1)δ j1l2δ (E2− ε1)G j1 (ε1,τ1− τ) . (2.92)Going to the frequency space, performing Matsubara sums and analytic continuation:Ω(1),VC,ring,1j j′(x,x′;ω+)=− e2(2pi)2 ∑j1 j2 j′1 j′2∫dε1dε2dε ′1dε′2×[ei (ε2−ε1)vF jxS∗j j1S j j2− e−i (ε2−ε1)vF j xδ j j1δ j j2][ei (ε ′2−ε ′1)vF j′x′S∗j′ j′1S j′ j′2− e−i (ε′2−ε ′1)vF j′x′δ j′ j′1δ j′ j′2]×∑n∫dygn2 (y)1(2pi)2 v2Fnρnj2 j1 j′2 j′1(ε2,ε1,ε ′2,ε′1;y) f (ε2)− f (ε1)ε2− ε1−ω+f (ε ′2)− f (ε ′1)ε ′2− ε ′1+ω+. (2.93)Integrating over the energies as before, we findΩ(1),VC,ring,1j j′(x,x′;ω+)=− e22pi∫dε1dε ′1 (−)[f(ε1+ω+)− f (ε1)][ f (ε ′1−ω+)− f (ε ′1)]×∑n∫dyαn (y1)1vFn12[∣∣S jn∣∣2 δ j′neiω+(xvF j− x′vF j′)e2iω+ yvFn H(y− x′)−δ jnδ j′ne−iω+(xvF j+ x′vF j′)e2iω+ yvFn H (y− x)H (y− x′)−δ jnδ j′neiω+(xvF j+ x′vF j′)e−2iω+ yvFn H (x− y)H (x′− y)− ∣∣S jn∣∣2 ∣∣Sn j′∣∣2 eiω+(xvF j+ x′vF j′)e2iω+ yvFn +δ jn∣∣Sn j′∣∣2 e−iω+(xvF j− x′vF j′)e2iω+ yvFn H (y− x)]. (2.94)There exists an analogous term with all electron lines reverted. Upon substitution into Eq. (3.31)these two terms produce37G(1),VCj j′(x,x′)=e22pii∑n∫ ∞0dyvFnαn (y)× limω→0limηω→0+ω[∣∣S jn∣∣2 δ j′neiω+(xvF j− x′vF j′)e2iω+ yvFn H(y− x′)−δ jnδ j′ne−iω+(xvF j+ x′vF j′)e2iω+ yvFn H (y− x)H (y− x′)−δ jnδ j′neiω+(xvF j+ x′vF j′)e−2iω+ yvFn H (x− y)H (x′− y)− ∣∣S jn∣∣2 ∣∣Sn j′∣∣2 eiω+(xvF j+ x′vF j′)e2iω+ yvFn+δ jn∣∣Sn j′∣∣2 e−iω+(xvF j+ x′vF j′)e2iω+ yvFn H (y− x)]. (2.95)Integration by parts gives us2iω+vFn∫ yuyldyαn (y)e2iω+ yvFn = αn (yu)e2iω+ yuvFn −αn (yl)e2iω+ ylvFn −∫ yuyldye2iω+ yvFndαn (y)dy, (2.96)where yu can be vFnx/vF j, vFnx′/vF j′ or ∞, and yl can be vFnx/vF j, vFnx′/vF j′ or 0. We can let xand x′ be sufficiently large so that yu > Ln is always satisfied; thus in the dαn/dy term in Eq. (2.96),yu may be replaced by Ln.If yu → ∞, the αn (yu) term damps out due to the small imaginary part ηω , and Eq. (2.96)becomes in the ω → 0 and ηω → 0 limit2iω+vFn∫ yuyldyαn (y)e2iω+ yvFn =−αn (yl)−∫ Lnyldydαn (y)dy=−αn (Ln) =−αn (∞) . (2.97a)On the other hand, if yu is finite, the αn (yu) term will survive the ω → 0 and ηω → 0 limit:2iω+vFn∫ yuyldyαn (y)e2iω+ yvFn = αn (yu)−αn (yl)−∫ Lnyldydαn (y)dy= αn (yu)−αn (Ln) = 0. (2.97b)Therefore, taking the dc limit explicitly in Eq. (2.95), we find wire n contributes to the vertexcorrection only when it is attached to a TLL lead, and the interaction inside the wire is immaterial:G(1),VCj j′(x,x′)=e22pi∑n12αn (∞)(δ jn−∣∣S jn∣∣2)(δn j′− ∣∣Sn j′∣∣2) . (2.98)When αn (∞) = 0, as is the case for any wire n attached to an FL lead, the vertex correction due to nvanishes.382.6 First-order CS formulation of RGIn this section, we analyze the result of Section 2.5 from the perspective of the CS formulation ofRG[12], and present a modified Landauer formula involving the renormalized S-matrix in the caseof FL leads, supplemented by vertex corrections from TLL leads.Beginning from the simplest case where all leads are FL leads, i.e. αn (∞) = 0 for all n, the vertexcorrection Eq. (2.98) vanishes, and the full linear dc conductance to O(α) is given by Eq. (2.88).Reducing the cutoff from D to D−δD and demanding the right-hand side of Eq. (2.88) be a scalinginvariant, we have∫δDdε[− f ′ (ε)]∣∣∣Sd(1)j j′ (ε,D)∣∣∣2+∫ D−D dε [− f ′ (ε)][(Sd(1)j j′ (ε,D))∗δSd(1)j j′ (ε,D)+ c.c.]= 0(2.99)where∫δD =∫ D(D−δD)+∫ −(D−δD)−D stands for integration over fast modes, and δSd(1) is the renormal-ization of Sd(1):δSd(1)j j′ (ω,D)≡ Sd(1)j j′ (ω,D)−Sd(1)j j′ (ω,D−δD) . (2.100)Here Sd(1)j j′ (ε,D) is Eq. (2.90) with the ε′ integral going from−D to D. We have made the cutoff de-pendence explicit, and all S-matrix elements are understood to be cutoff-dependent, S j j′→ S j j′ (D).Since the derivative of the Fermi function is peaked at the Fermi energy with width T , the∫δDintegral in Eq. (2.99) approximately vanishes while D& T ; Eq. (2.99) is thus automatically satisfiedif Eq. (2.100) vanishes. The implication is that, at least in the case of FL leads, the renormalizationof the conductance can be fully accounted for by the renormalization of the S-matrix.To the lowest order in δD, the condition that δSd(1)j j′ (ω,D) = 0 is equivalent toδS j j′ (ω,D)≡ S j j′ (ω,D)−S j j′ (ω,D−δD)=∑n∫δDdε ′f (ε ′)2(ε ′−ω)(S jnS∗nnSn j′[(αn (∞)−αn (0))e2i(ω−ε′) LnvFn +αn (0)]−δ jnSnnδ j′n[(αn (∞)−αn (0))e−2i(ω−ε′) LnvFn +αn (0)]). (2.101)When we assume D & max{|ω| ,T}, and apply the same considerations below Eq. (2.68), fromEq. (2.101) we recover none other than Eq. (2.71). Thus to the first order in interaction the CSapproach and the Wilsonian approach predict the same S-matrix renormalization, Eq. (2.72).Once the cutoff D is reduced to the order of T , the perturbative correction to the S-matrixSd(1)j j′ (ε,D)−S j j′ (D) vanishes to the scaling accuracy; thus S j j′ (D= T )may be used to approximatethe dressed S-matrix in Eq. (2.88), and the conductance for a junction connected to FL leads is given39Figure 2.7: Dressing of the first order vertex correction diagrams by the first order self-energydiagrams.by the modified Landauer formula,GFLj j′ =−e22pi(δ j j′−∣∣S j j′ (T )∣∣2) , (2.102)where the S-matrix is now fully renormalized according to Eq. (2.72), with the cutoff reduced to thetemperature T . This is the Landauer-type formula invoked in Refs. [70, 77, 132].When some of the leads are TLL leads, corrections of Eq. (2.98) must also be taken into account.It is important to note, however, that in a CS analysis of the total conductance, Eq. (2.100) remainsvalid to O(α). This is because as the cutoff is lowered, Eq. (2.98) contributes additional terms ofthe form of α (∞)S∗δS to Eq. (2.99). However, by Eq. (2.100), δS is O(α); hence α (∞)S∗δS isO(α2), and is negligible to O(α).To calculate the total conductance at D= T with TLL leads, we go slightly beyond the first orderand dress the O(α) vertex correction diagrams with O(α) self-energy diagrams, shown in Fig. 2.7.The bare S-matrix in Eq. (2.98) is then replaced by the dressed S-matrix, Sd(1):Gd(1),VCj j′(x,x′)=e22pi∑n12αn (∞)(δ jn−∫dε1[− f ′ (ε1)]∣∣∣Sd(1)jn (ε1)∣∣∣2)×(δn j′−∫dε2[− f ′ (ε2)]∣∣∣Sd(1)n j′ (ε2)∣∣∣2) . (2.103)This allows us to repeat our previous analysis for the case of FL leads, and further approximate Sd(1)j j′by S j j′ (D= T ). Thus the TLL leads contribute an additional conductance of40GTLLj j′ −GFLj j′ =e22pi∑nαn (∞)2(δ jn−∣∣S jn (T )∣∣2)(δn j′− ∣∣Sn j′ (T )∣∣2) . (2.104)Eqs. (2.72), (2.102) and (2.104) provide a comprehensive first-order picture for non-resonanttunneling through a junction: the interaction renormalizes the S-matrix, the renormalized S-matrixdetermines the conductance through a Landauer-type formula if the junction is connected to FLleads, and the residual interaction further modifies the conductance if the junction is attached toTLL leads[111]. As will be demonstrated in Section 2.7, this picture is by no means limited to thefirst order.2.7 RPAIn this section, we extend our first-order RG analysis in Section 2.6 to arbitrary order in interactionunder the RPA[11, 13, 15]. The correlation function Eq. (2.74) is perturbatively evaluated for bothself-energy diagrams and vertex corrections by the same procedures, except that the interaction isdressed with ring diagrams; see Fig. 2.8. We subsequently find the S-matrix RG equation in the CSscheme and express the conductance in terms of the renormalized S-matrix. This is once more astraightforward calculation, and we present the result first before laying out the details.Introduce the shorthand Wj j′ (D) ≡∣∣S j j′ (D)∣∣2. The RPA self-energy diagrams give rise to amodified Landauer formula:GFLj j′ =−e22pi[δ j j′−Wj j′ (T )], (2.105)where the renormalization of the S-matrix is governed by a generalization of Eq. (2.72)[111],− dS j j′ (D)d lnD=−12 ∑n1n2[S jn1 (D)Πn1n2 (D)S∗n2n1 (D)Sn2 j′ (D)−δ j′n1Π∗n1n2 (D)Sn2n1 (D)δn2 j].(2.106)The RPA-dressed interaction isΠ(D)≡ 2 [Q (D)−W (D)]−1 , (2.107)whereQ j j′ = Q j (D)δ j j′ , Q j (D) =1+K j (D)1−K j (D) , (2.108)with K j (D) =√(1−α j (D))/(1+α j (D)) being the cutoff-dependent “Luttinger parameter” forwire j; α j (D) is given in Eq. (2.70). To lowest order in α j, Πi j = δi jα j. When all wires of thejunction are attached to FL leads, in parallel with the O(α) calculation, Eq. (2.105) captures the41Figure 2.8: The RPA diagrammatics: (a) effective interaction in the RPA represented by thickwavy lines; (b) dressed propagator in the RPA, to O(δD/D) in RG, represented by thickstraight lines; and (c) diagrams contributing to the Kubo conductance in the RPA. Thedressed propagator in (b) is calculated to O(δD/D) only, because higher order termsin δD/D do not contribute to the renormalization of the S-matrix [Eq. (2.106)]— seeSection 2.6 for an explanation in the first order context. (a) and (c) do not involve trun-cation at O(δD/D) because any renormalization of the interaction [Eq. (2.107)] and theconductance [Eqs. (2.105) and (2.109)] can be attributed to the renormalization of theS-matrix. Note that (c) features a thin interaction line (rather than a thick one) to avoiddouble-counting.entirety of the conductance. This is in agreement with the Kubo formula calculation in Refs. [11–13, 15] in the language of chiral fermion densities.When some wires are attached to TLL leads, they again provide important corrections to the dcconductance. All RPA vertex correction diagrams dressed with RPA self-energy evaluate toGTLLj j′ −GFLj j′ =e22pi∑n1n2 [δ jn1−Wjn1 (T )]12ΠLn1n2[δn2 j′−Wn2 j′ (T )], (2.109)where the residual effective interaction isΠL = 2[QL−W (T )]−1 , (2.110)and QL is given by Eq. (2.108) with K j replaced by KLj =√(1−α j (∞))/(1+α j (∞)), the Lut-tinger parameter of the lead.Remarkably, if we follow Eq. (2.30) and introduce the dc “contact resistance” tensor betweenthe wires and leads,42Figure 2.9: Schematic representation of the relation between the conductance of a junc-tion with FL leads and that of a junction with TLL leads, through the “contact resis-tance”Eq. (2.30).(G−1c)j j′ =(e22pi)−1 12[(KLj)−1−1]δ j j′ , (2.111)then Eq. (2.109) can be formally recast asGTLL =(1−GFLG−1c)−1 GFL, (2.112)where 1 is the N×N identity matrix. The same relation has been derived in Refs. [51, 93], whichassume that the dc contact resistance between a finite TLL wire and an FL lead is not affected by thejunction at the other end of the TLL wire. This assumption is reinforced by our RPA calculations.While it is tempting to further simplify Eq. (2.112) into(GTLL)−1+G−1c =(GFL)−1, this simplifi-cation cannot be rigorous since neither GTLL nor GFL is invertible. Nevertheless, it does provide uswith an intuitive understanding of Eq. (2.112); see Fig. 2.9.We emphasize that the inclusion of the vertex correction diagrams does not change the RG equa-tion of the S-matrix, Eq. (2.106). [The TLL leads do change the renormalization of the S-matrixthrough the scale-dependent interaction, Eq. (2.70).] The reason for this is as follows. Eq. (2.106)results from dressing the single particle propagator as shown in panel (b) of Fig. 2.8. The conduc-tance is calculated in perturbation theory by replacing all bare single particle propagators (the thinlines) with the dressed ones (the thick lines) in the basic bubble diagram and the vertex correctiondiagrams; or equivalently, by replacing all bare S-matrix elements with the ones dressed with theRPA self-energy. As with the case at the first order, the RPA vertex correction diagrams do notintroduce additional cutoff-sensitive integrals, and all cutoff-sensitive integrals originate from thedressed S-matrix. Therefore, the dressed S-matrix should be a cutoff-independent quantity when43we apply the CS scheme to the conductance, regardless of whether the vertex correction diagramscontribute to the conductance. The form of Eq. (2.106) is thus independent of vertex corrections.An immediate consequence of the robustness of the S-matrix renormalization is that, in thetemperature range above the inverse lengths of the wires, the leading universal scaling exponents ofthe conductance versus temperature are the same for FL leads and TLL leads to the accuracy of theRG method. Since the temperature dependence of the residual interaction ultimately results fromthat of the renormalized W matrix, Taylor-expanding Eq. (2.109) in the vicinity of a fixed point,we find that the leading scaling exponents of the TLL lead conductance are none other than thoseof the W matrix, i.e. those of the FL lead conductance; in this temperature range the TLL leadsmerely modify the non-universal multiplicative coefficients to the power law. Therefore, for thetemperature dependence of the conductance in the introduction, we have directly quoted the FL leadresults from Refs. [13, 15, 70].Eqs. (2.105)–(2.112) are the central results of this chapter. They show that at least in the RPA,in addition to the Landauer-type formula, TLL leads give rise to important corrections to the lineardc conductance which are also given in terms of the renormalized S-matrix[111]. In the remainingsections of this chapter, we will implement these results in non-resonant tunneling through 2-leadjunctions and Y-junctions.We now expound the RPA calculations that lead to Eqs. (2.106) and (2.109).2.7.1 Details of the RPA conductanceThe RPA self-energy beyond the first order involves a new type of Matsubara sum. For instance, atthe third order in interaction, we need1β ∑ipmf (E4)− f (E3)ipm+E4−E3f (E8)− f (E7)ipm+E8−E71i(pm+ωn)−E2= f (E2)f (E4)− f (E3)E2− iωn+E4−E3f (E8)− f (E7)E2− iωn+E8−E7+∫ dε˜2piinB (ε˜)1ε˜+ iωn−E2(f (E4)− f (E3)(ε˜+ i0)+E4−E3f (E8)− f (E7)(ε˜+ i0)+E8−E7− f (E4)− f (E3)(ε˜− i0)+E4−E3f (E8)− f (E7)(ε˜− i0)+E8−E7)(2.113)where nB (ε) = 1/(eβε −1) is the Bose distribution. To derive Eq. (2.113) we again wrap theintegration contour around the branch cut at the real axis. The fraction with numerator f (E3)−f (E4) originates from the fermion loop with loop energy E3 and E4; at the lth order there will bel− 1 loops present. ipm is the bosonic frequency carried by the interaction lines; after ipm, iωn isalso summed over following Eq. (2.83).44After we perform analytic continuation and integrate over the loop momenta, as x, x′→ ∞, thethree most important terms in the correlation function at the third order areΩ(3),SE,RPA,E2j j′(x,x′;ω+)=e22pieiω+(xvF j+ x′vF j′) ∫dε ′1[f(ε ′1)− f (ε ′1−ω+)]S j j′× ∑n1n2n3∫ ∞0dy˜1dy˜2dy˜3αn1 (vFn1 y˜1)αn2 (vFn2 y˜2)αn3 (vFn3 y˜3)∫dE2 f (E2) E˜22×[S∗jn3Sn1n3S∗n1 j′ |Sn2n1 |2 |Sn3n2 |2 e2iE˜2(y˜1+y˜2+y˜3)+S∗jn3Sn1n3S∗n1 j′δn3n2δn2n1H (y˜1− y˜2)H (y˜3− y˜2)e2iE˜2(y˜1−y˜2+y˜3)+δ jn3δn3n1S∗n1 j′δn3n2 |Sn2n1 |2H (y˜2− y˜3)e2iE˜2(y˜1+y˜2−y˜3)+S∗jn3δn3n1δn1 j′ |Sn3n2 |2 δn2n1H (y˜2− y˜1)e2iE˜2(−y˜1+y˜2+y˜3)+δ jn3S∗n3n1δn1 j′δn3n2δn2n1H (y˜2− y˜1)H (y˜2− y˜3)e2iE˜2(−y˜1+y˜2−y˜3)](2.114a)where we have substituted y˜n = yn/vFn and E˜2 = E2− ε ′1+ω+,Ω(3),SE,RPA,ε˜+j j′(x,x′;ω+)=e22pieiω+(xvF j+ x′vF j′) ∫dε ′1[f(ε ′1)− f (ε ′1−ω+)]S j j′× ∑n1n2n3∫ ∞0dy˜1dy˜2dy˜3αn1 (vFn1 y˜1)αn2 (vFn2 y˜2)αn3 (vFn3 y˜3)∫dε˜nB (ε˜) ε˜2×[δ jn3δn3n1S∗n1 j′δn3n2 |Sn2n1 |2H (y˜2− y˜3)H (y˜3− y˜1)e2i(ε˜+i0)(y˜1+y˜2−y˜3)+S∗jn3δn3n1δn1 j′ |Sn3n2 |2 δn2n1H (y˜2− y˜1)H (y˜1− y˜3)e2i(ε˜+i0)(−y˜1+y˜2+y˜3)+δ jn3S∗n3n1δn1 j′δn3n2δn2n1H (y˜2− y˜1)H (y˜2− y˜3)e2i(ε˜+i0)(−y˜1+y˜2−y˜3)](2.114b)and finallyΩ(3),SE,RPA,ε˜−j j′(x,x′;ω+)=e22pieiω+(xvF j+ x′vF j′) ∫dε ′1[f(ε ′1)− f (ε ′1−ω+)]S j j′× ∑n1n2n3∫ ∞0dy˜1dy˜2dy˜3αn1 (vFn1 y˜1)αn2 (vFn2 y˜2)αn3 (vFn3 y˜3)∫dε˜nB (ε˜) ε˜2×[−δ jn3δn3n1S∗n1 j′δn1n2 |Sn2n3 |2H (y˜2− y˜1)H (y˜3− y˜1)e2i(ε˜−i0)(y˜1−y˜2−y˜3)−δ jn3S∗n3n1δn1 j′δn1n2δn2n3H (y˜1− y˜2)H (y˜3− y˜2)e2i(ε˜−i0)(−y˜1+y˜2−y˜3)−S∗jn3δn3n1δn1 j′ |Sn1n2 |2 δn2n3H (y˜2− y˜3)H (y˜1− y˜3)e2i(ε˜−i0)(−y˜1−y˜2+y˜3)−δ jn3S∗n3n1δn1 j′ |Sn1n2 |2 |Sn2n3 |2 e2i(ε˜−i0)(−y˜1−y˜2−y˜3)](2.114c)45plus similar terms with all electron lines reverted. y˜ j ≡ y j/vF j runs between 0 and ∞, j = 1, 2, 3.These three terms come from lines 2, 3 and 4 of Eq. (2.113) respectively.In the dc limit, the zeroth order contribution and the self-energy corrections to the conductanceagain constitute a Landauer-type formula with a dressed S-matrix, similar to Eq. (2.88). Now wereduce the cutoff and demand the conductance be cutoff-independent. Once the y˜ integrals areperformed, it is obvious that the cutoff-sensitive integrals are the E2 integral and the ε˜ integral.We are in a position to discuss the real space integrals. We first focus on the simplest case wherethe interactions in wires and leads are uniform and identical, αn1 (y) = αn1 for any n1, so that all α’sfactor out. At the third order, we find the following integrals:I1(E+)≡ ∫ ∞0dy˜1dy˜2dy˜3eiE+(y˜1+y˜3)eiE+(y˜1−2y˜2+y˜3)H (y˜1− y˜2)H (y˜3− y˜2) (2.115)which appears alongside the factors δn1n2δn2n3 , and∫ ∞0dy˜1e2iE+y˜1 =i2E+(2.116)which appears alongside, for example, Wn1n2Wn2n3 . (More accurately, Eq. (2.116) comes with each“node” n2 as long as n2 is not sandwiched between two Kronecker δ factors.) Here E+ ≡ E + i0may be replaced by E˜2 or (±ε˜+ i0). At higher orders, we need to evaluate the integralIM(E+)≡ 2M+1∏l=1(∫ ∞0dy˜l)eiE+(y˜1+y˜2M+1)M∏j=1[eiE+(y˜2 j−1+y˜2 j+1−2y˜2 j)H (y˜2 j−1− y˜2 j)H (y˜2 j+1− y˜2 j)](2.117)This is accompanied by a string of 2M consecutive δ factors uninterrupted byW factors, δn1n2δn2n3 · · ·δn2Mn2M+1 .We will prove in Section 2.7.2 thatIM(E+)=(i2E+)2M+1CM (2.118)where CM = (2M)!/ [M!(M+1)!] is the Mth Catalan number[120, 121]. The first few Catalannumbers 0≤M ≤ 5 are 1, 1, 2, 5, 14, 42.At this stage we can combine the (E+)−1 factors in Eqs. (2.116) and (2.118) with the E˜2 or ε˜factors. At each order there will be a single (E+)−1 factor left unpaired, which gives the leading-logrenormalization δD/D as the cutoff is reduced from D to D−δD. Collecting terms of all orders wesee the S-matrix RG equation is of the form of Eq. (2.106), but the interaction Π(D) is given by46Π j j′2=α j2δ j j′+α j2α j′2{Wj j′+∑n1αn12[δ jn1δn1 j′+Wjn1Wn1 j′]+ ∑n1n2αn12αn22[δ jn1δn1n2Wn2 j′+Wjn1δn1n2δn2 j′+Wjn1Wn1n2Wn2 j′]+ ∑n1n2n3αn12αn22αn32[2δ jn1δn1n2δn2n3δn3 j′+δ jn1δn1n2Wn2n3Wn3 j′+Wjn1δn1n2δn2n3Wn3 j′+Wjn1Wn1n2δn2n3δn3 j′+Wjn1Wn1n2Wn2n3Wn3 j′]+ · · ·}(2.119)The rules to write down terms in Eq. (2.119) are as follows. At O(αm), there is a total number of(m−1) factors of δ and W . The δ factors always appear in even-length strings separated by theW factors. Each string of δ of length 2M is associated with a multiplicative coefficient of the MthCatalan number CM. For instance, at O(α17)there is a term WδδδδδδWWWδδδδWW , whoseprefactor will be C3C2 = 5×2 = 10.We can resum Eq. (2.119) by observing that we can uniquely construct every term containing aleast one factor of W , by adding to an existing term a (possibly empty) even-length string of δ fol-lowed by one factor ofW ; e.g. the term δδδδWWδδW is uniquely constructed as δδδδ /W /WδδW .In other words, Π satisfies the relationΠ j j′2=Π¯ j j′2+∑l1l2Π¯ jl12Wl1l2Πl2 j′2. (2.120)Here Π¯ is the part of Π which does not contain any factors of W :Π¯ j j′2=α j2δ j j′+α j2α j′2[∑n1αn12δ jn1δn1 j′+ ∑n1n2n3αn12αn22αn322δ jn1δn1n2δn2n3δn3 j′+ · · ·]=α j2δ j j′∞∑M=0CM(α j2)2M=α j1+√1−α2jδ j j′ . (2.121)In the last line we have used the generating function of Catalan numbers[120],∞∑M=0CMxM =21+√1−4x . (2.122)Inserting Eq. (2.121) into Eq. (2.120) and solving forΠ, we obtain Eq. (2.107) in the case of spatiallyuniform interactions, αn (y) = αn.We now argue that the cutoff-dependence of the Luttinger parameter is through Eq. (2.70) asis the case with the first order calculation. To this end, notice that it is values of y˜n between 0 and47O(1/E+) that dominate the integral in Eq. (2.117). Therefore, when D = ReE+ & vFn/Ln, theintegral is governed by vFny˜n . Ln; in this range of y˜n, αn (vFny˜n) = αn (0). On the other hand,when D vFn/Ln, the integral is controlled mainly by vFny˜n Ln, where αn (vFny˜n) = αn (∞).This justifies the crossover behavior given by Eqs. (2.70) and (2.108), and concludes the calculationof the self-energy terms in the RPA conductance.Calculations of the RPA vertex corrections, or the ring diagrams, are completely in parallel withthe first-order vertex corrections except Eq. (2.117) appears in the real space integrals. Here E+ inEq. (2.117) should be substituted for ω+. At the mth order, all m factors of 1/ω+ in Eqs. (2.116) and(2.118) can be paired with the m+1 factors of ω+ from loop energy integrals; the single unpairedω+ will be combined with the 1/ω factor in Eq. (3.31) so that the conductance is finite in the dclimit. Also, all interaction strengths appearing here are those in the leads αn (∞); this is because inthe dc limit ω . vFn/Ln for any lead n, and we may refer to our argument in the previous paragraphfor D vFn/Ln. Eventually, taking into account the dressing of the electron lines, we recoverEq. (2.109).2.7.2 Real space integral Eq. (2.118)To prove Eq. (2.118), we adopt the following change of variables in Eq. (2.117): z0 = y˜1, z2 j−1 =y˜2 j−1− y˜2 j, z2 j = y˜2 j+1− y˜2 j, 1 ≤ j ≤ M. The absolute value of the Jacobian of this change ofvariables is simply∣∣∣(−1)M∣∣∣ = 1. We also introduce the shorthand s j = ∑ jl=0 (−1)l zl . Eq. (2.117)then becomesIM(E+)=∫ ∞0dz0M∏l=1(∫ s2l−20dz2l−1∫ ∞0dz2l) M∏j=0e2iE+z2 j (2.123)Now consider the auxiliary object,I˜M(E+,z0)≡ M∏l=1(∫ s2l−20dz2l−1∫ ∞0dz2l) M∏j=0e2iE+z2 j ≡ (2iE+)−2M e2iE+z0 M∑l=0TM,ll!(−2iE+z0)l(2.124)where TM,l are dimensionless coefficients; obviously I˜0 (E+,z0) = e2iE+z0 and T0,0 = 1. I˜M obeys therecurrence relationI˜M+1(E+,z0)=∫ z00dz1e2iE+z1∫ ∞0dz2I˜M(E+,z0− z1+ z2). (2.125)Inserting Eq. (2.124) into Eq. (2.125), we find that TM,l satisfies the simple recurrence relationTM+1,l =∑Mj=l−1TM, j, and that TM+1,0 = 0 (M ≥ 0). Such a recurrence relation leads to the Catalan’striangle[121],48TM,l =(2M− l−1)!lM!(M− l)! (M ≥ 1) . (2.126)Therefore,IM(E+)=∫ ∞0dz0I˜M(E+,z0)=−(2iE+)−2M−1 M∑l=0TM,l (2.127)Noting that ∑Ml=0TM,l = CM, which is a property of Catalan’s triangle, we immediately recoverEq. (2.118).2.8 2-lead junctions and Y-junctionsIn this section we evaluate the conductance at several established fixed points of 2-lead junctionsand Y-junctions attached to FL leads and TLL leads. The analysis is carried out at the first order ininteraction [Eqs. (2.102) and (2.104)] and then in the RPA [Eqs. (2.105) and (2.112)]. In particular,we will examine the conductance of the maximally open M fixed point in the RPA for the Z3 sym-metric Y-junction. A more detailed discussion on RG fixed point S-matrices and their stability canbe found in Appendix A.For simplicity, the interactions are once more modeled by Eq. (2.67). We write α j (0), theinteraction strength in wire j, simply as α j; also when the junction is connected to TLL leads, weassume the interactions in wires and leads are uniform and identical, i.e. α j (∞) = α j. Of course,by definition α j (∞) = 0 for FL leads.2-lead junctionIn a 2-lead junction of spinless fermions away from resonance, solving the S-matrix RG equations[Eq. (2.72) at the first order and Eq. (2.106) in the RPA], we find that the only fixed points are thecomplete reflection fixed point (the N fixed point) and the perfect transmission fixed point (the Dfixed point)[11, 77, 132].At the N fixed point W12 = 0, the two wires are decoupled from each other, and we find theobvious result that the conductance GN,FLj j′ = GN,TLLj j′ = 0, irrespective of what leads the junction isattached to.On the other hand, at the D fixed point W12 = 1, the backscattering between the two wiresvanishes. With FL leads GD,FLj j′ =(e2/2pi)(1−2δ j j′), as predicted by the naive Landauer formula;with TLL leads, Eq. (2.104) predictsGD,TLLj j′ =(1− α1+α22)e22pi(1−2δ j j′)(2.128)at the first order, and Eq. (2.112) predicts49GD,TLLj j′ =2K1K2K1+K2e22pi(1−2δ j j′)(2.129)in the RPA. Here the RPA has recovered the famous result for the conductance of two semi-infiniteTLL wires, which takes the form of Eq. (2.29).Y-junctionEven at the first order in interaction, the RG flow portrait for a Y-junction is more complicated thanthe two-lead junction[70]. Solving Eq. (2.72), we find a “non-geometrical” M fixed point whoseexistence and transmission probabilities generally depend on the interaction strengths, in additionto the “geometrical” fixed points N, A j and χ±. Provided the interactions are not too strong, theseare also the only fixed points allowed in the RPA[15]. N (complete reflection) and A j (asymmetric)can be obtained by adding a third decoupled wire with label j to the N and D fixed points of thetwo-lead junction respectively. The conductances at N and A j are therefore a trivial generalizationof the two-lead case, and we will focus on χ± and M alone.At the chiral fixed points χ±, in the absence of interaction, an electron incident from wire j isperfectly transmitted to wire j±1 (here we identify j+3≡ j); thus the time-reversal symmetry isbroken. The W matrix is given by Wj j′ =(1−δ j j∓ ε j j′)/2, where the anti-symmetric tensor ε j j′ isdefined by ε12 = ε23 = ε31 = 1, ε21 = ε32 = ε13 =−1 and ε j j = 0. At the first order, inserting theWmatrix into Eqs. (2.102) and (2.104), we find Gχ±,FLj j′ =−(e2/2pi)(3δ j j′−1± ε j j′)/2, andGχ±,TLLj j′ −Gχ±,FLj j′ =e22pi12[(α j+α j′)(32−δ j j′)+12(α1+α2+α3)(1−δ j j′± ε j j′)]. (2.130)In the RPA, on the other hand, Eq. (2.112) gives the conductance at χ± with TLL leads asGχ±,TLLj j′ =−2e22piK j (K1+K2+K3)δ j j′+(±K1K2K3ε j j′−K jK j′)K1+K2+K3+K1K2K3, (2.131)which again agrees with the bosonization result Eq. (2.50).The presence of the M fixed point can be inferred in a Z3 symmetric time-reversal invariant Y-junction with attractive interactions: in this system, N is unstable, A j is forbidden by Z3 symmetry,and χ± are forbidden by time-reversal symmetry, so there must be at least one stable fixed point.The W matrix has generally interaction-dependent elements at M. At the first order,Wj j′ ={ ( α1α2α3/α jα1α2+α2α3+α3α1)2, j = j′(1− α1α2α3/α jα1α2+α2α3+α3α1)(1− α1α2α3/α j′α1α2+α2α3+α3α1), j 6= j′. (2.132)We see explicitly that M obeys time-reversal symmetry, Wj j′ =Wj′ j. Demanding 0 ≤Wj j′ ≤ 1, we50find that at the first-order M can only exist in the following situations: 1) α1, α2, α3 > 0; 2) α1, α2,α3 < 0; 3) α1 > 0, α2 > 0, α3 < −α1, α3 < −α2; 4) α1 < 0, α2 < 0, α3 > −α1, α3 > −α2; andsituations equivalent to 3) and 4) up to permuted subscripts (e.g. (α1,α2,α3)→ (α3,α1,α2)).Substituting Eq. (2.132) into Eq. (2.104), we find that at the first-order the conductance at MobeysGM,TLLj j′ −GM,FLj j′ ={ e22pi(α1+α2)(α2+α3)(α1+α3)2(α1α2+α2α3+α3α1)3α2j (α1+α2+α3−α j)2 , j = j′− e22pi (α1+α2)(α2+α3)(α1+α3)2(α1α2+α2α3+α3α1)3[α jα j′ (α1α2+α2α3+α3α1)− (α1α2α3)2α jα j′], j 6= j′.(2.133)Note that for Z3 symmetric interactions (α j = α), Wj j′ = 1/9+δ j j′/3 becomes independent ofthe interaction strength. Now Wj j′ produces the maximal transmission probability 8/9 allowed byunitarity in a Z3 symmetric S-matrix, and at the first order GM,TLLj j′ −GM,FLj j′ =−(8/27)α(e2/2pi)(1−3δ j j′).Compared to FL leads, TLL leads enhance conductance for attractive interactions and reduce con-ductance for repulsive interactions, as with the two-lead D fixed point.In the RPA, the W matrix of the M fixed point is generally cumbersome, but reduces to theaforementioned maximally transmitting W matrix for Z3 symmetric interactions. Eq. (2.112) thengives[111]GM,TLLj j′ =4K3K+6e22pi(1−3δ j j′). (2.134)This result supports the findings of Ref. [104]. There the M fixed point conductance of a Y-junctionof infinite TLL wires is computed numerically using density matrix renormalization group, andconjectured to beG j j′ =2Kγ2K+3γ−3Kγe22pi(1−3δ j j′), (2.135)where it is suggested that the dimensionless parameter γ is 4/9 based on the non-interacting limitK = 1.2.9 Conclusion and discussionsWe would like to discuss some of the questions left open in our approach.First, we have assumed that scattering by the junction is fully described by operators which arequadratic in conduction fermions and independent of other degrees of freedom. Local operatorsquartic in fermions are ignored, among others. This does not pose a threat to the first-order calcula-tions, because any quartic local operator has a scaling dimension of at least 4×1/2 = 2 in the non-interacting case, and is necessarily highly irrelevant. However, it has been shown that sufficiently51strong attractive bulk interactions can render quartic boundary operators relevant[93]. An exampleis the electron pair hopping operator at the Z3 symmetric Y-junction, ψ†1Lψ†1Rψ3∂xψ3 (x= 0)+h.c.:it is of dimension 3/K at the asymmetric fixed point A3, where K is the Luttinger parameter of allthree wires, and A3 sees wire 3 decoupled from perfectly connected wires 1 and 2. Apparently,for very strong interactions K > 3, this operator becomes relevant and can potentially dominate thephysical properties of the stable fixed point. Unfortunately, the present RPA analysis does not pre-dict a scaling exponent consistent with this operator[15]; it is hence incomplete in this regard, andshould not be carried too far into the regime of strongly attractive bulk interactions.A related issue is the existence of the D fixed points in the Y-junction. Predicted by the bosonicapproaches[51, 86, 93] but not the fermionic ones[13, 15], these fixed points are only stable forstrong attractive interactions. They are most notably characterized by Andreev reflections, evenwhen electron-electron interaction is absent in the bulk. This hints at multi-particle scatteringat the junction, and rules out the possibility to represent the D fixed points by single-particle S-matrices. (Single-particle S-matrices with particle-hole channels are not feasible either since theD fixed points respect particle number conservation[93].) The D fixed points are not predicted bypurely fermionic approaches, because the latter are based on the ansatz that the junction is alwaysdescribed by a single-particle S-matrix along the RG flow; but such an ansatz will likely be invali-dated if, for instance, relevant quartic boundary operators are present. We are thus led to believe thatthe lack of D fixed points in the present RPA analysis does not refute their possible stability whenthe bulk interactions are strongly attractive. Indeed, the refermionization method adopted by Ref.[45] may be successfully used to describe the crossover from the “pair tunneling” D fixed point tothe χ± fixed points in the vicinity of Luttinger parameter K = 3, with an S-matrix of free fermionswhich are not the original electrons.On the other hand, even when the bulk interactions are relatively weak, it is not a priori clear towhat extent the RPA is successful. In the Tomonaga-Luttinger model (which we have adopted in ourbulk quantum wires), the RPA is known to be exact due to the interaction which separately conservesthe numbers of right- and left-movers[34]. This is no longer the case once right- and left-moversbecome mixed up by the scattering at the junction. It has been pointed out that going beyond the RPAchanges the renormalization of the S-matrix away from the “geometrical” fixed points, although alluniversal scaling exponents stay the same[11–13]. As for the “non-geometrical” M fixed point in theY-junction, its position is generally shifted when we go beyond the RPA. Remarkably, however, ifthe interaction is Z3 symmetric, not only the W matrix but also the scaling exponents at the M fixedpoint remain identical with the RPA results up to the third order in interaction[13]. The agreementof our RPA result with the numerics of Ref. [104] is suggestive, but more work on vertex correctionsis required to verify the validity of our RPA conductance at the Z3 symmetric M fixed point with TLLleads, Eq. (2.134).As a conclusion to this chapter, using the fermionic RG formalism, we calculated the linear dc52conductance tensor of a junction of multiple quantum wires. We showed, both at the first order andin the RPA, that a junction attached to FL leads has a conductance tensor which obeys a linearizedLandauer-type formula with a renormalized S-matrix. TLL leads modify the conductance throughvertex corrections, and the conductance with FL leads can be heuristically related to the conductancewith TLL leads through the contact resistance between leads.53Chapter 3Conductance of long ABK ringsThis chapter tackles the problem of long ABK rings[112]. We begin by reviewing the spin-1/2single-channel Kondo effect in Section 3.1, focusing on the size of the Kondo screening cloud andthe pi/2 phase shift from scattering by the Kondo singlet. To treat ABK rings, we introduce a gen-eralized Anderson model with an interacting QD in Section 3.2; the screening channel is separatedfrom the non-screening ones, and an effective Kondo model in the local moment regime is obtained.In Section 3.3 the linear dc conductance is calculated using Kubo formula. Disconnected and con-nected contributions are examined separately, and the approximate elimination of the connectedcontribution is discussed. Perturbation theories in the weak-coupling and FL regimes are employedin Section 3.4; weak-coupling results applicable at high temperatures formally resemble the shortring case. In Section 3.5, we make contact with earlier results by applying our formalism to a shortABK ring as well as a QD connected to two finite non-interacting quantum wires. Section 3.6 thenapplies the formalism to the closed long ring, and Section 3.7 studies open long rings and theirpotential utilization as two-path interferometers. Conclusions and open questions are presented inSection 3.8. Appendix B consists of details related to the calculation of disconnected contributions.Finally, Appendix C include technical details of the two perturbation theories, explicitly calculatingthe screening channel T-matrix and the connected contribution to the conductance.3.1 The spin-1/2 single-channel Kondo effectLet us go over properties of the spin-1/2 single-channel Kondo model[1]. For concreteness, weconsider an impurity spin in a three-dimensional electron gas. If the system is spherically sym-metric, only the s-wave harmonic is coupled to the impurity spin. Hence the low-energy effectiveHamiltonian is one-dimensional:H =∫ ∞0dx(ivF)[ψ†L (x)∂xψL (x)−ψ†R (x)∂xψR (x)]+ Jψ†L~σ2ψL (0) ·~S, (3.1)54where vF is again the Fermi velocity of the conduction band, with the dispersion εk = vFk (wemeasure energies relative to the Fermi energy), ~S is the impurity spin, and J is the Kondo couplingconstant. The right- and left-movers ψR/L are respectively outgoing and incoming s-wave compo-nents, and obey the anticommutation relation{ψL/R (x) ,ψ†L,R(x′)}= δ(x− x′) . (3.2)Here we only let ψL (0) couple to ~S, because in absence of J, ψR and ψL satisfy the boundarycondition at x= 0:ψR (0) = ψL (0) ; (3.3)thus coupling ψR (0) to ~S merely amounts to redefining J. We can again unfold the system so thatfor x> 0, ψR (x) = ψL (−x); the Hamiltonian then becomesH =∫ ∞−∞dx(ivF)ψ†L (x)∂xψL (x)+ Jψ†L~σ2ψL (0) ·~S. (3.4)The perturbation theory in J is infrared divergent; the energy scale associated with this diver-gence is the Kondo temperature TK . To find TK we can, for instance, calculate the β -function of Jwhich describes how J renormalizes as the high-energy cutoff decreases. This is done by restrictingthe energy of ψL to (−D,D), where the reduced bandwidth D is much smaller than the originalbandwidth, and subsequently integrating out in the action the “fast” degrees of freedom with en-ergies between (−D,−D+δD) and (D−δD,D), where δD D. The high-energy cutoff in theresultant theory is thus reduced from D to D−δD.The O(J2)fast-mode correction to the Euclidean action isJ22∫dτdτ ′〈Tτψ†L (τ)σa2ψL (τ)ψ†L(τ ′) σb2ψL(τ ′)〉f〈TτSa (τ)Sb(τ ′)〉(3.5)where both expectation values are calculated in the interaction picture, 〈〉 f indicates that only fastmodes are integrated over (or, equivalently, contracted), all ψL operators are at x= 0, and sums overrepeated indices are implied. The impurity spin operator has no dynamics of its own in the absenceof J; thus〈TτSa (τ)Sb(τ ′)〉= H(τ− τ ′)SaSb+H (τ ′− τ)SbSa = 14δab+12iεabcSc sgn(τ− τ ′) , (3.6)where H (τ) is again the Heaviside function, and εabc is the Levi-Civita symbol. It is the signfunction term that makes the spin-flip Hamiltonian very different from normal potential scattering.Meanwhile, each ψL can be decomposed into a “fast” component ψL f and a “slow” one ψLs, so55the correlation function of ψL has 24 = 16 terms. Of these, the only two terms contributing to therenormalization of J are〈Tτψ†L f (τ)σaψLs (τ)ψ†Ls (τ′)σbψL f (τ ′)〉fand〈Tτψ†Ls (τ)σaψL f (τ)ψ†L f (τ′)σbψLs (τ ′)〉f.The slow modes are not affected by the contraction, while the fast modes are contracted as follows:−〈TτψL f (τ)ψ†L f(τ ′)〉f=−(∫ −D+δD−D+∫ DD−δD)dε2pivFe−ε(τ−τ′) [H (τ− τ ′)− f (ε)]=− δD2pivFe−D|τ−τ′| sgn(τ− τ ′) . (3.7)In the second line we are considering the zero temperature case (we expect the only effect of finitetemperatures to be cutting off the RG flow). The sign function here cancels the sign function fromtime-ordering the impurity spin. Noting that σaσb = δab+ iεabcσc, and that εabcεabd = 2δcd , we findthe overall O(J2)contribution to the action to be− J22∫dτdτ ′ψ†L (τ)~σ2ψL(τ ′) ·~S δD2pivFe−D|τ−τ′|≈−J2 12pivFδDD∫dτψ†L (τ)~σ2ψL (τ) ·~S; (3.8)in the second line we have neglected the slight retardation when integrating over τ ′, since only|τ ′− τ|. 1/D contributes significantly to the integral.Therefore, the renormalization of J is given by δJ = (δ lnD)J2/(2pivF), and the RG equationfor the dimensionless coupling constant λ ≡ J/(2pivF) is− dλd lnD= λ 2. (3.9)With the boundary condition λ (D= D0) = λ0, this equation has the solutionλ =1ln DTK,where the Kondo temperature is defined asTK ≡ D0 exp(− 1λ0). (3.10)For ferromagnetic coupling λ0 < 0, λ simply flows to zero as the cutoff is reduced. However, forantiferromagnetic coupling λ > 0, TK is the energy scale at which λ becomes O(1) and perturbationtheory breaks down.To obtain a qualitative picture of what happens in the antiferromagnetic case well below TK ,where apparently the Kondo coupling J flows to +∞, it is useful to consider a lattice model. The56impurity spin is coupled to the end of a semi-infinite tight-binding chain of hopping strength t, andthe Hamiltonian takes the formH =−∞∑n=0t(c†ncn+1+h.c.)+ Jc†0~σ2c0 ·~S. (3.11)In the naive strong coupling limit J t, due to the second term, the impurity spin should be“screened”, and form a singlet with one electron on site 0. Other electrons are essentially freeexcept they are now forbidden to visit site 0, as doing so requires an energy penalty ∼ J. Hence ourstrong coupling boundary condition is that the overall wave function should vanish at site 0; or, interms of right- and left-movers,ψR (0)+ψL (0) = 0. (3.12)Comparing Eq. (3.12) with Eq. (3.3), we see that the strong coupling boundary condition corre-sponds to an additional phase factor of−1= exp(2ipi/2), i.e. a pi/2 phase shift due to the scatteringby the Kondo singlet (together with the removal of the impurity spin from the effective Hamilto-nian).A question not addressed by the lattice model is the true length scale of the wave function of thescreening electron LK (or size of the “Kondo screening cloud”) in the strong coupling limit[2, 85].This length scale is one lattice spacing in the J t limit of the lattice model, but there is no latticein our original effective model Eq. (3.1); the lattice of the underlying material has already droppedout upon taking the continuum limit. Since the screening cloud forms in the crossover from weakcoupling to strong coupling, it is reasonable to expect that LK is related to the crossover energy scaleTK , LK = vF/TK .3.2 Anderson model and Kondo modelStarting in this section, we study a generalized tight-binding Anderson model[112], in order to makecontact with the long ABK ring geometries which will be our main interest. This model describesN FL leads meeting at a junction containing a QD with an on-site Coulomb repulsion. In addition tothe QD, the junction comprises an arbitrary configuration of non-interacting tight-binding sites. Thefull Hamiltonian contains a non-interacting part, a QD part, and a coupling term between the two:H = H0+HT +Hd . (3.13a)The non-interacting part is made up of two terms,H0 = H0,leads+H0,junction; (3.13b)57the lead termH0,leads =−tN∑j=1∞∑n=0∑σ(c†j,n,σc j,n+1,σ +h.c.)(3.13c)models the FL leads as semi-infinite nearest-neighbor tight-binding chains with hopping t, wherej is the lead index, n is the site index and σ = ↑,↓ is the spin index. For simplicity all leadsare assumed to be identical. H0,junction is the non-interacting part of the junction; it glues all leadstogether and often includes additional sites (e.g. representing the arms of an ABK ring), but does notinclude coupling to the interacting QD. In a typical open ABK ring with electron leakage, two of theleads serve as source and drain electrodes, while the remaining N−2 leads mimic the base contactsthorough which electrons escape the junction. In experiments usually the current flowing throughthe source or the drain is monitored, but the leakage current can also be measured.Assume that there are M sites in the junction to which the QD is directly coupled; hereafter werefer to these sites as the coupling sites. The coupling to the QD can be written asHT =−M∑r=1∑σ[trc†C,r,σdσ +h.c.](3.13d)where dσ annihilates an electron with spin σ on the QD, and c†C,r,σ creates a spin-σ electron on therth coupling site. cC,r may coincide with c j,0 (although not with c j,n for n≥ 1). In the simplest ABring, there is only one physical AB phase, which may be incorporated in either H0,junction or HT . Inmore complicated models both H0,junction and HT can depend on AB phases.Finally, the Hamiltonian of the interacting QD is given byHd =∑σεdndσ +Und↑nd↓ (3.13e)where ndσ = d†σdσ . We assume SU (2) spin symmetry throughout the chapter.A generic system with N = 5 and M = 3 is sketched in Fig. 3.1, with details of the mesoscopicjunction hidden. We will analyze more concrete realizations of this model, including ABK rings andQD attached to quantum wires[66, 75, 116], in Sections 3.5, 3.6 and 3.7.3.2.1 Screening and non-screening channelsWhile it is H0,junction that ultimately determines the properties of the junction, its details are actuallynot important in our formalism. Instead, in the following we characterize the model by its back-ground scattering S-matrix and coupling site wave functions. Both quantities are easily obtainedfrom a given H0,junction, and as we show in Section 3.3, they play a central role in our quest for thelinear dc conductance.To recast our model into the standard form of an interacting QD coupled to a continuum of states,58Figure 3.1: Sketch of a generic system which allows the application of our formalism. HereN = 5 and M = 3.it is convenient to first diagonalize the non-interacting part of the Hamiltonian H0 by introducingthe scattering basis q j,k,σ :H0 =∫ pi0dk2piN∑j=1∑σεkq†j,k,σq j,k,σ , (3.14)where εk = −2t cosk is the dispersion relation in the leads, and for simplicity we let the latticeconstant a = 1. In addition to the scattering states q j,k,σ , there may exist a number of bound stateswith their energies outside of the continuum, but since their wave functions decay exponentiallyaway from the junction region, they do not affect linear dc transport properties.The scattering basis operator q j,k,σ annihilates a scattering state electron incident from lead jwith momentum k and spin σ , and obeys the anti-commutation relation{q j,k,σ ,q†j′,k′,σ ′}= 2piδ j j′δσσ ′δ (k− k′).The corresponding wave function has the following form on site n in lead j′,χ j,k(j′,n)= δ j j′e−ikn+S j′ j (k)eikn; (3.15a)and on coupling site r,χ j,k (r) = Γr j (k) . (3.15b)In other words, for an electron incident from lead j, S j′ j is the background reflection or transmissionamplitude in lead j′, and Γr j is the wave function on coupling site r. The scattering S-matrix S isunitary: S†S= 1.59From its wave function, q j,k,σ can be related to c j,n,σ and cC,r,σ :c j,n,σ =∫ pi0dk2piN∑j′=1[δ j j′e−ikn+S j j′ (k)eikn]q j′,k,σ , (3.16a)andcC,r,σ =∫ pi0dk2piN∑j′=1Γr j′ (k)q j′,k,σ . (3.16b)We now express HT in the scattering basis. Inserting Eq. (3.16b) into Eq. (3.13d), we find the QD isonly coupled to one channel in the continuum, i.e. the screening channel:HT =−∑σ∫ pi0dk2piVk(ψ†k,σdσ +h.c.), (3.17)where the screening channel operator ψ†k,σ is defined byψ†k,σ =1VkN∑j=1M∑r=1trΓ∗r, j (k)q†j,k,σ , (3.18)and the normalization factor Vk > 0 is defined byV 2k =N∑j=1M∑r,r′=1trt∗r′Γ∗r j (k)Γr′ j (k) = tr[Γ† (k)λΓ(k)]. (3.19)This ensures{ψk,σ ,ψ†k′,σ ′}= 2piδσσ ′δ (k− k′). Here we also introduce the M×M Hermitian QDcoupling matrix λ ,λrr′ = trt∗r′ . (3.20)It will be useful to define a series of non-screening channels φl,k,σ orthogonal to ψ , where l = 1,· · · , N− 1. The φ channels are decoupled from the QD. In a compact notation we can write thetransformation from the scattering basis to the screening–non-screening basis asΨk,σ ≡ψk,σφ1,k,σ· · ·φN−1,k,σ= Ukq1,k,σq2,k,σ· · ·qN,k,σ , (3.21)where U is a unitary matrix. The first row of U is known:U1, j,k =1VkM∑r=1t∗r Γr j (k) . (3.22)60As long as U stays unitary, its remaining matrix elements can be chosen freely without affectingphysical observables. Ψ now also diagonalizes H0,H0 =∑σ∫ pi0dk2piεk(ψ†k,σψk,σ +N−1∑l=1φ †l,k,σφl,k,σ); (3.23)we shall also need the inverse transformation,q j,k,σ = U∗1, j,kψk,σ +N−1∑l=1U∗l+1, j,kφl,k,σ . (3.24)3.2.2 Kondo modelIn the local moment regime of the Anderson model[49, 67], for TU we can perform the Schrieffer-Wolff transformation[107] on ψ to obtain an effective Kondo model with a reduced bandwidth anda momentum-dependent coupling:H = H0+∫ kF+Λ0kF−Λ0dkdk′(2pi)2(Jkk′ψ†k~σ2ψk′ ·~Sd+Kkk′ψ†kψk′), (3.25a)where the spin on the QD is given by ~Sd = d† (~σ/2)d. Λ0 kF is the initial momentum cutoff, andthe dispersion is linearized near the Fermi wave vector kF as εk = vF (k− kF); for the tight-bindingmodel vF = 2t sinkF .The interaction consists of a spin-flip term J,Jkk′ =VkVk′ ( jk+ jk′) , (3.25b)jk =1εk− εd +1U+ εd− εk ≈ j, (3.25c)and a particle-hole symmetry breaking potential scattering term K,Kkk′ =14VkVk′ (κk+κk′) , (3.25d)κk =1εk− εd −1U+ εd− εk ≈ κ . (3.25e)The energy cutoff is initially D0 ≡ vFΛ0. When we reduce the running energy cutoff from D toD+ dD (0 < −dD D) to integrate out the high-energy degrees of freedom in the narrow stripsof energy (−D,−D−dD) and (D+dD,D), K is exactly marginal in the RG sense, whereas J ismarginally relevant and obeys the following RG equation:61− d (νJkk′)d lnD=12(νJk,kF+ΛνJkF+Λ,k′+νJk,kF−ΛνJkF−Λ,k′), (3.26)or equivalently− d jd lnD= ν j2(V 2kF+Λ+V2kF−Λ), (3.27)where ν is the density of states per channel per spin, ν = 1/(2pivF). Therefore, renormalization ofthe Kondo coupling is controlled by the momentum-dependent normalization factor V 2k , defined inEq. (3.19). j is the only truly independently renormalized coupling constant despite the appearanceof Eq. (3.26); this follows from the fact that the screening channel is the only channel coupled tothe QD[116].The prototype Kondo model possesses a momentum-independent coupling function, Jkk′ ≈2 jV 2kF . As a result, spin-charge separation occurs and the Kondo interaction is found to be ex-clusively in the spin sector[3]. The charge sector is nothing but a non-interacting theory with aparticle-hole symmetry breaking phase shift due to the potential scattering term K, while at verylow energy scales the spin sector renormalizes to a local FL theory with pi/2 phase shift.On the other hand, in a mesoscopic geometry V 2k often exhibit fluctuations on a mesoscopicenergy scale EV . (More precisely, EV can be defined as the energy corresponding to the largestFourier component in the spectrum of V 2k , but for both specific models discussed in this chapter wecan simply read it off the analytic expression.) In the presence of a characteristic length scale L,EV may be of the order of the Thouless energy vF/L, as is the case for the closed long ABK ringin Section 3.6; however this is not always true, with a counterexample provided by the open longABK ring in Section 3.7 where EV is of the order of the bandwidth 4t. Well above EV , V 2k appearsfeatureless and can be approximated by its mean valueV 2k with respect to k. The Kondo temperatureTK can be loosely defined as the energy cutoff at which the dimensionless coupling 2ν jV 2k becomesO(1). As briefly sketched in Section 1.3, there are two very different parameter regimes of theKondo temperature[112, 116, 131]:a) The small Kondo cloud regime TK EV . For EV ∼ vF/L, the size of the Kondo screeningcloud LK ≡ vF/TK L; hence the name. In this regime, the bare Kondo coupling is sufficientlylarge, so that 2ν jV 2k renormalizes to O(1) before it “senses” any mesoscopic fluctuation. By ap-proximating V 2k ≈V 2k , Eq. (3.27) has a solutionj (D)≈ j01+2ν j0V 2k lnDD0, (3.28)where j0 is the bare Kondo coupling constant at the initial energy cutoff D0. Eq. (3.28) gives the“background” Kondo temperature62TK ≈ T 0K ≡ D0 exp(− 12ν j0V 2k), (3.29)independent of the mesoscopic details of the geometry. For T TK , the low-energy effective theoryis also conjectured to be a FL, but the T-matrix (or the phase shift) of the screening channel is notyet known with certainty[71, 116].b) The very large Kondo cloud regime TK EV . For EV ∼ vF/L, LK L. In this regime,the bare Kondo coupling is very small, and j does not begin to renormalize significantly until theenergy cutoff is well below EV . The variation of V 2k is hence negligible in the resulting low energytheory, V 2k ≈ V 2kF , but V 2kF may be significantly different from V 2k , which means Kondo temperatureis thus highly sensitive to the mesoscopic details of V 2kF . Because V2k is almost independent of k, wemay map the low-energy theory in question onto the conventional Kondo model, where conductionelectrons are scattered by a point-like spin in real space. (We stress that this mapping would notbe possible for a strongly k-dependent V 2k , which is the case for the small cloud regime.) Follow-ing well-known results in the conventional Kondo model,[49] we see that the low-energy effectivetheory is a local FL theory, with parameters also sensitive to mesoscopic details.3.3 dc conductanceIn this section we calculate the dc conductance tensor of the system in linear response theory, gen-eralizing the results in Ref. [66] to our multi-terminal setup. The result is presented as the sum of adisconnected contribution and a connected one (Fig. 3.2). By “disconnected” and “connected”, weare referring to the topology of the corresponding Feynman diagrams: a disconnected contributionoriginates from a Feynman diagram without any cross-links, and can always be written as the prod-uct of two two-point functions. The disconnected contribution has a simple Landauer form, andis quadratic in the T-matrix of the screening channel ψ . The connected contribution is also shownto depend on properties near the Fermi surface only, but it is usually difficult to evaluate analyti-cally except at high temperatures, or at low temperatures if the FL perturbation theory is applicable.Nevertheless, just as with the short ABK ring, the connected contribution can be approximatelyeliminated at temperatures low compared to another mesoscopic energy scale T Econn.3.3.1 Kubo formula in terms of screening and non-screening channelsThe linear dc conductance tensor G j j′ is defined through〈I j〉= ∑ j′G j j′Vj′ , where I j is the currentoperator in lead j, andVj′ is the applied bias voltage on lead j′. I j is given by I j =−edN j/dt¯, wheret¯ is the real time variable, andN j ≡∑σ∞∑n=0c†j,n,σc j,n,σ (3.30)63Figure 3.2: Disconnected (self-energy) and connected (vertex correction) contributions to thedensity-density correlation function, which is directly related to the conductance throughthe Kubo formula Eq. (3.31). The dashed lines represent external legs at times t¯ and0, the solid lines represent fully dressed Ψ fermion propagators, and the hatched circlerepresents all connected 4-point vertices of the screening channel.is the density operator in lead j. G j j′ is then given by the Kubo formulaG j j′ =e2hlimΩ→0(−2piiΩ)G′j j′ (Ω) , (3.31)where the retarded density-density correlation function isG′j j′ (Ω)≡−i∫ ∞0dt¯eiΩ+t¯ 〈[N j (t¯) ,N j′ (0)]〉 (3.32)and Ω+ ≡Ω+ i0+. The retarded correlation function can be obtained by means of analytic contin-uation iωp→Ω+ from its imaginary time counterpart,G ′j j′ (iωp) =−∫ β0dτeiωpτ〈TτN j (τ)N j′ (0)〉, (3.33)where ωp = 2ppi/β is a bosonic Matsubara frequency, β = 1/T is the inverse temperature, and Tτis the imaginary time-ordering operator.To calculate the correlation function we need the density operator N j written as bilinears of Ψ.This is achieved by the insertion of Eq. (3.16a) and then Eq. (3.24) into Eq. (3.30). We findN j =∑σ∫ pi0dk1dk2(2pi)2Ψ†k1,σMjk1k2Ψk2,σ , (3.34)where for l1, l2 = 1, ..., N,(M jk1k2)l1l2= ∑j1 j2Ul1, j1,k1U∗l2, j2,k2[δ j j1δ j j211− ei(k1−k2+i0) +S∗j j1 (k1)δ j j211− e−i(k1+k2−i0)+δ j j1S j j2 (k2)11− ei(k1+k2+i0) +S∗j j1 (k1)S j j2 (k2)11− ei(k2−k1+i0)]. (3.35)64The matrix M obeys M jk1k2 =(M jk2k1)†which ensures N j is Hermitian.3.3.2 Disconnected partWe substitute Eq. (3.34) into Eq. (3.32). The disconnected part of the conductance is obtained bypairing up Ψ and Ψ† operators to form two two-point Green’s functions:G′Dj j′ (Ω)=−2i∫ ∞0dt¯eiΩ+t¯∫ pi0dk1dk2(2pi)2dq1dq2(2pi)2tr[M jk1k2G>k2q1 (t¯)Mj′q1q2G<q2k1 (−t¯)−M j′q1q2G>q2k1 (−t¯)Mjk1k2G<k2q1 (t¯)].(3.36)Here the factor of 2 in the second line is due to the spin degeneracy. The greater and lesser Green’sfunctions in the screening–non-screening basis are defined asG>kq (t¯)≡−i〈ψk (t¯)ψ†q (0)〉 〈φ1,k (t¯)φ †1,q (0)〉· · · 〈φN−1,k (t¯)φ †N−1,q (0)〉 , (3.37a)andG<kq (t¯)≡+i〈ψ†q (0)ψk (t¯)〉 〈φ †1,q (0)φ1,k (t¯)〉· · · 〈φ †N−1,q (0)φN−1,k (t¯)〉 . (3.37b)(The correlation functions are for either spin.) In equilibrium, fluctuation-dissipation theorem re-quires thatG>kq (ω) = 2i [1− f (ω)] ImGRkq (ω), andG<kq (ω) =−2i f (ω) ImGRkq (ω), where f (ω) =1/(eβω +1)is the Fermi function. These equilibrium relations result from the fact that GRkq (ω) =GRqk (ω) for the Anderson model[66] [see Eq. (3.41) below]. With these relations Eq. (3.36) becomesG′Dj j′ (Ω)= 8∫ dωdω ′(2pi)2f (ω)− f (ω ′)ω−ω ′+Ω+∫ pi0dk1dk2(2pi)2dq1dq2(2pi)2tr[M jk1k2 ImGRk2q1(ω ′)M j′q1q2 ImGRq2k1 (ω)].(3.38)We note that, in contrast to the case of Ref. [66], the momentum integral here is not necessarily real.Instead, its complex conjugate takes the same form but with ω and ω ′ interchanged:65∫ pi0dk1dk2(2pi)2dq1dq2(2pi)2tr[M jk1k2 ImGRk2q1(ω ′)M j′q1q2 ImGRq2k1 (ω)]∗=∫ pi0dk1dk2(2pi)2dq1dq2(2pi)2tr[M jk1k2 ImGRk2q1 (ω)Mj′q1q2 ImGRq2k1(ω ′)]. (3.39)Making use of this property, we can show that[G′Dj j′ (−Ω)]∗= G′Dj j′ (Ω). Thus the disconnectedcontribution to the dc conductance can be written asGDj j′ =e2hlimΩ→0(2piΩ) ImG′Dj j′ (Ω) . (3.40)We should realize, however, that taking the imaginary part of G′Dj j′ is generally not equivalent totaking the δ -function part of 1/(ω−ω ′+Ω+) in Eq. (3.38).For the Anderson model, since the interaction is restricted to the d electrons, it is not difficult tofind the Dyson’s equation for the retarded Green’s function of ψ and φ by the equation-of-motiontechnique:GRk2q1 (ω) = 2piδ (k2−q1)gRk2 (ω)+ τψgRk2 (ω)Tk2q1 (ω)gRq1 (ω) , (3.41)where the free retarded Green’s function for ψ and φ isgRk (ω) =1ω+− εk , (3.42)and τψ is the projection operator onto the screening channel subspace. Again, only the Green’sfunction of the screening channel is modified by coupling to the QD. The retarded T-matrix of thescreening channel in the single-particle sector is related to the retarded two-point function of the QDbyTk2q1 (ω) =Vk2GRdd (ω)Vq1 , (3.43)where GRdd (ω)≡−i∫ ∞0 dt¯eiω+t¯〈{dσ (t¯) ,d†σ (0)}〉is again the same for σ = ↑,↓.From Eqs. (3.38) and (3.41) we may express the disconnected contribution to the linear dcconductance in the Landauer form[112]:GDj j′ =−2e2h∫ 2t−2tdεp[− f ′ (εp)]T Dj j′ (εp) , (3.44)where the disconnected “transmission probability” T Dj j′ is written in terms of the absolute square ofa “transmission amplitude”,66T Dj j′ (εp) = δ j j′−∣∣∣{S (p)[1−2ipiνpTpp (εp)U†pτψUp]} j j′∣∣∣2= δ j j′−∣∣∣∣S j j′ (p)+ 2iV 2p [S (p)Γ† (p)λΓ(p)] j j′ [−piνpTpp (εp)]∣∣∣∣2 . (3.45)Again λ is the QD coupling matrix defined in Eq. (3.20) and νp is the density of states per channelper spin for the tight-binding modelνp =14pit sin p. (3.46)The detailed derivation of Eq. (3.45) by contour methods is left for Appendix B.At zero temperature, when the single-particle sector of the screening channel T-matrix is a purephase shift, there is no connected contribution and Eq. (3.44) yields the full linear dc conductance[66].In this case a clear picture emerges from Eq. (3.45): the conductance is given by the Landauer for-mula with an effective single-particle S-matrix, which is obtained from the original S-matrix simplyby imposing a phase shift on the screening channel, corresponding to the particle-hole symmetrybreaking potential scattering and the scattering by the Kondo singlet[75, 88].Another useful representation of the disconnected probability, similar to that in Ref. [66], isobtained by expanding Eq. (3.45):T Dj j′ (εp) =T0, j j′ (εp)+ZR, j j′ (εp)Re [−piνpTpp (εp)]+ZI, j j′ (εp) Im [−piνpTpp (εp)]+Z2, j j′ (εp) |−piνpTpp (εp)|2 , (3.47a)with a background transmission termT0, j j′ (εp) = δ j j′−∣∣S j j′ (p)∣∣2 , (3.47b)a term linear in the real part of the T-matrix, proportional toZR, j j′ (εp) =4V 2pIm{[S (p)Γ† (p)λΓ(p)]j j′ S∗j j′ (p)}, (3.47c)a term linear in the imaginary part, proportional toZI, j j′ (εp) =4V 2pRe{[S (p)Γ† (p)λΓ(p)]j j′ S∗j j′ (p)}, (3.47d)and a term quadratic in the T-matrix, proportional to67Z2, j j′ (εp) =− 4V 4p∣∣∣[S (p)Γ† (p)λΓ(p)] j j′∣∣∣2 . (3.47e)In the dc limit, the total current flowing out of the junction is zero, and a uniform voltageapplied to all leads does not result in any current; hence the linear dc conductance satisfies currentand voltage Kirchhoff’s laws ∑ jG j j′ = ∑ j′G j j′ = 0. As a comparison it is interesting to considerthe sum of the disconnected transmission probability, Eq. (3.47a), over j or j′. Using the unitarityof S and Eq. (3.20) it is not difficult to find that∑jT Dj j′ (εp) ={Im [−piνpTpp (εp)]−|−piνpTpp (εp)|2} 4V 2p[Γ† (p)λΓ(p)]j′ j′ , (3.48)and∑j′T Dj j′ (εp) ={Im [−piνpTpp (εp)]−|−piνpTpp (εp)|2} 4V 2p[S (p)Γ† (p)λΓ(p)S† (p)]j j . (3.49)As mentioned in Ref. [66], the quantity in curly brackets in Eqs. (3.48) and (3.49) measures thedeviation of the single-particle sector of the T-matrix from the optical theorem[134]. In the case ofa non-interacting QD or the T = 0 FL theory of the Kondo limit, where the connected contributionto the conductance vanishes, these row/column sum formulas conform to our expectations: the T-matrix obeys the optical theorem, leading to ∑ jT Dj j′ =∑ j′TDj j′ = 0, so that ∑ jG j j′ =∑ j′G j j′ = 0 isensured.3.3.3 Connected part and its low-temperature eliminationIn this subsection we show that the connected contribution to the conductance is again a Fermi sur-face contribution, and discuss how it can be approximately eliminated at low temperatures. Follow-ing Ref. [66] we construct a transmission probability for this contribution. After a partial insertionof Eq. (3.34) into Eq. (3.33), the connected part of the density-density correlation function can bewritten asG ′Cj j′ (iωp) =∫ β0dτeiωpτPj j′ (τ,τ) , (3.50)where the connected four-point function Pj j′ with two temporal arguments isPj j′ (τ1,τ2)≡−∫ pi0dk1dk2(2pi)2(M jk1k2)11∑σ〈Tτψ†k1σ (τ1)ψk2σ (τ2)N j′ (0)〉C; (3.51)68the subscript C denotes connected diagrams. Note that only the screening channel contributes tothe connected part, as the non-screening channels are free fermions. Using the equation-of-motiontechnique, it is easy to relate Pj j′ to a partially amputated quantity:Pj j′ (τ,τ)=∫ pi0dk1dk2(2pi)2(M jk1k2)11Vk1Vk2∫dτ1dτ2gk1 (τ1− τ)gk2 (τ− τ2)∑σ〈Tτd†σ (τ1)dσ (τ2)N j′ (0)〉C ,(3.52)wheregk (τ)≡ [ f (τ)−H (τ)]e−εkτ (3.53)is the imaginary time free Green’s function and H (τ) is the Heaviside unit-step function. [Here“amputation” refers to the removal of the outermost V(ψ†d+h.c.)vertices from the〈ψ†ψψ†ψ〉Ccorrelation function; recall that the ψ electrons only become interacting due to their interactionwith d electrons.] With τ only appearing in free propagators, we can perform the Fourier transformexplicitly,G ′Cj j′ (iωp) =1β ∑ωmPj j′ (iωm, iωm+ iωp)=1β ∑ωm∫ pi0dk1dk2(2pi)2(M jk1k2)11gk1 (iωm)gk2 (iωm+ iωp)Vk1Vk2×∫dτ1dτ2e−iωmτ1ei(ωm+ωp)τ2∑σ〈Tτd†σ (τ1)dσ (τ2)N j′ (0)〉C . (3.54)One may now use the contour integration argument in Ref. [66].[74] The final result is that theconnected contribution to the dc conductance is expressed in terms of a transmission probabilityT C related to Pj j′ :GCj j′ =−2e2h∫ 2t−2tdω[− f ′ (ω)]T Cj j′ (ω) , (3.55)whereT Cj j′ (ω) = limΩ→0Ω28Pj j′ (ω− iη1,ω+Ω+ iη2)+ c.c. (3.56)and69Pj j′ (ω− iη1,ω+Ω+ iη2)=∫ pi0dk1dk2(2pi)2(M jk1k2)11gAk1 (ω)gRk2 (ω+Ω)Vk1Vk2∫dτ1dτ2e−iω−τ1ei(ω++Ω)τ2∑σ〈Tτd†σ (τ1)dσ (τ2)N j′ (0)〉C .(3.57)Here η1, η2→ 0+ are positive infinitesimal numbers.It is in fact possible to do the k1 and k2 integrals. Using Eqs. (3.22), (3.35) and finally (B.6), weobtain∫ pi0dk1dk2(2pi)2(M jk1k2)11gAk1 (εp)gRk2 (εp+Ω)Vk1Vk2= ∑r1r2t∗r1tr2∫ pi−pidk1dk2(2pi)2gAk1 (εp)gRk2 (εp+Ω)Γr1 j (k1)Γ∗r2 j (k2)11− ei(k1−k2+i0) . (3.58)Here domains of the momentum integrals are extended to (−pi,pi) according to Eq. (B.6), whichfacilitates the application of the residue method. As explained in Appendix B, the poles of Γ(k1) andΓ∗ (k2) are not important in the dc limit Ω→ 0. Therefore, the O(1/Ω) contribution is dominatedby the poles of the free Green’s functions, and is given by∫ pi0dk1dk2(2pi)2(M jk1k2)11gAk1 (εp)gRk2 (εp+Ω)Vk1Vk2 =2piiΩνp[S(p′)Γ†(p′)λΓ(p)S† (p)]j j+O(1) .(3.59)where εp+Ω≡ εp′ , 0≤ p, p′ ≤ pi . This leads toT Cj j′ (εp)= νp[S (p)Γ† (p)λΓ(p)S† (p)]j j[ipi4limΩ→0Ω∫dτ1dτ2e−iω−τ1ei(ω++Ω)τ2∑σ〈Tτd†σ (τ1)dσ (τ2)N j′ (0)〉C+ c.c.].(3.60)A similar manipulation can be done for the N j′ part of the correlation function.One can again consider the row and column sums of the tensorT C. Tracing over j immediatelyyields∑jT Cj j′ (εp) = νpV2p[ipi4limΩ→0Ω∫dτ1dτ2e−iω−τ1ei(ω++Ω)τ2∑σ〈Tτd†σ (τ1)dσ (τ2)N j′ (0)〉C+ c.c.];(3.61)combining the last two equations, we have70T Cj j′ (εp)−1V 2p[S (p)Γ† (p)λΓ(p)S† (p)]j j∑j′′T Cj′′ j′ (εp) = 0. (3.62)Let us now define Econn as the characteristic energy scale below which both S (p) and Γ(p) varyslowly. By definition Econn . EV ; while Econn is not necessarily the same as EV , for the two ABKring geometries considered in this chapter EV ∼Econn. For a mesoscopic structure with characteristiclength scale L, Econn is usually the Thouless energy, Econn ∼ vF/L; however, this is again not alwaysthe case, and the open long ABK ring in Section 3.7 provides a counterexample where Econn iscomparable to the bandwidth. Below Econn, the function[S (p)Γ† (p)λΓ(p)S† (p)]j j /V2p is onlyweakly dependent on p.Eq. (3.62) suggests that we can approximately eliminate the connected part of G j j′ , provided[66]the temperature is low compared to Econn. Consider the linear combinationG j j′− 1V 2kF[S (kF)Γ† (kF)λΓ(kF)S† (kF)]j j∑j′′G j′′ j′ ≡ G j j′ ; (3.63)this corresponds to measuring the conductance by measuring the current in lead j, plus a constanttimes the total current in all leads. (Note that here we include both disconnected and connectedcontributions.) By Kirchhoff’s law, this linear combination must equal G j j′ itself. We write it as asum of disconnected and connected contributions:G j j′ =(GDj j′−1V 2kF[S (kF)Γ† (kF)λΓ(kF)S† (kF)]j j∑j′′GDj′′ j′)−∫dεp[− f ′ (εp)]{T Cj j′ (εp)− 1V 2kF[S (kF)Γ† (kF)λΓ(kF)S† (kF)]j j∑j′′T Cj′′ j′ (εp)}. (3.64)For T Econn, by Eq. (3.62), the quantity in curly brackets approximately vanishes for |εp− εkF |.T , whereas the Fermi factor approximately vanishes for |εp− εkF | T . ThereforeG j j′ ≈ GDj j′−1V 2kF[S (kF)Γ† (kF)λΓ(kF)S† (kF)]j j∑j′′GDj′′ j′ ; (3.65)in other words, at T Econn it is possible to write the conductance in terms of disconnected contri-butions alone.Since Eq. (3.65) contains only the disconnected contribution, we may calculate it explicitlyusing Eqs. (3.47a) and (3.48). Since both S (p) and Γ(p) are slowly varying below the energy scaleEconn, we find the conductance is approximately linear in the T-matrix,71G j j′ ≈−2e2h∫dεp[− f ′ (εp)]{T0, j j′ (εkF )+ZR, j j′ (εkF )Re [−piνpTpp (εp)]+[ZI, j j′ (εkF )+Z2, j j′ (εkF )]Im [−piνpTpp (εp)]}, (3.66)provided T Econn. Eq. (3.66) can also be obtained by eliminating the connected part with thecolumn sum Eq. (3.49) instead of the row sum,G j j′− 1V 2kF[Γ† (kF)λΓ(kF)]j′ j′∑j′′G j j′′ , (3.67)which corresponds to measuring the conductance by applying a small uniform bias voltage in allleads, in addition to the small bias voltage in lead j′.Eq. (3.66) is the first central result of this chapter. It generalizes the result of the two-leadshort ABK ring in Ref. [66] to an arbitrary ABK ring, and expresses the linear dc conductance as alinear function of the scattering channel T-matrix, as long as the temperature is low compared to themesoscopic energy scale Econn at which S (p) and Γ(p) varies significantly[112].3.4 Perturbation theories3.4.1 Weak coupling perturbation theoryAlthough we now understand that the connected part of the conductance can be eliminated at lowtemperatures, this procedure may not be applicable in the weak-coupling regime T TK . In thissubsection we calculate the linear dc conductance perturbatively in powers of V 2k /U , again gener-alizing the short ring results of Ref. [66]; we expect the result to be valid in both small and largeKondo cloud regimes as long as T TK and the renormalized Kondo coupling constant remainsweak.Disconnected partWe first find the disconnected part; the result is already given in Ref. [66], but for completeness wereproduce it here. (Technical notes can be found in Appendix C.) As implied by Eq. (3.45), our taskamounts to calculating the retarded T-matrix of the screening channel in the single-particle sector,which is in turn achieved by calculating the two-point Green’s function −〈Tτψkσ (τ)ψ†k′σ (0)〉HinHeisenberg picture. The pertinent Feynman diagrams to O(J2)and O(K2)are depicted in Fig. 3.3,and we findνTkk′ (Ω) = νKkk′+ν2∫dεq1Ω+− εq(KkqKqk′+316JkqJqk′), (3.68)72Figure 3.3: Diagrammatics of weak-coupling perturbation theory. a) The vertices correspond-ing to the Kondo coupling and the potential scattering in Eq. (3.25a). b) Diagrams con-tributing to the T-matrix of the screening channel ψ electrons up to O(J2) ∼ O(K2).We have traced over the impurity spin so that the double dashed lines (impurity spinpropagators) form loops, and arranged the internal time variables from left to right inincreasing order. c) Connected diagrams contributing to the linear dc conductance up toO(J2).where again ν = 1/(2pivF) for the model with a reduced band. The factor of 3/16 results fromtime-ordering and tracing over the impurity spin, where we have used the following identity〈TτSad (τ1)Sbd (τ2)〉=14δ ab. (3.69)The O(K) and O(K2)terms, accounting for the particle-hole symmetry breaking potential scat-tering due to the QD, clearly obey the optical theorem Im [−piνpTpp (εp)] = |−piνpTpp (εp)|2. If κis comparable to the renormalized value of j then the O(K) term dominates the T-matrix.On the other hand, if we tune the QD to be particle-hole symmetric, εd =−U/2 and κ = 0, bothterms containing K will vanish, and the O(J2)term becomes the lowest order contribution to theT-matrix. For this term, one should also make a distinction between the real principal value part andthe imaginary δ -function part. There are two different ways in which a screening channel electronincident onto the QD can be scattered[134]: either (i) elastically, where the energy and spin of theelectron as well as the spin state of the QD are unchanged, or (ii) inelastically, where the scatteredelectron leaves behind particle-hole excitations and/or spin excitations. As noticed in Ref. [75]73and reiterated in Ref. [66], the principal value part of the O(J2)term introduces non-universalitiesdue to its dependence on all energies in the reduced band (−D,D); nevertheless it is merely anelastic potential scattering term that respects the optical theorem, and we neglect it in the following.Meanwhile, the δ -function part is an inelastic effect stemming from the Kondo physics, as can beseen from its violation of the optical theorem in the single-particle sector. (The T-matrix apparentlydisobeys the optical theorem because it is restricted to the single-particle sector, and the sum overintermediate states excludes many-particle states which appear in inelastic scattering.) Therefore,for a particle-hole symmetric QD, to O(J2)we have[66]−piνTpp (εp) = i3pi216ν2J2pp. (3.70)The weak-coupling perturbation theory is famous for being infrared divergent[67, 74] at O(J3),but as long as T TK , to logarithmic accuracy we verify in Appendix C that the O(J3)correctionsto the T-matrix can be absorbed into our O(J2)result by reinterpreting the bare Kondo couplingconstant Jpp as a renormalized one. The renormalization is governed by Eq. (3.26), and cut offat either the “electron energy” |εp| or the temperature T , whichever is larger. In other words, theKondo coupling Jpp in Eq. (3.70) should be replaced by Jpp (max{|εp| ,T}), where the argument inround brackets stands for the energy cutoff D in Eq. (3.26) where the running coupling constant isevaluated.Connected partWe now calculate the connected part to O(J2); the calculation follows Ref. [66] closely. InsertingEq. (3.34) into Eq. (3.33), we write the connected part of the density-density correlation function interms of a four-point correlation function of ψ:G ′Cj j′ (iωp) =∫ pi0dk1dk2dq1dq2(2pi)4(M jk1k2)11(M j′q1q2)11GCk1k2q1q2 (iωp) , (3.71)whereGCk1k2q1q2 (iωp)≡−∫ β0dτeiωpτ ∑σσ ′〈Tτψ†k1σ (τ)ψk2σ (τ)ψ†q1σ ′ (0)ψq2σ ′ (0)〉C. (3.72)We insert Eqs. (3.22) and (3.35) into Eq. (3.71), and take the continuum limit, which is appropriatefor the Kondo model. Because in the wide band limit the most divergent contribution to G′Cj j′ (Ω) isfrom k1 ≈ k2 and q1 ≈ q2, we can expand the integrand around these points,74G ′Cj j′ (iωp)= O(1)+∫ dk1dk2dq1dq2(2pi)41Vk1Vk2Vq1Vq2GCk1k2q1q2 (iωp) ∑r1r2r′1r′2t∗r1tr2t∗r′1tr′2×{Γr1 j (k1)Γ∗r2 j (k2)1−i(k1− k2+ i0) +[ΓS† (k1)]r1 j[ΓS† (k2)]∗r2 j1−i(k2− k1+ i0)}×{Γr′1 j′ (q1)Γ∗r′2 j′(q2)1−i(q1−q2+ i0) +[ΓS† (q1)]r′1 j′[ΓS† (q2)]∗r′2 j′1−i(q2−q1+ i0)}. (3.73)The only non-vanishing diagrams at O(J2)are shown in panel c) of Fig. 3.3:GCk1k2q1q2 (iωp) =38Jq2k1Jk2q1∫ β0dτeiωpτ∫ β0dτ1dτ2gq2 (−τ1)gk1 (τ1− τ)gk2 (τ− τ2)gq1 (τ2)=38Jq2k1Jk2q11β ∑ωn1gq2 (iωn1)gk1 (iωn1)gk2 (iωn1 + iωp)gq1 (iωn1 + iωp) ; (3.74)we have again used Eq. (3.69). The frequency summation is performed by deforming the complexplane contour and wrapping it around the lines Imz = 0 and Imz = −ωp. Analytic continuationyieldsGCk1k2q1q2 (Ω) =−3Jq2k1Jk2q116pii∫dω f (ω){[gRq2 (ω)gRk1 (ω)−gAq2 (ω)gAk1 (ω)]gRk2 (ω+Ω)gRq1 (ω+Ω)+gAq2 (ω−Ω)gAk1 (ω−Ω)[gRk2 (ω)gRq1 (ω)−gAk2 (ω)gAq1 (ω)]}. (3.75)Substituting Eqs. (3.75) and (3.25b) into Eq. (3.73), we are able to evaluate the momentumintegrals in the Λ→∞ limit by contour methods. The RRRR and AAAA terms vanish, and the AARRterms combine to produce a Fermi surface factor f ′ (ω):G′Cj j′ (Ω)= O(1)− 1Ω∫dω[− f ′ (ω)] 3(2 j)216pii ∑r1r2r′1r′2t∗r1tr2t∗r′1tr′2[ΓS†(kF +ωvF)]r1 j×[ΓS†(kF +ω+ΩvF)]∗r2 jΓr′1 j′(kF +ω+ΩvF)Γ∗r′2 j′(kF +ωvF)1v2F= O(1)+1ipiΩ∫dεp[− f ′ (εp)] 3pi2ν2J2pp16 Z2, j j′ (εp) , (3.76)75where we used Eq. (3.47e). The connected contribution to the conductance is now clearly a Fermisurface property:GCj j′ =−2e2h∫dεp[− f ′ (εp)]T C(2)j j′ (εp) , (3.77)where the connected transmission probability isT Cj j′ (εp) =Z2, j j′ (εp)3pi216ν2J2pp. (3.78)This is formally identical to the short ring result, and is of the same order of magnitude [O(J2)] asthe disconnected contribution for a particle-hole symmetric QD[66]. In fact, if leads j and j′ are notdirectly coupled to each other (i.e. they become decoupled when their couplings with the QD areturned off; the simplest example is a QD embedded between source and drain leads[102]), we haveS j j′ = 0, and the disconnected contribution for a particle-hole symmetric QD is O(J4). In this casethe O(J2)connected contribution dominates.Just as with the T-matrix, when we calculate the connected part further to O(J3)to logarithmicaccuracy (as is done in Appendix C), the result can be absorbed into Eq. (3.78) if the couplingconstant J is understood as fully renormalized according to Eq. (3.26), with its renormalization cutoff by |εp| or T .Total conductanceWe write the total conductance at T TK as a background term and a correction due to the QD:G j j′ =−2e2h∫dεp[− f ′ (εp)][T0, j j′ (εp)+δT j j′ (εp)] . (3.79)If the QD is well away from particle-hole symmetry, κ can be of the same order of magnitude asj (T ) even when the latter is fully renormalized to the given temperature. In this case, the T TKcorrection to the background conductance will be dominated by the potential scattering term; theconnected contribution is negligible. The expression for δT isδT j j′ (εp)≈−ZR, j j′ (εp)piνKpp. (3.80)If, however, the QD is particle-hole symmetric, the connected contribution becomes important.Inserting Eq. (3.70) into Eq. (3.47a) and combining with (3.78), we find the Kondo-type correctionto O(J2)at T TKδT j j′ (εp) =[ZI, j j′ (εp)+Z2, j j′ (εp)] 3pi216ν2J2pp. (3.81)Again, in the RG improved perturbation theory, Jpp in Eq. (3.81) should be replaced by Jpp (max(|εp| ,T )),76indicating that Jpp is the renormalized value at the running energy cutoff max(|εp| ,T ). This expres-sion is valid as long as T TK , irrespective of whether the system is in small or large Kondo cloudregime.Eq. (3.81) is formally similar to the previously obtained short ring result[66]. It should be noted,however, that the energy dependence ofZI ,Z2 and J2 is possibly much stronger than the short ringcase, and the thermal averaging in Eq. (3.79) can lead to very different results in small and largeKondo cloud regimes. For instance, if Econn TK T (which may happen in the small cloudregime), the Fermi factor in Eq. (3.79) averages over many peaks in T , ZI , Z2 and V 2 that areassociated with the underlying mesoscopic structure. In this case connected part elimination is notapplicable. On the other hand, if TK T Econn, the variation of T ,ZI ,Z2 orV 2 is negligible onthe scale of T , and the Fermi factor in Eq. (3.79) may be approximated by a δ function. This leadstoG j j′ =−2e2h{T0, j j′ (εkF )+[ZI, j j′ (εkF )+Z2, j j′ (εkF )] 3pi216ν2J2kFkF}, (3.82)which agrees with our prescription of eliminating the connected part, Eq. (3.66).We mention that Eq. (3.81) can be interpreted as a completely inelastic contribution to theconductance, since in obtaining it we have retained only the O(J2)inelastic part of the single-particle T-matrix Eq. (3.70), and the connected contribution comes from two-particle processeswhich are again inelastic.3.4.2 FL perturbation theoryIt is also interesting to consider temperatures low compared to the Kondo temperature T TK .Since our formalism does not by itself provide a low-energy effective theory of the small cloudregime for TK EV , we focus on the very large Kondo cloud regime TK EV , where as explainedin Section 3.2 the low-energy effective theory is simply a FL theory. If we further assume T Econn,then we can simply eliminate the connected contribution to the conductance with Eq. (3.66).To use Eq. (3.66) we need the low-energy T-matrix for the screening channel in the single-particle sector in the FL regime, which is again well known[3, 5]. As discussed in Section 3.1, thestrong-coupling single particle wave function at zero temperature is obtained by imposing a phaseshift on the weak-coupling wave function. This phase shift δψψ results from both elastic scatteringoff the Kondo singlet and particle-hole symmetry breaking potential scattering:δψψ,σ = σpi2+δP, (3.83)where σ = ±1 for spin-up/spin-down electrons. To the lowest order in potential scattering O(K),we have[75]77tanδP =−piνKkFkF . (3.84)Let us introduce the phase-shifted screening channel ψ˜ , which is then related to the originalscreening channel ψ via a scattering basis transformation:ψk,σ =∫ kF+ΛkF−Λdp2pi(ik− p+ i0 − e2iδψψ,σ ik− p− i0)ψ˜p,σ . (3.85)Using the definition of the T-matrix in the single-particle sector Eq. (3.41) and the transformationEq. (3.85) one can show that retarded T-matrices for ψ and ψ˜ are related byTkk′,σ (ω) =i2piν(e2iδψψ,σ −1)+ e2iδψψ,σ T˜kk′,σ (ω) . (3.86)Since ψ˜ diagonalizes the strong-coupling fixed point Hamiltonian, by definition T˜kFkF ,σ (ω = 0) = 0at zero temperature.The leading irrelevant operator perturbing the strong-coupling fixed point is localized at the QD(with a spatial extent of vF/TK), and quadratic in spin current[3, 47]. It is most conveniently writtenin terms of ψ˜:Hint =2piv2FTK: ψ˜†α ψ˜α ψ˜†β ψ˜β : (x= 0)− v2FTK[: ψ˜†α(iddx− kF)ψ˜α +(−i ddx− kF)ψ˜†α ψ˜α :](x= 0) . (3.87)Here :: denotes normal ordering, and sums over repeated spin indices α and β are implied. The ψ˜operators have been unfolded, so that their wave functions are now defined on the entire real axisinstead of the positive real axis. The two terms in Hint are illustrated in panel a) of Fig. 3.4 as a four-point vertex and a two-point one. Both terms share a single coupling constant of O(1/TK), becausethe leading irrelevant operator written in the ψ˜ basis must be particle-hole symmetric by defini-tion. The on-shell retarded T-matrix for ψ˜ in the single-particle sector is calculated to O(1/T 2K)inRef. [3]:−piνT˜pp (εp) = εpTK + i3ε2p+pi2T 22T 2K. (3.88)For completeness we give a derivation of this result in Appendix C. It is diagramatically representedby Fig. 3.4 panel b).Substituting Eqs. (3.83), (3.86) and (3.88) into Eq. (3.66), we eliminate the connected contribu-tion, and obtain the T TK conductance in the very large Kondo cloud regime[112]:78Figure 3.4: Diagrammatics of FL perturbation theory. a) The two vertices given by the leadingirrelevant operator, Eq. (3.87). b) Diagrams contributing to the T-matrix of ψ˜ electrons upto O(1/T 2K). The propagators are those of the phase-shifted screening channel operatorsψ˜ .G j j′ ≈−2e2h{T0, j j′ (εkF )−ZR, j j′ (εkF )(12− pi2T 2T 2K)sin2δP+[ZI, j j′ (εkF )+Z2, j j′ (εkF )](cos2 δP− pi2T 2T 2Kcos2δP)}. (3.89)We note that the connected contribution can in fact be evaluated directly in the FL theory. This is alsodone in Appendix C, and provides further verification of our scheme of eliminating the connectedcontribution.Predictions of the conductance at high temperatures [Eq. (3.79)] and at low temperatures [Eq. (3.89)]together constitute the second main result of this chapter. We emphasize once more that, whileEq. (3.79) is valid as long as T TK , Eq. (3.89) is expected to be justified provided T TK EV ,so that the FL theory applies, and also T Econn, so that the connected contribution can be elimi-nated.For clarity we tabulate various regimes of energy scales discussed so far (Table 3.1). Note againthat the connected contribution to conductance can be eliminated when T Econn. In general wehave Econn . EV , but we assume in this table that Econn ∼ EV , which is the case with the systems tobe discussed in this chapter.79Figure 3.5: The short ABK ring studied in Refs. [66, 75].3.5 Comparison with early results3.5.1 Short ABK ringOur formalism can be applied to the short ABK ring[66, 75] shown in Fig. 3.5. There are twoleads (N = 2) and two coupling sites (M = 2); H0,junction =−t ′(c†1,0c2,0+h.c.). The coupling sitescoincide with the 0th sites of the leads, cC,r=1 ≡ c1,0, cC,r=2 ≡ c2,0; also the AB phase is on thecouplings to the QD, t1 = tLeiϕ2 and t2 = tRe−iϕ2 . We again let τ˜ = t ′/t.It is straightforward to obtain the background S-matrix and coupling site wave function matrix:S (k) =− 11− τ˜2e2ik(e2ik(1− τ˜2) eikτ˜ (e2ik−1)eikτ˜(e2ik−1) e2ik (1− τ˜2)), (3.90)Γ(k) =− 11− τ˜2e2ik(e2ik−1 2ie2ikτ˜ sink2ie2ikτ˜ sink e2ik−1). (3.91)With Eqs. (3.27), (3.81) and (3.89), one reproduces all analytic results in Refs. [66, 75], includingthe Kondo temperature, the high- and low-temperature conductance, and the elimination of con-nected contribution at low temperatures.The limit t ′ = 0 is useful as a benchmark against long ring geometries so we study it in somemore detail. In this limit we recover the simplest geometry where a QD is embedded between sourceand drain leads[102]. The normalization factor isV 2k = 4(t2L+ t2R)sin2 k, (3.92)and the zero-temperature transmission amplitude through the QD is given by Eqs. (3.45), (3.83),80Figure 3.6: The finite quantum wire geometry studied in Ref. [116].(3.86) and (3.88):tQD ≡ 2iV 2k[S (k)Γ† (k)λΓ(k)]12 [−piνTkk (εk)]= e2ik2tLtRt2L+ t2R12(e2iδP +1). (3.93)3.5.2 Finite quantum wireAnother special case is the finite wire (or semi-transparent Kondo box) geometry in Fig. 3.6 wherethe reference arm is absent[116]; again N =M = 2. The left and right QD arms and coupling sitesare subject to gate voltages:H0,junction =−t(dL−1∑n=1c†L,ncL,n+1+dR−1∑n=1c†R,ncR,n+1+h.c.)+(εLWdL∑n=1c†L,ncL,n+ εRWdR∑n=1c†R,ncR,n)−(tLLW c†L,dLc1,0+ tRLW c†R,dRc2,0+h.c.). (3.94)The coupling sites are the first sites of the QD arms, cC,r=1 ≡ cL,1, cC,r=2 ≡ cR,1; t1 = tLWD andt2 = tRWD.The two leads are decoupled without the QD, so S and Γ are both diagonal. In this system wehaveS11 (k) =− eik sinkL (dL+1)− γ2L sinkLdLe−ik sinkL (dL+1)− γ2L sinkLdL, (3.95)81andΓ11 (k) =− 2iγL sink sinkLe−ik sinkL (dL+1)− γ2L sinkLdL, (3.96)where kL is determined by the gate voltage εLW ,−2t coskL+ εLW =−2t cosk, (3.97)and γL = tLLW/t. S22 and Γ22 can be obtained simply by substituting L with R. Again, these resultsallow us to reproduce the (weak coupling) Kondo temperature, the high-temperature conductance, aswell as the low-temperature conductance in the large Kondo cloud regime. (We do not quantitativelydiscuss the low-temperature conductance in the small cloud regime in this thesis; see Section 3.8.)3.6 Closed long ringIn this section we apply our general formalism to the simplest model of a closed long ABK ring,studied in Ref. [131] (Fig. 3.7): the QD is coupled directly to the source and drain leads, and a longreference arm connects the two leads smoothly. A weak link with hopping t ′ splits the referencearm into two halves of equal length dre f /2 where dre f is an even integer. As opposed to Ref. [131],however, we use gauge invariance to assign the AB phase to the QD tunnel couplings rather thanthe weak link: t1 ≡ tLeiϕ/2 and t2 ≡ tRe−iϕ/2. We assume no additional non-interacting long armsconnecting the QD with the source and drain leads, because multiple traversal processes in suchlong QD arms will lead to interference effects[84] independent of the AB phase, complicating theproblem[131]. The Hamiltonian representing this model takes the formH0,junction =−t[(dre f /2−1∑n=1+dre f−1∑n=dre f /2+1)c†re f ,ncre f ,n+1+h.c.]− t(c†re f ,1c1,0+ c†re f ,dre f c2,0+h.c.)− t ′(c†re f ,dre f /2cre f ,dre f /2+1+h.c.); (3.98)the coupling sites are cC,r=1 ≡ c1,0 and cC,r=2 ≡ c2,0.We first repeat the Kondo temperature analysis in Ref. [131] in order to distinguish betweensmall and large Kondo cloud regimes, then carefully study the conductance at high and low temper-atures, taking into account the previously neglected connected contribution.82Figure 3.7: Geometry of the long ABK ring with short upper arms and a pinched referencearm.3.6.1 Kondo temperatureThe background S-matrix for this model is identical to the short ABK ring[75] up to overall phases,due to the smooth connection between reference arm and leads:S (k) = eikdre f(r˜ (k) t˜ (k)t˜ (k) r˜ (k)); (3.99)where the S-matrix elements r˜ and t˜ for the weak link arer˜ (k) =− 1− τ˜2e−2ik− τ˜2 , t˜ (k) =−2iτ˜ sinke−2ik− τ˜2 , (3.100)and we introduce the shorthand τ˜ = t ′/t. The wave function is also straightforward to find:Γ j j′ (k) = δ j j′+S j j′ (k) . (3.101)In the wide band limit, r˜ and t˜ are approximately independent of k in the reduced band kF−Λ0 <k< kF+Λ0 where the momentum cutoffΛ0 1. This allows us to approximate them by their Fermisurface values, r˜ (k)≈ r˜ = |r˜|eiθ and t˜ (k)≈ t˜ =±i |t˜|eiθ (the ±pi/2 phase difference is required byunitarity of S); without loss of generality we focus on the t˜ = i |t˜|eiθ case.From Eq. (3.101) one conveniently obtains the normalization factor83V 2k = 2(t2L+ t2R)[1+ |r˜|cos(kdre f +θ)− γ |t˜|cosϕ sin(kdre f +θ)]= 2(t2L+ t2R)[1+√1−|t˜|2 (1− γ2 cos2ϕ)cos(kdre f +θ ′)] , (3.102)where γ = 2tLtR/(t2L+ t2R)measures the degree of symmetry of coupling to the QD. In the secondline we have used |t˜|2+ |r˜|2 = 1 and introduced another phase θ ′, where θ ′−θ is a function of γ , |t˜|and ϕ but independent of k. We note that this expression is also applicable in the continuum limit,where the lattice constant a→ 0 (we have previously set a= 1) but the arm length dre f a is fixed. Inthat case dre f should be understood as the arm length dre f a.For long rings and filling factors not too small kFdre f 1, V 2k oscillates around 2(t2L+ t2R)asa function of k, and has its extrema at kn = (npi−θ ′)/dre f where n takes integer values. The onlycharacteristic energy scale for V 2k is therefore the peak/valley spacing ∆ = vFpi/dre f , and EV ∼Econn ∼ ∆. As in Ref. [131] we define the reduced band such that ∆ D0 ≡ vFΛ0, and the reducedband initially contains many oscillations.In the small Kondo cloud regime TK∆, one may assume the oscillations ofV 2k are smeared outwhen the energy cutoff is being reduced from D0, which is still well above TK : V 2k 'V 2k = 2(t2L+ t2R).This means TK in this regime is approximately the background Kondo temperature T 0K defined inEq. (3.29), independent of the position of the Fermi level at the energy scale ∆, and also independentof the magnetic flux.On the other hand, in the large cloud regime TK . ∆, now that the Kondo temperature is largelydetermined by the value of V 2k in a very narrow range of energies around the Fermi level, the meso-scopic k oscillations become much more important. When the running energy cutoff D is abovethe peak/valley spacing ∆, the renormalization of j is controlled by V 2k as in Eq. (3.28). Once D isreduced below ∆, we may approximate the renormalization of j as being dominated by V 2kF . Thisleads to the following estimation of the Kondo temperature:TK ' ∆exp[− 12V 2kFν j (∆)]= ∆(T 0K∆)[1+√1−|t˜|2(1−γ2 cos2ϕ)cos(kFdre f+θ ′)]−1. (3.103)It is clear that TK can be significantly dependent on the AB phase ϕ in this regime. In particular,TK varies from ∼√T 0K∆ (“on resonance”) to practically 0 (“off resonance”) as ϕ is tuned between0 and pi[131], when the Fermi energy is located on a peak or in a valley kF = kn, the backgroundtransmission is perfect |t˜|= 1, and coupling to the QD is symmetric γ = 1; see Fig. 3.8. (The specialcase V 2kF = 0 corresponds to a pseudogap problem νV2k ∝ (k− kF)2, and the stable RG fixed pointcan be the local moment fixed point or the asymmetric strong coupling fixed point, depending on84Figure 3.8: Kondo temperature TK for the closed long ABK ring, calculated by numerical in-tegration of the weak coupling RG equation Eq. (3.27), plotted against the AB phaseϕ . TK (ϕ) is an even function of ϕ and has a period of 2pi , so only 0 ≤ ϕ ≤ pi isshown. System parameters are: dre f = 60, θ = pi/2, |r˜| = 0, tL = tR, D0 = 10. Thecurves with TK ∆ (small Kondo cloud regime) have a large bare Kondo coupling(t2L+ t2R)j0/pi = 0.15, whereas the curves with TK ∆ (large Kondo cloud regime) havea much smaller bare Kondo coupling(t2L+ t2R)j0/pi = 0.02. In the small cloud regimeTK is almost independent of ϕ and kF , as the curves are flat and overlapping with eachother. In the large cloud regime, however, TK highly sensitive to both ϕ and kF .the degree of particle-hole symmetry[39, 128].) As a general rule, stronger transmission throughthe pinch |t˜| and greater symmetry of coupling γ result in stronger interference between the twotunneling paths through the device, and hence increases the tunability of the Kondo temperature bythe magnetic flux.3.6.2 High-temperature conductanceWe now calculate the conductance at T TK by perturbation theory. Following the discussion inRef. [131], we consider the case of a particle-hole symmetric QD κ = 0 and Kkk′ = 0, and also ignorethe elastic real part of the potential scattering generated[66] at O(J2). These assumptions allow usto adopt Eq. (3.81) for the O(J2)correction to the transmission probability:δT j j′ (εk) = 3pi2ν2 j2(V 2k Re{[S (k)Γ† (k)λΓ(k)]j j′ S∗j j′ (k)}−∣∣∣[S (k)Γ† (k)λΓ(k)] j j′∣∣∣2) ,(3.104)85where we have used Eqs. (3.47d) and (3.47e).Note that Eq. (3.104) does not depend on details of the non-interacting part of the ring Hamilto-nian H0,junction. For a parity-symmetric geometry with two leads and two coupling sites (N=M= 2),when coupling to the QD is also symmetric (tL = tR) and time-reversal symmetry is present (ϕ = 0 orpi), we can further show that the sign of the O(J2)transmission probability correction is determinedby the sign of 1− 2 |T0,12|, a property discussed in Ref. [66] at the end of Section IV C. Indeed,parity symmetry implies that S11 = S22, S12 = S21, Γ11 = Γ22, Γ12 = Γ21; hence it is not difficult tofind from Eq. (3.104) that14[δT 11 (εk)+δT 22 (εk)−δT 12 (εk)−δT 21 (εk)] = 38pi2ν2J2kk[1−2 |S12 (k)|2]. (3.105)The left-hand side correspond to a particular way to measure the conductance, namely parity-symmetric bias voltage and parity-symmetric current probes, or y= 1/2 in Section V of Ref. [66].We now return to the long ring geometry without assumptions about tL, tR and ϕ . PluggingEqs. (3.99) and (3.101) into Eq. (3.104) we findδT 11 (εk) =−3pi2ν2 j2 [C0 (k)+C1 (k)cosϕ+C2 (k)cos2ϕ] , (3.106)where the coefficients C0 (k), C1 (k) and C2 (k) are independent of ϕ but are usually complicatedfunctions of k:C0 (k) =(t4L+ t4R) |t˜|2 [1+2 |r˜|cos(kdre f +θ)+ |r˜|2 cos2(kdre f +θ)]−2t2Lt2R[3−4 |t˜|2+4 |r˜|3 cos(kdre f +θ)+(|r˜|4+ |t˜|4)cos2(kdre f +θ)], (3.107a)C1 (k) = 4 |t˜| tLtR(t2L+ t2R)sin(kdre f +θ)×[|r˜|2 cos2(kdre f +θ)+ |r˜|(|r˜|2−|t˜|2)cos(kdre f +θ)−|t˜|2], (3.107b)C2 (k) = 2 |t˜|2 t2Lt2R{1+2 |r˜|cos(kdre f +θ)+ |r˜|2 cos2(kdre f +θ)}. (3.107c)In the special case of a smooth reference arm |r˜|= 0 and |t˜|= 1, the Kondo-type correction becomesespecially simple:86δT 11 (εk) =−3pi2ν2 j2[t2L+ t2R−2tLtR sin(kdre f +θ +ϕ)]× [t2L+ t2R−2tLtR sin(kdre f +θ −ϕ)] . (3.108)As in Refs. [66, 131], only the first and the second harmonics of the AB phase ϕ appear in thecorrection to the transmission probability δT 11.We may perform the thermal averaging in Eq. (3.79) at this stage. The Fermi factor − f ′ (εk)ensures only the energy range |εk| . T contributes significantly to the conductance; in this energyrange the renormalization of j is cut off by T .In the small Kondo cloud regime, T TK means T ∆ so that we can average over manypeaks of δT j j′ (εk), so we may drop all rapidly oscillating Fourier components in Eq. (3.106). Thisleads toδG≈−2e2h3pi2ν2 [ j (T )]2[(t4L+ t4R) |t˜|2−2t2Lt2R(3−4 |t˜|2)+2 |t˜|2 t2Lt2R cos2ϕ] . (3.109)We see that the first harmonic in ϕ approximately drops out upon thermal averaging.On the other hand, in the large Kondo cloud regime, for T TK it is possible to have eitherT ∆ or T ∆. In the former case Eq. (3.109) continues to hold. In the latter case δT j j′ (εk) haslittle variation in the energy range |εk|. T , so it is appropriate to replace − f ′ (εk) with a δ functionat the Fermi level; thusδG≈−2e2h3pi2ν2 [ j (T )]2 [C0 (kF)+C1 (kF)cosϕ+C2 (kF)cos2ϕ] . (3.110)Fig. 3.9 illustrates these two different cases for the large Kondo cloud regime[112]. We note thatour T TK results, Eq. (3.109) for T ∆ and Eq. (3.110) for T ∆, are different from thoseof Ref. [131]. We believe the discrepancy is due to the fact that only single-particle scatteringprocesses are taken into consideration by Ref. [131]; the connected contribution to the conductanceis omitted, despite being of comparable magnitude with the disconnected contribution.It should be noted that our T TK results do not constitute evidence in favor of the Kondoscreening cloud per se, as they do not involve a direct comparison between TK and ∆. Indeed,the difference between T ∆ and T ∆ is also present even if we replace the Kondo QD witha non-interacting QD. Nevertheless, our results may be experimentally relevant as they reveal thequalitatively different interplay between the AB flux and the Kondo scattering.87Figure 3.9: Kondo-type correction to the conductance δG at T TK for the closed long ABKring with a particle-hole symmetric QD, calculated by RG improved perturbation theoryEq. (3.106), plotted against the AB phase ϕ . Again only 0 ≤ ϕ ≤ pi is shown. Systemparameters are: dre f = 60, θ = pi/2, |r˜| = 0, tL = tR,(t2L+ t2R)j0/pi = 0.02 at D0 = 10(i.e. the system is in the large cloud regime). T/∆= 0.0955 in panel a) and T/∆= 19.1in panel b). For T ∆ the conductance shows considerable kF dependence, while forT ∆ such dependence essentially vanishes and curves at different kF overlap. Also,for T ∆ the first harmonic cosϕ drops out as predicted by Eq. (3.109), and δG(ϕ) hasa period of pi .883.6.3 FL conductanceIt is also interesting to calculate the conductance in the T TK limit in the very large Kondo cloudregime, starting from Eq. (3.89). We make the assumption that the particle-hole symmetry breakingpotential scattering is negligible, δP = 0, as in Ref. [131]. Inserting Eqs. (3.99) and (3.101) intoEqs. (3.47d) and (3.47e), we find the total conductance has the formG=2e2h[Ts+(|t˜|2−Ts)(piTTK)2], (3.111)where the T = 0 transmission probability isTs =∣∣∣∣∣eikFdre f t˜− 2V 2kF[tLeiϕ2(eikFdre f r˜+1)+ tRe−iϕ2 eikFdre f t˜]×[tLe−iϕ2 eikFdre f t˜+ tReiϕ2(1+ eikFdre f r˜)]∣∣∣2 . (3.112)While Eq. (3.111) is ostensibly in agreement with Eq. (69) of Ref. [131], we suspect that there aretwo oversights in the derivation of the latter: at finite temperature, Ref. [131] neglects the connectedcontribution to the conductance, and also replaces the thermal factor − f ′ (εp) with a δ function inEq. (3.66). These two discrepancies cancel each other, leading to the same T TK result as ours.3.7 Open long ringWe turn to the open long ABK ring, with strong electron leakage due to side leads coupled to thearms of the ring, where our multi-terminal formalism shows its full capacity.In our geometry shown in Fig. 3.10, the source lead branches into two paths at the left Y-junction, a QD path of length dL+dR and a reference path of length dre f . These two paths convergeat the right Y-junction at the end of the drain lead. An embedded QD in the Kondo regime separatesthe QD path into two arms of lengths dL and dR. To open up the ring we attach additional non-interacting side leads to all sites inside the ring other than the two central sites in the Y-junctionsand QD[6, 7, 23, 118]. The side leads, numbering dL+dR+dre f in total, are assumed to be identicalto the main leads (source and drain), except that the first link on every side lead (connecting site 0 ofthe side lead to its base site in the ring) is assumed to have a hopping strength tx which is generallydifferent than the bulk hopping t. The Hamiltonian representing this model is therefore89H0,junction =−t(dL−2∑n=0c†L,ncL,n+1+dR−2∑n=0c†R,ncR,n+1+dre f−1∑n=1c†re f ,ncre f ,n+1+h.c.)−[(tLJLc†1,0+ tLJQc†L,dL−1+ tLJRc†re f ,1)cJL+h.c.]−[(tRJLc†2,0+ tRJQc†R,dR−1+ tRJRc†re f ,dre f)cJR+h.c.]− tx(dL−1∑n=0c†L,nc(L)n,0 +dR−1∑n=0c†R,nc(R)n,0 +dre f∑n=1c†re f ,nc(re f )n,0 +h.c.), (3.113)where cJL(R) is the annihilation operator on the central site of the left (right) Y-junction, and c(α)n,0 isthe annihilation operator on site 0 of the side lead attached to the nth site on arm α , α = L, R andre f . The coupling sites are cC,r=1 ≡ cL,0 and cC,r=2 ≡ cR,0, and again we let the couplings to the QDbe t1 = tLeiϕ2 and t2 = tRe−iϕ2 .Our hope is that in certain parameter regimes the open long ring provides a realization of thetwo-path interferometer, where the two-slit interference formula applys:Gsd = Gre f +Gd+2√ηv√Gre fGd cos(ϕ+ϕt) , (3.114)where Gre f is the conductance through the reference arm with the QD arm sealed off, and Gd is theconductance through the QD with the reference arm sealed off. ϕ is as before the AB phase, and ϕtis the accumulated non-magnetic phase difference of the two paths (including the pi/2 transmissionphase through the QD). ηv is the unit-normalized visibility of the AB oscillations; ηv = 1 at zerotemperature if all transport processes are coherent[23]. In the two-path interferometer regime, ϕtreflects the intrinsic transmission phase through the QD, provided that the geometric phases of thetwo paths are the same (e.g. identical path lengths in a continuum model), no external magneticfield is applied to the QD, and the particle-hole symmetry breaking phase shift is zero.For non-interacting embedded QDs well outside of the Kondo regime, small transmission throughthe lossy arms is known to suppress multiple traversals of the ring and ensure that the transmissionamplitudes in two paths are mutually independent[7]. We show below that in our interferometerwith a Kondo QD, the same criterion renders the mesoscopic fluctuations of the normalization factorV 2k negligible, and paves the way to the two-slit condition tsd = tre f + tdeiϕ . If we additionally havesmall reflection by the lossy arms, then both the Kondo temperature of the system and the intrinsictransmission amplitude through the QD are the same as their counterparts for a QD directly embed-ded between the source and the drain. At finite temperature T TK , we recover and generalizethe single-channel Kondo results of Ref. [23] for the normalized visibility ηv and the transmissionphase ϕt .90Figure 3.10: Geometry of the open long ABK ring. Side leads are appended to the QD armsand the reference arms, which are all of comparable lengths.3.7.1 Wave function on a single lossy armTo solve for the background S-matrix S and the wave function matrix Γ of the open ring, we firstanalyze a single lossy arm attached to side leads[7], depicted in Fig. 3.11.Consider an arbitrary site labeled n on this arm; let the wave function on this site be φn, andlet incident and scattered amplitudes on the side lead attached to this site be Asn and Bsn. The wavefunction on site l (l≥ 0) on the side lead (also with bulk hopping t) is then written as Asne−ikl+Bsneikl ,which gives the usual energy −2t cosk. The Schroedinger’s equations are(−2t cosk)(Asn+Bsn) =−txφn− t(Asne−ik+Bsneik), (3.115a)(−2t cosk)φn =−tφn−1− tφn+1− tx (Asn+Bsn) . (3.115b)91Figure 3.11: A single lossy arm attached to side leads.Eliminating Bsn, we find(−2cosk+ t2xt2eik)φn =−φn−1−φn+1+ eik (2isink) txt Asn. (3.116)This means if Asn= 0, i.e. no electron is incident from the side lead n, we can write the wave functionon the nth site on the arm asφn =CLηn1 +CRηn2 , (3.117)where CL,R are constants independent of n and k. η1,2 are roots of the characteristic equationη2+(−2cosk+ t2xt2eik)η+1 = 0, (3.118)so that η1η2 = 1. Hereafter we choose the convention |η1| < 1. When tx/t sink, to the lowestnontrivial order in tx/t,η1 ≈ eik(1− t2xt2eik2isink), (3.119)and thus |η1|2 ≈ 1− t2x /t2.Eq. (3.117) bypasses the difficulty of solving for each φn individually: on the same arm theconstants CL and CR only change where the side lead incident amplitude Asn 6= 0.Let us now quantify the conditions of small transmission and small reflection. Connecting92external leads smoothly to both ends of a lossy arm of length dA 1, we may write the scatteringstate wave function incident from one end ase−ikn+ R˜eikn (left lead, n= 0,1,2, · · ·)CLηn1 +CRη−n1 (lossy arm, n= 1, · · · ,dA)T˜ eikn (right lead, n= 0,1,2, · · ·); (3.120)the Schroedinger’s equation then yields1+ R˜=CL+CR, (3.121a)eik+ R˜e−ik =CLη1+CRη−11 , (3.121b)T˜ =CLηdA+11 +CRη−dA−11 , (3.121c)T˜ e−ik =CLηdA1 +CRη−dA1 . (3.121d)It is now straightforward to find the transmission and reflection coefficients:T˜ =eik(e2ik−1)ηdA1 (η21 −1)1−η2dA+21 +2eikη1(η2dA1 −1)+ e2ik(η21 −η2dA1) , (3.122a)R˜=eik(η2dA1 −1)[eik(1+η21)− (e2ik+1)η1]1−η2dA+21 +2eikη1(η2dA1 −1)+ e2ik(η21 −η2dA1) . (3.122b)At k = 0 or pi we always have trivially∣∣R˜∣∣= 1 and ∣∣T˜ ∣∣= 0; we therefore focus on energies thatare not too close to the band edges, so that sink is not too small. In this case, under the long armassumption dA 1, the small transmission condition∣∣T˜ ∣∣ 1 is satisfied if and only if |η1|dA 1,and the small reflection condition∣∣R˜∣∣ 1 is satisfied if and only if tx t[7].3.7.2 Background S-matrix and coupling site wave functionsWe now return to the open long ring model to solve for S and Γ with the aid of Eq. (3.117). Let usdenote the incident amplitude vector by(A1,A2,A(L)0 , · · · ,A(L)dL−1,A(re f )1 , · · · ,A(re f )dL ,A(R)0 , · · · ,A(R)dR−1)T; (3.123)here A(α)n is the incident amplitude in the side lead attached to the nth site on arm α . We areinterested in the normalization factor V 2k and the source-lead component of the conductance tensor93G12; for this purpose, according to Eqs. (3.19) and (3.89), the first two rows of the S-matrix S andthe full coupling site wave function matrix Γ must be found. In other words, we need to express thescattered amplitudes in the main leads B1, B2, as well as the wave functions at the coupling sites Γ1and Γ2, in terms of incident amplitudes. Because the Schroedinger equation is linear and all incidentamplitudes are independent, we can let all but one of the incident amplitudes be zero at a time, andobtain the full solution by means of linear superposition.When the incident amplitudes from the side leads are all zero A(α)n = 0, according to Eq. (3.117)the wave function at wave vector k takes the formA je−ikn+B jeikn (main lead j = 1,2, n= 0,1,2, · · ·)D(L)L ηn1 +D(L)R η−n1 (left QD arm, n= 0,1, · · · ,dL−1)D(re f )L ηn1 +D(re f )R η−n1 (reference arm, n= 1,2, · · · ,dre f )D(R)L ηn1 +D(R)R η−n1 (right QD arm, n= 0,1, · · · ,dR−1). (3.124)To characterize the two Y-junctions in the AB ring, it is convenient to introduce the auxiliary objectsS′L and S′R: B1D(L)R η−dL+11D(re f )L η1= S′L A1D(L)L ηdL−11D(re f )R η−11 , (3.125) B2D(R)R η−dR+11D(re f )R η−dre f1= S′R A2D(R)L ηdR−11D(re f )L ηdre f1 . (3.126)The 3× 3 matrices S′L and S′R are generally not unitary. They are properties of the Y-junctions,and are independent of the amplitudes (A, B etc.) and arm lengths (dL, dR and dre f ). In the limittx/t → 0, η1 → eik, and S′L and S′R turn into the usual unitary S-matrices, which we denote asSL,R ≡ S′L,R (tx→ 0). For our model we can find S′L and S′R explicitly by solving the Schroedinger’sequations below:tLJLφL = t(A1eik+B1e−ik), (3.127a)tLJQφL = t(D(L)L ηdL1 +D(L)R η−dL1), (3.127b)tLJRφL = t(D(re f )L +D(re f )R), (3.127c)94(−2t cosk)φL =−tLJL (A1+B1)− tLJQ(D(L)L ηdL−11 +D(L)R η−dL+11)− tLJR(D(re f )L η1+D(re f )R η−11), (3.127d)where φL is the wave function on the central site of the left Y-junction. We can solve for S′R in asimilar fashion.For given A1 and A2, Eq. (3.124) has 8 unknowns. Now that S′L and S′R are known, Eqs. (3.125)and (3.126) provide us with 6 equations. The remaining two equations are the open boundaryconditions at the ends of the two QD arms, appropriate when the QD is decoupled:D(L)L η−11 +D(L)R η1 = 0, (3.128)D(R)L η−11 +D(R)R η1 = 0. (3.129)It is straightforward to solve the closed set of equations.On the other hand, when we let one of the incident amplitudes in the side leads be nonzero,there are two additional amplitudes in the wave function. For instance, if A(L)m 6= 0 for a given m, weneed to effectuate the following changes to the wave function on the left QD arm in Eq. (3.124):{D(L1)L ηn1 +D(L1)R η−n1 (left QD arm, n= 0,1, · · · ,m)D(L2)L ηn1 +D(L2)R η−n1 (left QD arm, n= m,m+1, · · · ,dL−1); (3.130)D(L)L,R should be replaced by D(L1)L,R in Eq. (3.128) and by D(L2)L,R in Eq. (3.125). Furthermore, we shouldhave two boundary conditions at site m:D(L1)L ηm1 +D(L1)R η−m1 = D(L2)L ηm1 +D(L2)R η−m1 , (3.131)−(η1+1η1)(D(L1)L ηm1 +D(L1)R η−m1)=−(D(L1)L ηm−11 +D(L1)R η−m+11)−(D(L2)L ηm+11 +D(L2)R η−(m+1)1)+ eik (2isink)txtA(L)m , (3.132)thus closing the set of equations. The last equation is none other than Eq. (3.116).We are now in a position to present the solutions for B1, B2 and Γ1, Γ2 in terms of incidentamplitudes. If we assume dL ∼ dR ∼ dre f /2 1 (comparable arm lengths and path lengths in thelong ring) and |η1|dL 1 (small transmission criterion), we have to O(|η1|dL)95B1 = S′L11A1+S′L13S′R31ηdre f−11 A2−dL−1∑n=0eik (2isink)txtηn+11 −η−n−11η1−η−11S′L12ηdL1 A(L)n+dre f∑n=1eik (2isink)txtS′R33ηdre f−n1 +η−dre f+n1η1−η−11S′L13ηdre f−11 A(re f )n , (3.133a)B2 = S′L31S′R13ηdre f−11 A1+S′R11A2+dre f∑n=1eik (2isink)txtη−n+11 +S′L33ηn−11η1−η−11S′R13ηdre f−11 A(re f )n−dR−1∑n=0eik (2isink)txtηn+11 −η−n−11η1−η−11S′R12ηdR1 A(R)n , (3.133b)Γ1 = S′L21η(dL−1)1(1−η21)A1−dL−1∑n=0eik (2isink)txt(η−dL+n+11 +S′L22ηdL−n−11)ηdL1 A(L)n−dre f∑n=1eik (2isink)txt(S′R33ηdre f−n1 +η−dre f+n1)S′L23ηdL1 ηdre f−11 A(re f )n , (3.133c)Γ2 = S′R21η(dR−1)1(1−η21)A2−dre f∑n=1eik (2isink)txt(η−n+11 +S′L33ηn−11)S′R23ηdR1 ηdre f−11 A(re f )n−dR−1∑n=0eik (2isink)txt(η−dR+n+11 +S′R22ηdR−n−11)ηdR1 A(R)n . (3.133d)3.7.3 Kondo temperature and conductanceTo the lowest nontrivial order in |η |dL , Eq. (3.133d) leads to the following simple results after somealgebra:V 2k =−(η1−η∗1 )(2isink)(t2L+ t2R), (3.134)S12 (k) = S′L13S′R31ηdre f−11 , (3.135)[S (k)Γ† (k)λΓ(k)]12 = tLtReiϕ (2isink)(η1−η−11)S′L12S′R21ηdL+dR1 . (3.136)In obtaining Eq. (3.136) we have used the algebraic identity96S′∗L21S′L11∣∣(1−η21)∣∣2 = (4sin2 k)( txt )2[|η1|2η∗1S′L12η1−η∗1− |η1|21−|η1|2(S′∗L22S′L12+S′∗L23S′L13)];(3.137)in the limit tx→ 0 this is just a statement of the S-matrix unitarity.Eq. (3.134) tells us that, in the small transmission limit, the normalization factor V 2k exhibitslittle mesoscopic fluctuation, so that EV ∼ t; furthermore, it does not depend on the AB phase ϕat all (see Fig. 3.12). When we also impose the small reflection condition tx t, η1 ≈ eik and wefind V 2k ≈(4sin2 k)(t2L+ t2R); this is precisely the normalization factor for a QD embedded betweensource and drain leads. We recall from Eq. (3.27) that the normalization of Kondo coupling isgoverned by V 2k . Therefore, at least for our simple model of an interacting QD, the conditions ofsmall transmission and small reflection combine to reduce the Kondo temperature of the open longABK ring to that of the simple embedded geometry, independent of the details of the ring or the ABflux.Proceeding with the small transmission assumption, we observe that since EV ∼ t, the distinctionbetween small and large Kondo cloud regimes is no longer applicable. This is presumably becausethe Kondo cloud leaks into the side leads in the open ring, and is no longer confined in a mesoscopicregion as in the closed ring. The low-energy theory of our model is therefore the usual local FL. Atzero temperature, the connected contribution to the conductance vanishes, and the conductance G12is proportional to the disconnected transmission probability Eq. (3.45) at the Fermi energy:−T D12 (εkF ) =∣∣∣∣∣S′L13S′R31ηdre f−11 + 2tLtRt2L+ t2R eiϕ η1−η−11η1−η∗1S′L12S′R21ηdL+dR112(e2iδP +1)∣∣∣∣∣2, (3.138)where η1, S′L and S′R are all evaluated at the Fermi surface, and we have used Eqs. (3.83) and (3.86).In the small reflection limit, Eq. (3.138) becomes−T D12 (εkF ) =∣∣∣SL13SR31ηdre f−11 + eiϕSL12SR21ηdL+dR−21 tQD∣∣∣2 , (3.139)where SL,R ≡ S′L,R (tx→ 0) are the aforementioned S-matrices of the Y-junctions, andtQD ≡ e2ik 2tLtRt2L+ t2R12(e2iδP +1)(3.140)is the transmission amplitude through an embedded QD in the Kondo limit [see Eq. (3.93)].It is clear from Eq. (3.138) that the two-slit condition tsd = tre f + tdeiϕ is satisfied at zerotemperature. Furthermore, under the small reflection condition, both tre f = SL13SR31ηdre f−11 andtd = SL12SR21ηdL+dR−21 tQD have straightforward physical interpretations; in particular td can be fac-97Figure 3.12: Normalization factorV 2k from Eq. (3.19) as a function of k for different AB phasesϕ in the open long ABK ring, obtained by solving the full tight-binding model. We focuson a small slice of momentum |k−pi/3|< 0.05. Two values of tx are considered: tx = 0corresponding to the closed ring without electron leakage, and tx = 0.3t correspondingto strong leakage along and small transmission across the arms. System parametersare: dL = dR = dre f /2 = 100, tL,RJL = tL,RJQ = tL,RJR = t, and symmetric QD coupling tL =tR. For comparison we have also plotted the analytic prediction Eq. (3.134) for tx =0.3t, which agrees quantitatively with the full tight-binding solution. While V 2k for theclosed ring is extremely sensitive to kF and ϕ , the sensitivity is strongly suppressed byelectron leakage, and curves for different ϕ overlap when tx = 0.3t. Since V 2k controlsthe renormalization of the Kondo coupling, the Kondo temperature of the open longABK ring is not sensitive to mesoscopic details in the small transmission limit.98torized into a part tQD which is the intrinsic transmission amplitude through QD, and a part duecompletely to the rest of the QD arm and the two Y-junctions.We now consider the finite temperature conductance, assuming realistically that T t andTK t. If we further assume that the two Y-junctions are non-resonant, so that S′L and S′R changesignificantly as functions of energy only on the scale of the bandwidth 4t, then mesoscopic fluctua-tions are entirely absent from Eq. (3.136), i.e. Econn ∼ t. (It is worth mentioning that Econn can bemuch less than t if the Y-junctions allow resonances, e.g. when the central site of each Y-junctionis weakly coupled to all three legs; however, EV ∼ t even in this case.) Since T Econn, we cancomfortably eliminate the connected contribution and use Eq. (3.89). At T TK the total FL regimeconductance G(T,ϕ)≡−G12 is found to O(T/TK)2[112]:G(T,ϕ)≡ Gre f +Gd+2√Gre fGd{cos(ϕ+θ +δP)−(piTTK)2×[cos(ϕ+θ +δP)2cos2 δP− tanδP sin(ϕ+θ +δP)]}. (3.141a)Here the conductance through the reference path is defined asGre f ≡ 2e2h∣∣∣S′L13S′R31ηdre f−11 ∣∣∣2 , (3.141b)the conductance through the QD path is defined asGd (T ) = G(0)d[cos2 δP−(piTTK)2cos2δP](3.141c)with its T = 0 and δP = 0 valueG(0)d ≡2e2h4t2Lt2R(t2L+ t2R)2∣∣∣∣∣η1−η−11η1−η∗1 S′L12S′R21ηdL+dR1∣∣∣∣∣2, (3.141d)and finally the non-magnetic phase difference between the QD path and the reference path (includingthe QD) in the absence of δP isθ = arg(η1−η−11η1−η∗1ηdL+dR−dre f+11S′L12S′R21S′L13S′R31). (3.141e)Once again, η1, S′L and S′R are all evaluated at the Fermi surface.For T TK , we discuss two different scenarios: the particle-hole symmetric case and thestrongly particle-hole asymmetric case. In the particle-hole symmetric case, as explained in Sec-tion 3.4, the potential scattering term K vanishes, and the O(J2)connected contribution plays an99important role. Inserting Eqs. (3.134)–(3.136) into Eqs. (3.81) and (3.79), we find the total high-temperature conductance in the particle-hole symmetric case to beG(T,ϕ) = Gre f +Gd+2√34piln TTK√Gre fGd cos(ϕ+θ) . (3.142)Here the conductance through the reference path Gre f is given as before, while the conductancethrough the QD path has the usual logarithmic temperature dependence,Gd (T )≡ 316pi2ln2 TTKG(0)d . (3.143)We have taken into account the renormalization of the Kondo coupling, Eq. (3.28); thermal averag-ing cuts off the logarithm at T . For our slowly varying V 2k given by Eq. (3.134), V2k is simply theFermi surface value V 2kF , and the Kondo temperature is defined by Eq. (3.29).Comparing Eqs. (3.141a) and (3.142) and noting that δP = 0, we find that there is no phase shiftbetween T TK and T TK in the presence of particle-hole symmetry, which is consistent withe.g. Fig. 4(d) of Ref. [43]. We also observe that the particle-hole symmetric normalized visibilityηv, defined in Eq. (3.114), has a characteristic logarithmic behavior at T TK :ηv =316pi2ln2 TTK. (3.144)On the other hand, to demonstrate the pi/2 phase shift due to Kondo physics, it is more usefulto consider the case of relatively strong particle-hole asymmetry κ ∼ j (T ) at T TK . The leadingcontribution to the conductance from potential scattering is O(K), and the leading contributionfrom the Kondo coupling is O(J2); therefore, κ ∼ j (T ) indicates that we may neglect the Kondocoupling altogether at temperature T . To the lowest order in potential scattering O(K), Eq. (3.80)applies; also, since T t, the thermal averging in Eq. (3.79) becomes trivial. Using the relationbetween K and δP, Eq. (3.84), we finally obtainG(T,ϕ) = Gre f +2√Gre fG(0)d tanδP sin(ϕ+θ) . (3.145)Comparing Eqs. (3.141a) and (3.145), it becomes evident that transmission through the QD under-goes a pi/2+ δP phase shift from T TK to T TK as the Kondo correlations are switched on;see Fig. 3.13. We remark that the strongly particle-hole asymmetric case represents the situationwithout Kondo correlation whereas the particle-hole symmetric case does not. This is because inthe latter case the leading QD contribution to the conductance is O(J2), which is of Kondo originas we have discussed above Eq. (3.70).We can again make a direct comparison with Eq. (3.114). While θ itself is not necessarily pi/2,ϕt is experimentally observed with respect to its value with Kondo correlations turned off, so we100Figure 3.13: Low-temperature and high-temperature conductances G as functions of AB phaseϕ in the open long ABK ring with a particle-hole asymmetric QD, calculated withEqs. (3.141a) and (3.145). We assume TK t so that the thermal averaging inthe high temperature case is trivial. System parameters are: tx = 0.3t, kF = pi/3,dL = dR = dre f /2 = 100, tL,RJL = tL,RJQ = tL,RJR = t, and particle-hole symmetry breakingphase shift δP = 0.1. A phase shift of approximately pi/2 is clearly visible as the tem-perature is lowered and Kondo correlations become important.should define the reference value ϕ(0)t by e.g. comparing with Eq. (3.145):ϕ(0)t = θ −pi2. (3.146)Therefore, to O(T/TK)2 we readily obtain the following results for the T TK transmission phaseand the normalized visibility[112]:ϕt −ϕ(0)t =pi2+δP−(piTTK)2tanδP, (3.147)101ηv = 1−(piTTK)2 1cos2 δP. (3.148)These are in agreement with the |δP| 1, T = 0 and δP = 0, T TK results of Ref. [23], whichassumes ϕ(0)t = 0, i.e. the non-magnetic phase difference between the two paths is zero withoutKondo correlations. Note that in obtaining the T dependence in Eq. (3.148) it is crucial to includethe connected contribution to conductance.We stress that our O(T/TK)2 results for the transmission phase across the QD and the normalizedvisibility, Eqs. (3.147) and (3.148), are both exact in δP, which is non-universal and encompasses theeffects of all particle-hole symmetry breaking perturbation. In particular, the (T/TK)2 coefficientswere not reported previously.3.8 Conclusion and open questionsOne question we have not so far addressed is the low-energy physics in the small Kondo cloudregime, T TK and EV TK ; here EV is the energy scale below which the normalization factorV 2k controlling the Kondo temperature varies slowly, and EV & Econn. We assume EV and Econn areof the same order of magnitude, a condition satisfied by both the closed ring (EV ∼ Econn ∼ vF/L)and the open ring with non-resonant Y-junctions (EV ∼ Econn ∼ t). For temperatures above themesoscopic energy scale EV T TK , we are no longer able to eliminate the connected contri-bution. However, because T EV one can argue that physics associated with the energy scale EVis smeared out by thermal fluctuations, and the mesoscopic system behaves as a bulk system withparameters showing no mesoscopic fluctuations[71]. On the other hand, below the mesoscopic en-ergy scale T EV TK , since for T Econn our formalism predicts that the connected part canbe eliminated, the knowledge of the screening channel T-matrix in the single-particle sector alone isadequate for us to predict the conductance. Ref. [71] again offers an appealing hypothesis: the low-energy effective theory is again a FL theory, with the T-matrix governed by Kondo physics at shortrange∼O(LK) =O(vF/TK) and modulated by mesoscopic fluctuations at long range∼O(L). Thisscenario leads to a quasiparticle spectrum which is in qualitative agreement with slave boson meanfield theory[71]. The FL picture is often analyzed by a renormalized perturbation theory of the quasi-particles, where the bare parameters of the QD are replaced by renormalized values; in particular thelarge U between bare electrons is replaced by a small renormalized U˜ between quasiparticles[49].In the small Kondo cloud regime, we expect that the real space geometry in the renormalized per-turbation theory resembles that of the bare theory[116]; thus a perturbation theory calculation inU in our formalism is potentially useful in understanding the low-energy physics, as long as U isinterpreted as the effective U˜ . It will be interesting to test this EV TK picture, along with ourperturbative predictions on conductance in other parameter regimes in this chapter, against resultsobtained from the numerical RG algorithm[5, 43, 50].102There is also an issue regarding the assumption of a single-level QD in the Kondo regime. Toexperimentally detect the pi/2 phase shift in an AB interferometer, one typically sweeps the plungergate voltage on the QD, and monitors the phase shift between consecutive Coulomb blockade peaks.A pi/2 plateau should be observed at T . TK near the center of each Coulomb valley deep in thelocal moment regime, with an odd number of electrons on the QD[43, 123]. However, one needsto adopt a multi-level QD model to quantitatively reproduce the experimental results, in particularthe phase shift lineshape asymmetry relative to the center of a valley, and also possibly a phaselapse inside the valley[113, 123]. A generalization of the current formalism to the multi-level caseis necessary in order to quantify the influence of the interferometer on the measured transmissionphase shift through a realistic QD.Another natural open problem is the extension to the multi-channel Kondo physics. In ourgeneralized Anderson model, separation of screening and non-screening channels is achieved in asingle-level QD, and there is only one effective screening channel. Exotic physics emerges in thepresence of two or more screening channels, realizable in e.g. a many-QD system[3, 92, 101]. Inthe 2-channel Kondo effect with identical couplings to two channels, for example, the low-energyphysics is governed by a non-FL RG fixed point: at zero temperature a single particle scattered bythe impurity can only enter a many-body state, and there are no elastic single-particle scatteringevents[134]. Ref. [23] discusses the manifestations of the 2-channel Kondo physics in the two-pathinterferometer, but again makes the two-slit assumption without examining its validity. Therefore anextension of our approach to the multi-channel case will be useful to justify the two-slit assumptionin the open long ring and thus the 2-channel predictions of Ref. [23].To conclude this chapter, we generalized the method developed in Ref. [66] to calculate thelinear dc conductance tensor of a generic multi-terminal Anderson model with an interacting QD.The linear dc conductance of the system has a disconnected contribution of the Landauer form, anda connected contribution which is also a Fermi surface property. At temperatures low comparedto the mesoscopic energy scale below which the background S-matrix and the coupling site wavefunctions vary slowly, T Econn, the connected contribution can be approximately eliminated usingproperties of the conductance tensor; the elimination procedure physically corresponds to probingthe current response or applying the bias voltages in a particular manner. At temperatures highcompared to the Kondo temperature T TK this connected part is computed explicitly to O(J2),and found to be of the same order of magnitude as the disconnected part in the case of a particle-holesymmetric QD.With this method we scrutinize both closed and open long ABK ring models. We find modifica-tions to early results on the closed ring with a long reference arm of length L: the high-temperatureconductance at T TK should have qualitatively distinct behaviors for T vF/L and T vF/L.In the open ring we conclude that the two-path interferometer is realized when the arms on the ringhave weak transmission and weak reflection, and demonstrate the possibility to observe in this de-103vice the pi/2 phase shift due to Kondo physics, and the suppression of AB oscillation visibility dueto inelastic scattering.104Table 3.1: Different regimes of energy scales discussed in this chapter. T , TK and EV arerespectively the temperature, the Kondo temperature, and the energy scale over whichV 2k varies significantly. We also assume Econn ∼ EV , where Econn is the energy scale overwhich S and Γ vary significantly. For the low-temperature conductance in the small Kondocloud regime, see discussion in Section 3.8.Weak-coupling pertur-bation theory appliesTK depends on meso-scopic detailsConnected part elimi-nation possibleT TK EV Yes No NoT EV TK Yes Yes NoEV T TK Yes Yes YesFL perturbation theoryappliesTK depends on meso-scopic detailsConnected part elimi-nation possibleT TK EV Yes Yes YesT EV TK ? No YesEV T TK ? No No105Chapter 4ConclusionsIn this thesis, we have discussed the transport properties of two types of multi-terminal interactingsystems: junctions of interacting quantum wires attached to TLL leads, and closed and open longABK rings. In both cases, we obtain corrections to the non-interacting Landauer formula, arisingfrom interactions in the TLL leads and the QD in the Kondo regime respectively.In Chapter 2 we examine junctions of multiple interacting quantum wires, focusing on the casewithout resonances. The significance of these systems derives from their pervasiveness in quantumcircuits and also their demonstration of strongly-correlated TLL physics. Working in the fermioniclanguage, we treat the interaction in the wires and the leads as a perturbation. In the absence ofinteractions, the Hamiltonian is fully determined by the single-particle scattering S-matrix, whichdefines the behavior of scattering state wave functions away from the junction. When the inter-actions are introduced, we may continue to diagonalize the quadratic part of the Hamiltonian andcharacterize its eigenstates (the scattering basis) in terms of a single-particle S-matrix. This alsoallows us to represent the interactions in the scattering basis. Treating the S-matrix elements asrunning coupling constants, it is possible to perform a Wilsonian RG calculation that determines therenormalization of the S-matrix.If the quantum wires are attached to TLL leads, i.e. the interaction does not vanish even at x→∞,it is known from bosonization that the Landauer description of the conductance is inadequate. Todirectly investigate the linear dc conductance with TLL leads in the fermionic formalism, we employthe CS formulation of the RG, computing the conductance as a function of the renormalized S-matrixusing the Kubo formula, and demanding that the conductance be independent of the ultravioletcutoff.We confirm that the S-matrix RG equation from the Wilsonian RG is reproduced in the CS ap-proach at the first order in interaction, and generalize the RG equation to the RPA in a form inde-pendent of the number of leads to which the junction is connected. Meanwhile, the conductanceitself depends on whether there is a residual interaction in the leads. If the wires are attached to FL106leads, i.e. the interaction becomes negligible far from the junction, our calculations justify the for-mal use of the Landauer formula, with the renormalized S-matrix elements as its input parameters.On the other hand, if the wires are attached to TLL leads, we find an additional contribution to theconductance, which depends on the interaction strength in the leads. This is true both at the firstorder in interaction and in the RPA; in particular, introducing a “contact resistance” between an FLlead and a TLL lead, we are able to justify the heuristic relation between the FL conductance and theTLL conductance, previously proposed in the context of bosonization. Our calculations recover theconductance of a single quantum wire connected to either FL leads or TLL leads, and furthermoreprovide an explanation for the conductance at the M fixed point of a spinless Y-junction connectedto TLL leads, which was obtained numerically through density matrix renormalization group.In Chapter 3 we discuss long ABK rings, where the interaction is assumed to be limited to theQD. Our motivation is rooted in the prospect of probing the Kondo physics in mesoscopic structures,such as the elusive Kondo screening cloud and the famous pi/2 phase shift associated with scatteringby the Kondo singlet. We introduce a very generic multi-terminal single-impurity Anderson model,with a background S-matrix and a coupling site wave function matrix as its input parameters. Ourmodel is not only suitable for describing closed and long ABK rings, but also capable of reproducingmany previously studied single-QD devices, including the QD embedded between source and drainleads, the QD embedded in a finite-length quantum wire, as well as the short ABK ring.We first rewrite the model in terms of non-screening channels and a single screening channel,where only the screening channel interacts with the Anderson impurity, and the non-screening chan-nels are merely free electrons. In the local moment regime, a Schrieffer-Wolff transformation leadsto a Kondo model for the screening channel; it contains a Kondo coupling term and a potential scat-tering term, both of which may be strongly momentum-dependent. This momentum dependencecan result in two different parameter regimes for the Kondo temperature TK : the small Kondo cloudregime TK EV and the large Kondo cloud regime TK EV , where EV is the characteristic energyscale of the function that controls the renormalization of the Kondo coupling.Turning to the linear dc conductance, we find that as in the short ABK ring case, the linear dcconductance consists of two parts: a “disconnected part” of the Landauer form coming from bothscreening and non-screening channels, and a “connected part” for the interacting screening channelonly. The disconnected contribution involves the screening channel single-particle T-matrix, and atlow temperatures can be intuitively understood as the result of imposing a QD-induced phase shift onthe screening channel. The connected contribution is shown to be a Fermi surface property (as is thedisconnected contribution), and can be eliminated at temperatures low relative to the characteristicenergy scale of the background S-matrix and the coupling site wave function matrix. We are alsoable to compute the linear dc conductance explicitly when perturbation theories apply; these includethe weak-coupling perturbation theory valid at high temperatures T TK , and the FL perturbationtheory valid in the large cloud regime at low temperatures T TK .107Applying our formalism to closed long ABK rings, where the electric current is conserved alongeach ring arm, we reproduce earlier results on the AB flux dependence of TK in both the small cloudregime and the large cloud regime. We also find that at high temperatures T TK , the conductanceshows qualitatively different flux and Fermi energy dependences for temperatures higher than andlower than the characteristic energy scale vF/L, where vF is the Fermi velocity of the conductionband and L is the circumference of the ring. On the other hand, in an open long ABK ring whereelectrons may leak out of the ring from side leads, we find that the model simulates an AB inter-ferometer which inherits the TK of the QD when it is embedded directly between source and drainleads, provided the lossy ring arms have both small transmission and small reflection. Furthermore,as we turn on Kondo correlations on the QD by changing the temperature from well above the Kondotemperature to well below, the AB-flux-dependent conductance of the interferometer is predicted totruthfully reflect the pi/2 Kondo phase shift.108Bibliography[1] I. Affleck. Conformal field theory approach to the kondo effect. 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The fixed pointsof these equations are then listed and their stability analyzed, for comparison with the bosonizationresults in Section 2.2[110]. For simplicity, unless otherwise stated, we assume throughout this ap-pendix that the interactions in the wires and leads are uniform and identical, αn (x) = αn for anyn.A.1 First order in interactionA.1.1 2-lead junctionFor the transmission amplitude S12, Eq. (2.72) becomes− dS12d lnD=−12(α1W11+α2W22)S12. (A.1)122In the 2-lead junction, unitarity implies W11 =W22 = 1−W12. Therefore, we have the followingequation for W12 ≡ |S12|2,− dW12d lnD=−(α1+α2)W12 (1−W12) . (A.2)In the vicinity of the complete reflection fixed point N (W12 = 0), linearizing Eq. (A.2), we find−dW12/d lnD≈−(α1+α2)W12; thus the N fixed point has a scaling exponent for the conductance−(α1+α2), and is stable if −(α1+α2)< 0 and unstable if −(α1+α2)> 0. Similarly, the perfecttransmission fixed point D (W12 = 1) has a conductance scaling exponent α1 +α2, and is stable ifα1+α2 < 0 and unstable if α1+α2 > 0.A.1.2 Y-junctionFor a Y-junction, from Eq. (2.72)− dS12d lnD=−12(α1W11S12+α2W22S12+α3S13S∗33S32) . (A.3)To reduce this to an equation involving W only, we need to relate the product S∗12S13S∗33S32 to W .This is achieved by taking advantage of unitarity of the S-matrix:S∗12S13S∗33S32+ c.c. =W23W22−W13W12−W33W32. (A.4)Thus, the RG equation obeyed by W12 takes the form− dW12d lnD=−(α1W11+α2W22)W12− 12α3 (W23W22−W13W12−W33W32) . (A.5)Unitarity dictates that there are only 4 independent matrix elements ofW . Following Refs. [13, 15],we parametrize the W matrix by four real numbers (a,b,c, c¯) as follows:W =16 2+3a+b−√3(c+ c¯) 2−3a+b−√3(c− c¯) 2(1−b+√3c)2−3a+b+√3(c− c¯) 2+3a+b+√3(c+ c¯) 2(1−b−√3c)2(1−b+√3c¯) 2(1−b−√3c¯) 2(1+2b) . (A.6)In the presence of time-reversal symmetry, c= c¯. If wires 1 and 2 are symmetrically coupled to thejunction, c=−c¯; we further find a= b if Z3 symmetry exists for the non-interacting system.Eq. (2.72) and Eq. (A.6) now lead to a closed set of equations for a, b, c and c¯,123− dad lnD=112[(α1+α2)(3+a+3b−6a2−ab− cc¯)− (α1−α2)√3(1−2a)(c+ c¯)]+13α3 (a−ab− cc¯) , (A.7a)− dbd lnD=112[(α1+α2)(1+3a+b−2b2−3ab−3cc¯)+(α1−α2)√3(1+2b)(c+ c¯)]+13α3(1+b−2b2) , (A.7b)− dcd lnD=112[−(α1+α2)(c(1+2b+6a)−3c¯)+√3(α1−α2)(−1−a+b+ab+2c2+ cc¯)]− 13α3 (1+2b)c, (A.7c)− dc¯d lnD=112[−(α1+α2)(c¯(1+2b+6a)−3c)+√3(α1−α2)(−1−a+b+ab+2c¯2+ cc¯)]− 13α3 (1+2b) c¯. (A.7d)After finding a fixed point (a0,b0,c0, c¯0) of Eq. (A.7d), we can again linearize the equations[13,15] by expanding in terms of small deviations from the fixed point, x≡ (a−a0,b−b0,c− c0, c¯− c¯0):− dxd lnD= Mx. (A.8)The 4×4 matrix M have eigenvalues λl with corresponding left eigenvectors vl , l = 1, 2, 3, 4. Foran RG flow starting in the vicinity of the fixed point in question, the solution to Eq. (A.8) takes theformx(D) =4∑j=1Cl(D0D)λlvl , (A.9)whereC j are constants and D0 is the ultraviolet cutoff. λl thus controls the stability of the fixed point:vl is a stable scaling direction if λl < 0, and an unstable one if λl > 0. For a junction attached to FLleads, replacing D by the temperature T in Eq. (A.9), we will find the low temperature conductanceat a stable fixed point with all λl < 0, or the high temperature conductance at a completely unstablefixed point with all λl > 0; thus λl are the scaling exponents of the conductance. (λl are generallynot the same with the S-matrix scaling exponents discussed in Ref. [70].)124In the following we list λl for the first order fixed points and discuss their physical meanings.N fixed point: (a0,b0,c0, c¯0) = (1,1,0,0); λN,1 = −(α2+α3), λN,2 = −(α3+α1), λN,3 =−(α1+α2) and λN,4 =−(α1+α2+α3), with eigenvectors vN,1 =(1,−1/3,0,0), vN,2 =(0,−2/√3,1,1),vN,3 =(0,2/√3,1,1)and vN,4 = (0,0,1,−1) respectively. λN,1 corresponds to the process wherea single electron tunnels between wires 2 and 3; thus we know from the 2-lead junction problemthat it controls the RG flow between N and A1. (By a flow “between” two fixed points, we referto a flow which, starting sufficiently close to either of the two fixed points, can come into arbitraryproximity to the other.) Similarly, depending on the attractive or repulsive nature of the interactions,λN,2 and λN,3 control the RG flow from/to A2 and A3, respectively. The flow between N and M isjointly controlled by λN,1, λN,2 and λN,3. On the other hand, along the direction of vN,4, c=−c¯ 6= 0;thus vN,4 represents a chiral perturbation, and λN,4 controls the flows from/to χ± which are the onlyfixed points breaking time-reversal symmetry at the first order.We note that a, b, c and c¯ are subject to additional constraints imposed by the S-matrix unitarity[13,15]. By considering physically allowed S-matrices, it can be shown that CN,4 = 0 in Eq. (A.9) (i.e.vN,4 is not allowed) unless CN,1, CN,2 and CN,3 are all nonzero. For this reason, λN,4 has not beenregarded as an independent conductance scaling exponent in Refs. [13, 15]. Intuitively, this can alsobe understood from the fact that the breaking of time-reversal symmetry (vN,4) requires the pres-ence of single electron tunneling between all three wires (vN,1, vN,2 and vN,3), so that a magneticflux threaded into the junction cannot be trivially gauged away. It should be also mentioned thatλN,4 is never the leading scaling exponent in either the high temperature or the low temperaturelimit. For instance, if we assume λN,4 is the leading exponent at low temperatures, then λN,4 mustbe greater than all remaining λ ’s, and we find all α’s are negative and N is unstable in all directions,which contradicts our assumption.A j fixed points: At A1,2, (a0,b0,c0, c¯0) =(1/2,−1/2,∓√3/2,∓√3/2); at A3, (a0,b0,c0, c¯0) =(−1,1,0,0). We now focus on A3 where wire 3 is decoupled, and wires 1 and 2 are perfectlyconnected.At A3, λA3,1 = λA3,2 =−α3, λA3,3 = α1+α2 and λA3,4 = (α1+α2−2α3)/2, with eigenvectorsvA3,1 = (0,0,1,−1) , (A.10a)vA3,2 =(0,−(2/√3)(α1+α2) ,α1−α2,α1−α2), (A.10b)vA3,3 =(−√3(α1+α2+2α3) ,−(α1+α2+2α3)/√3,α1−α2,α1−α2), (A.10c)125vA3,4 = (0,0,1,1) (A.10d)respectively. Again λA3,1 corresponds to the single-electron tunneling between wires 2 and 3, λA3,2to that between 3 and 1, and λA3,3 to that between 1 and 2. As before λA3,3 controls the flow from/toN, and we see from the eigenvector vA3,1 that λA3,1 = λA3,2 controls the flows from/to χ±.In the special case of 1-2 symmetric interaction, α1 = α2, it is clear that vA3,4 is the only scalingdirection breaking the 1-2 symmetry. Now the flows from/to A1 and A2 are controlled by λA3,4 alone,and the flow from/to M is controlled by λA3,1 = λA3,2 and λA3,3 together but not λA3,4.We observe that vA3,1 and vA3,4 break the time-reversal symmetry and the 1-2 symmetry of thejunction respectively without changing a or b. This is again forbidden by unitarity[13, 15] at A3, i.e.CA3,1 =CA3,4 = 0 in Eq. (A.9) unless either CA3,2 or CA3,3 is nonzero. Physically it reflects the factthat any time-reversal asymmetry or 1-2 asymmetry at the junction should introduce perturbationsthat interrupt the perfectly connected wire. λA3,4 is therefore not treated as a scaling exponent inRefs. [13, 15].χ± fixed points: At χ±, (a0,b0,c0, c¯0) =(−1/2,−1/2,±√3/2,∓√3/2); λχ±,1 = α1, λχ±,2 =α2, λχ±,3 = α3 and λχ±,4 = (α1+α2+α3)/2, with eigenvectorsvχ±,1 =(−√3α1,−(α1+2(α2+α3))/√3,(α1− (α2+α3)∓ (α2+α3)) ,(α1− (α2+α3)± (α2+α3))) , (A.11a)vχ±,2 =(√3α2,(α2+2(α1+α3))/√3,(α2− (α1+α3)± (α1+α3)) ,(α2− (α1+α3)∓ (α1+α3))) , (A.11b)vχ±,3 =(√3(α1+α2) ,−(α1+α2−4α3)/√3,±(α1+α2) ,∓(α1+α2)), (A.11c)vχ±,4 =(√3,√3,±1,∓1)(A.11d)respectively. λχ±,1 corresponds to the single-electron tunneling between wires 2 and 3, and controlsthe flows from/to A1; similarly λχ±,2 and λχ±,3 controls the flows from/to A2 and A3 respectively.λχ±,4 controls the RG flows from/to N and M.The scaling direction vχ±,4 is forbidden by unitarity at χ± (Cχ±,4 = 0 in Eq. (A.9) unlessCχ±,1, Cχ±,2 and Cχ±,3 are all nonzero), so once more λχ±,4 is not treated as a scaling exponent126in Refs. [13, 15]. In terms of the mapping to the dissipative Hofstadter model[93], χ± correspondto the localized phase of a quantum Brownian particle subject to a magnetic field and a triangular-lattice potential; λχ±,1, λχ±,2 and λχ±,3 are then due to instantons tunneling back and forth betweenthe three inequivalent nearest neighbor pairs of potential minima, while λχ±,4 arises from instantonstunneling along the edges of elementary triangles formed by the potential minima. It is thereforereasonable that vχ±,4 is allowed only if there exist deviations from χ± along all three remainingscaling directions vχ±,1, vχ±,2 and vχ±,3.M fixed point: Due to time-reversal symmetry c0 = c¯0 = 0; it is straightforward to find a0 andb0 from Eqs. (2.132) and (A.6). There are 4 scaling exponents at M:λM,1 =− α1α2α3α1α2+α2α3+α3α1 , (A.12a)λM,2 =α21 (α2+α3)+α22 (α3+α1)+α23 (α1+α2)α1α2+α2α3+α3α1, (A.12b)λM,3(4) =14(α1α2+α2α3+α3α1)−1 {(α1+α2)(α2+α3)(α3+α1)±√(α1+α2)2 (α2+α3)2 (α3+α1)2−8α1α2α3 (α1+α2)(α2+α3)(α3+α1)}.(A.12c)Whenever M exists (see the conditions below Eq. (2.132)), λM,3, λM,4 are both real. Note that,unlike the situations at N, A j and χ±, at M the four scaling directions are fully independent ofeach other. In the special case of Z3 symmetric interactions (α j = α), λM,1 = −α/3, λM,2 = 2α ,λM,3 = λM,4 = 2α/3, in agreement with the conductance predictions of Ref. [70]. The correspondingleft eigenvectors arevM,1 = (0,0,1,−1) , (A.13a)vM,2 =(−√3(α1+α2) ,−(α1+α2+4α3)/√3,α1−α2,α1−α2). (A.13b)We can infer from the form of vM,1 that λM,1 controls the flows from/to χ±. vM,3 and vM,4 aretoo complicated to be given here in general.In the case of 1-2 symmetric interactions (α1 = α2), significant simplifications occur:λM,1 =− α1α3α1+2α3 , λM,2 =2(α21 +α1α3+α23)α1+2α3; (A.14a)127λM,3 =α3 (α1+α3)α1+2α3, λM,4 =α1 (α1+α3)α1+2α3; (A.14b)vM,3 = (3α3 (α1+α3) ,−2α1 (α1+2α3) ,0,0) , (A.15a)vM,4 = (0,0,1,1) . (A.15b)In this case we find sgnλM,2 = sgnλM,3 = sgnα3 and sgnλM,4 = sgnα1. λM,2 and λM,3 control theflows from/to A3 and N, and λM,4 controls the flows from/to A1 and A2.We are now in a position to give the directions of RG flows based on the local scaling exponents.The results are summarized below.1. The flow between N and A3 is toward N if α1+α2 > 0, and toward A3 if α1+α2 < 0;2. The flows between N and χ± are toward N if α1+α2+α3 > 0, and toward χ± if α1+α2+α3 < 0;3. If α1 = α2, the flows between A3 and A1,2 are toward A3 if α1 < α3, and toward A1 and A2 ifα1 > α3;4. The flows between A3 and χ± are toward A3 if α3 > 0, and toward χ± if α3 < 0.In addition, if the non-geometrical fixed point M exists:5. The flows between M and χ± are toward M if λM,1 < 0, and toward χ± if λM,1 > 0;6. If α1 = α2, the flows between M and A1,2 are toward M if α1 < 0, and toward A1 and A2 ifα1 > 0.7. If α1 = α2, the flow between M and A3 is toward M if α3 < 0, and toward A3 if α3 > 0.8. If α1 = α2, the flow between M and N is toward M if α3 < 0, and toward N if α3 > 0.A.2 RPAA.2.1 2-lead junctionCalculating Eq. (2.107) explicitly we find the cutoff-dependent RPA interactionΠ j j′ =(K−11 −1)(K−12 −1)(2δ j j′−1)W12+2(K−1j −1)δ j j′2+(K−11 +K−12 −2)W12, (A.16)and the RPA RG equation for S12,− dS12d lnD=− 2(1−W12)γ−1+2W12 S12, (A.17)128whereγ =K−11 +K−12 +2K−11 +K−12 −2; (A.18)or, in terms of W12,− dW12d lnD=−4W12 (1−W12)γ−1+2W12 , (A.19)in agreement with the RPA RG equation for conductance in Refs. [11, 12, 14].As with the first order equation Eq. (A.2), Eq. (A.19) has two fixed points, the N fixed pointW12 = 0 and the D fixed point W12 = 1. In this case the N fixed point has a conductance scaling ex-ponent of −4/(γ−1) = 2− 2/Ke, and the D fixed point has a scaling exponent of 4/(γ+1) =2− 2Ke, where Ke = 2/(K−11 +K−12). Both exponents conform to the predictions of bosonicmethods[11, 12, 14] as has been verified in Section 2.2.A.2.2 Y-junctionStarting from Eq. (2.106) and following the same prescription which transforms Eq. (2.72) toEq. (A.7d), we find the RG equations obeyed by a, b, c and c¯ in the RPA:− dad lnD= Q−1A{2[(1−b)(1+a+b−3a2)− (1+3a)cc¯+ c2+ c¯2](Q1+Q2+Q3−3)+(3+a+3b−6a2−ab− cc¯)(Q1+Q2−2)(Q3−1)+√3(1−2a)(c+ c¯)(Q1−Q2)(Q3−1)+4 [a(1−b)− cc¯] (Q1−1)(Q2−1)}, (A.20a)− dbd lnD= Q−1A {2 [(1−b)(1+2b−3ab)−3(1+b)cc¯] (Q1+Q2+Q3−3)+[(1−b)(1+3a+2b)−3cc¯] (Q1+Q2−2)(Q3−1)−√3(1+2b)(c+ c¯)(Q1−Q2)(Q3−1)+4(1−b)(1+2b)(Q1−1)(Q2−1)},(A.20b)− dcd lnD= Q−1A{2[−c(1+2b−3ab)+ c¯(2−2b−3c2)](Q1+Q2+Q3−3)+[−c(1+6a+2b)+3c¯] (Q1+Q2−2)(Q3−1)+√3[(1+a)(1−b)−2c2− cc¯](Q1−Q2)(Q3−1)−4c(1+2b)(Q1−1)(Q2−1)} ,(A.20c)129− dc¯d lnD= Q−1A{2[−c¯(1+2b−3ab)+ c(2−2b−3c¯2)](Q1+Q2+Q3−3)+[−c¯(1+6a+2b)+3c] (Q1+Q2−2)(Q3−1)+√3[(1+a)(1−b)−2c¯2− cc¯](Q1−Q2)(Q3−1)−4c¯(1+2b)(Q1−1)(Q2−1)} ,(A.20d)whereQA = 2 [(1−a)(1−b)− cc¯] (Q1+Q2+Q3−3)+4(1−b)(Q1−1)(Q2−1)+(4−3a−b)(Q1+Q2−2)(Q3−1)−√3(c+ c¯)(Q1−Q2)(Q3−1)+6(Q1−1)(Q2−1)(Q3−1) . (A.20e)In the special case of 1-2 symmetric interaction K1 =K2, Eq. (A.20e) is reduced to the RG equationsof Ref. [15]. The fully 1-2 symmetric case with both K1 = K2 and c = −c¯ has been extensivelyanalyzed there.Eq. (A.20e) may once again be linearized to extract the scaling exponents. We first enumeratethe scaling exponents at the geometrical fixed points N, A j and χ±; their physical meanings areidentical to their first order counterparts, and have been explained in detail in Appendix A.1.N fixed point: At N, λN,1 = 2−K−12 −K−13 , λN,2 = 2−K−13 −K−11 , λN,3 = 2−K−11 −K−12 , andλN,4 = 3−K−11 −K−12 −K−13 .A j fixed points: At e.g. A3, λA3,1 = λA3,2 = 2−K−13 − (1+K1K2)/(K1+K2), λA3,3 = 2−4K1K2/(K1+K2), and λA3,4 = 3−K−13 − (1+3K1K2)/(K1+K2).χ± fixed points: At χ±,λχ , j = 2− 4(K1+K2+K3−K j)K jK1+K2+K3+K1K2K3 , j = 1, 2, 3; (A.21a)λχ ,4 = 3− 4(K1K2+K2K3+K3K1)K1+K2+K3+K1K2K3 . (A.21b)All scaling exponents above are in agreement with predictions of bosonization[51].As for the non-geometrical fixed points, on account of mathematical simplicity we followRef. [15] and only give their positions and scaling exponents in the fully 1-2 symmetric case,K1 = K2 and c0 =−c¯0. We introduce the quantitiesζ =3Q1Q3−Q1−2Q32Q1+Q3−3 , (A.22)130τ0 =√1+Q21+2ζ ; (A.23)where Q1 and Q3 are related to K1 and K3 by Eq. (2.108). ζ and τ0 are identical to Q1 and τ inRef. [15] respectively.M and Q fixed points: At these two fixed points(a0,b0,c0) =(13(Q1∓ τ0 sgnQ1) , 16((|Q1|∓ τ0)2−3),0), (A.24)where the upper signs are for M and the lower signs for Q. The two fixed points merge when τ0 = 0,or in terms of the Luttinger parameters,K3 =2K1(K21 −K1+1)(K1+1)(2K1−1) . (A.25)We note that Q also exists for Z3 symmetric interactions but its W matrix remains Z3 asymmetric;thus Q cannot be reached when the RG flow starts from a Z3 symmetric S-matrix. The M fixed pointagain corresponds to the maximally open S-matrix in the Z3 symmetric case, while the Q fixed pointonly appears when the interactions are sufficiently strongly attractive[13, 15]. The conditions for Mand Q to appear are τ20 = 1+Q21+2ζ ≥ 0 and ||Q1|∓ τ0| ≤ 3 (the latter is due to S-matrix unitarity),and for Z3 symmetric interactions Q only starts to exist when the Luttinger parameter of all threewires K ≥ 3. Both M and Q are time-reversal symmetric.For attractive interaction in wire 1 (K1 > 1), the 4 scaling exponents at either M or Q areλM(Q),1 =−3(Q1± τ0+3)(Q21+2Q1− τ20 ±2τ0−3)2(Q1± τ0)(2Q1∓ τ0)2, (A.26a)λM(Q),2 =−3((Q1± τ0)2+3)(Q1± τ0)(2Q1∓ τ0) , (A.26b)λM(Q),3 =−3(Q1± τ0−3)(Q21−2Q1− τ20 ∓2τ0−3)2(Q1± τ0)(2Q1∓ τ0)2, (A.26c)λM(Q),4 =∓3τ0((Q1± τ0)2−9)2(Q1± τ0)(2Q1∓ τ0)2; (A.26d)again the upper signs are for M and the lower signs for Q. For repulsive interaction in wire 1(K1 < 1), the lower signs should be taken to obtain the scaling exponents at M. When expanded tothe first order in α1 and α3, Eq. (A.26d) agrees with Eq. (A.12c).C± fixed points: As with the Q fixed point, the non-geometrical chiral fixed points C± onlyexist when the interaction is strongly attractive. At these fixed points131(a0,b0,c0) =(16(2Q1 (ζ +2)−ζ 2+1),16(ζ 2+4ζ +1),±(ζ −1)6√3− (2Q1−ζ )(2+ζ )).(A.27)The conditions for C± to appear are −5≤ ζ ≤ 1 and0≤ 3− (2Q1−ζ )(2+ζ )≤ 13 (5+ζ )2 . (A.28)In the Z3 symmetric Y-junction, these conditions are satisfied when the Luttinger parameter of allthree wires K ≥ 2. C± and χ± merge when ζ = −2, or in terms of the Luttinger parameters,K−11 +2K−13 = 1.There are again 4 scaling exponents at C±,λC,1 =12Q1(Q1−1)(3Q1−ζ −2) +12(Q1−1)(ζ −1) , (A.29a)λC,2 =− 123Q1−ζ −2 −12ζ −1 −6, (A.29b)λC,3(4) =−3(3Q1−ζ +4)(ζ +5)2(3Q1−ζ −2)(ζ −1)(ζ +1ζ +5± 13√1+8(ζ +2)(3Q1−ζ −2)(ζ −1)(3Q1−ζ +4)). (A.29c)We conclude this appendix with a discussion of the RG flows in the Y-junction.In the generic Z3 asymmetric case, for simplicity we only focus on the RG flows when K1, K2and K3 are all close to unity, so that the only allowed non-geometrical fixed point is M. We focuson the flows between the geometrical fixed points. The results are listed below.1. The flow between N and A3 is toward N if K−11 +K−12 > 2, and toward A3 if K−11 +K−12 < 2;2. The flows between N and χ± are toward N if K−11 +K−12 +K−13 > 3 and λχ ,4 > 0, and towardχ± if K−11 +K−12 +K−13 < 3 and λχ ,4 < 0.3. The flows between A3 and χ± are toward A3 if K−13 +(1+K1K2)/(K1+K2)> 2, and towardχ± if K−13 +(1+K1K2)/(K1+K2)< 2.Finally, following Ref. [15] we detail the RG flows in a fully Z3 symmetric junction with Lut-tinger parameter K for all three wires. Only the fixed points consistent with Z3 symmetry, namelyN, χ±, M, and C±, need to be considered.When 0 < K < 1, N is the most stable fixed point, M is stable against chiral perturbations butotherwise unstable, and χ± are completely unstable; C± do not exist. The flows are from χ± to Mor N, and from M to N.When 1< K < 2, χ± are the most stable fixed points, M is stable against time-reversal symmet-132ric perturbations and unstable against chiral perturbations, and N is completely unstable; C± do notexist. The flows are from N to M or χ±, and from M to χ±.When 2<K < 3, χ± remain stable, N remains completely unstable, while M becomes fully sta-ble. C± emerge as the unstable fixed points separating χ± and M, approaching χ± as K approaches3. The flows are from N to M, χ± or C±, and from C± to M or χ±.When K > 3, χ± become completely unstable, N remains completely unstable and M remainsfully stable. C± remain unstable, moving toward a0 = b0 =−1/3 and c0 =−c¯0 =±2/3 as K→∞.The flows are from N to M or C±, from χ± to M or C±, and from C± to M.133Appendix BDetails of the disconnected contributionIn this appendix we present the detailed derivation of Eq. (3.45) [or equivalently Eq. (3.47a)] fromEq. (3.38)[112]. The calculations are similar to those in Appendix B of Ref. [66], but an importantdifference is that here we cannot simply take the δ -function part and neglect the principal valuepart in Eq. (3.38). Instead, most of the momentum integrals are evaluated by means of contourintegration.From Eq. (3.41)−2ImGRk2q1 (ω) = (2pi)2 δ (k2−q1)δ (ω− εk2)+ iτψ× [gRk2 (ω)Vk2GRdd (ω)Vq1gRq1 (ω)−gAk2 (ω)Vk2GAdd (ω)Vq1gAq1 (ω)] . (B.1)We denote the three terms above as 0, R and A respectively. Inserting into Eq. (3.38), we find 3types of contributions to the disconnected part:G′Dj j′ (Ω) = G′Dj j′,00 (Ω)+[G′Dj j′,0R (Ω)+G′Dj j′,0A (Ω)+G′Dj j′,R0 (Ω)+G′Dj j′,A0 (Ω)]+[G′Dj j′,RA (Ω)+G′Dj j′,RR (Ω)+G′Dj j′,AR (Ω)+G′Dj j′,AA (Ω)]; (B.2)The 00 term is the background transmission, the first pair of square brackets is linear in the T-matrixof the screening channel, and the second pair of square brackets is quadratic in the T-matrix.Due to the multiplying factor ofΩ in Eq. (3.31), O(1/Ω) terms in G′Dj j′ contribute to the linear dcconductance, while O(1) and other terms which are regular in the dc limit Ω→ 0 do not contribute.(We can check explicitly that there are no O(1/Ω2)or higher-order divergences.) Therefore, in thedc limit we are only interested in the O(1/Ω) part of G′Dj j′ .134B.1 Properties of the S-matrix and the wave functionsBefore actually doing the calculations it is useful to examine the properties of the background S-matrix and the wave functions in our tight-binding model, since we rely on these properties totransform the momentum integrals into contour integrals and evaluate them.First consider the analytic continuation k→−k. The wave function “incident” from lead j atmomentum −k takes the following form on lead j′ [cf. Eq. (3.15a)],χ j,−k(j′,n)= δ j j′eikn+S j′ j (−k)e−ikn; (B.3)and on coupling site r,χ j,−k (r) = Γr j (−k) .This wave function should be a linear combination of the scattering state wave functions at momen-tum k which form a complete basis. The linear coefficients are obtained from S-matrix unitarity:χ j,−k(j′,n)=∑j′′S∗j j′′ (k)χ j′′,k(j′,n)= δ j j′eikn+S∗j j′ (k)e−ikn, (B.4)and the same coefficients apply to the coupling sites:χ j,−k (r) =∑j′′S∗j j′′ (k)Γr j′′ (k) .HenceS (−k) = S† (k) , (B.5)Γ(−k) = Γ(k)S† (k) . (B.6)Eq. (B.5) is known as the Hermitian analyticity of the S-matrix[20].Another useful property is the location of poles of S (k) ≡ S(z= eik) on the z complex plane.Our analysis closely follows Ref. [96] which deals with the case of quadratic dispersion.Consider one pole of the S-matrix k≡ k1+ ik2, where for certain values of j and j′,∣∣S j′ j (k)∣∣→∞. In the scattering state∣∣q j,k〉≡ q†j,k |0〉, where |0〉 is the Fermi sea ground state, the incident com-ponent of the wave function at momentum k becomes negligible relative to the scattered component.Therefore, the time-dependent wave function on lead j′ at site n readsχ j,k(j′,n, t¯)≈ S j′ j (k)eikne−iεk t¯ = S j′ j (k)eikne2itt¯ cosk1 coshk2e2tt¯ sink1 sinhk2 . (B.7)This expression is valid for any j′ where∣∣S j′ j (k)∣∣ is divergent; for other j′ the wave function is135negligible.We define the “junction area” to include any tight-binding site that is not part of a lead, togetherwith the 0th site of each lead. The total probability of the electron being inside the junction area,N (t¯), obeys the probability continuity equationddt¯N (t¯) = it∑j′(c†j′,0c j′,1− c†j′,1c j′,0)(t¯) , (B.8)where the right-hand side is the current operator between site 0 and site 1 of lead j′, summed overall leads. Taking the expectation value in the state∣∣q j,k〉, we find− (4t sink1 sinhk2)C j (k)e4tt¯ sink1 sinhk2 =∑j′(2t sink1)e−k2∣∣S j′ j (k)∣∣2 e4tt¯ sink1 sinhk2 . (B.9)For the left-hand side we have used the form of the time evolution e−iεk t¯ , andC j (k) is a positive time-independent constant proportional to the total probability in the junction area; C j (k) is divergentwhenever∣∣S j′ j (k)∣∣ is divergent. For the right-hand side, we have used Eq. (B.7) at n= 0 and n= 1;the summation is over any j′ where∣∣S j′ j (k)∣∣ is divergent.Eq. (B.9) implies that either sink1 = 0, in which case k1 = 0 or pi; or sinhk2 < 0, in which case∣∣ei(k1+ik2)∣∣ > 1. The poles of S (k) on the z = eik plane are therefore either outside the unit circleor located on the real axis. For the models we study in this chapter, the poles of S (k) and those ofΓ(k)/(sink) coincide; in other words, the poles of S (k) and Γ(k)/(sink) on the z = eik plane areeither outside the unit circle or on the real axis.We mention that similar results apply in the theory with a reduced bandwidth and a linearizeddispersion in the leads. Eqs. (B.5) and (B.6) continue to hold; on the other hand, the probabilitycurrent is proportional to vF instead of 2t sink1, and all poles of S (k) and Γ(k)/(sink) are locatedin the lower half of the k plane.B.2 Background transmissionThis part is independent of the QD and the result should be the famous Landauer formula:G′Dj j′,00 (Ω) = 2∫ pi0dk1dk2(2pi)2f (εk1)− f (εk2)εk1− εk2 +Ω+tr(M jk1k2Mj′k2k1). (B.10)Inserting Eq. (3.35), taking advantage of Eq. (B.5) and the unitarity of the U matrix, we find136G′Dj j′,00 (Ω) = 2∫ pi−pidk1dk2(2pi)2f (εk1)− f (εk2)εk1− εk2 +Ω+×{δ j j′11− ei(k1−k2+i0)11− ei(k2−k1+i0) +S∗j j′ (k1)δ j j′11− e−i(k1+k2−i0)11− ei(k2−k1+i0)+δ j j′S j j′ (k2)11− ei(k1+k2+i0)11− ei(k2−k1+i0) +S∗j j′ (k1)S j j′ (k2)1[1− ei(k2−k1+i0)]2}.(B.11)By residue theorem we can perform the k2 integral in the part proportional to f (εk1) and the k1integral in the part proportional to f (εk2). In the following we assume Ω > 0; the case Ω < 0 canbe dealt with similarly.We begin from the first term in curly brackets, which is proportional to δ j j′ . For the part pro-portional to f (εk1), making the substitution z2 = eik2 , and calculating the contour integral on thecounterclockwise unit circle, we find∫ pi−pidk22pif (εk1)εk1− εk2 +Ω+11− ei(k1−k2+iη)11− ei(k2−k1+iη)=f (εk1)2it sin p111− ei(k1−p1)11− ei(p1−k1) +f (εk1)Ω11− e−2η , (B.12a)where η → 0+. (η corresponds to the rate of switching on the bias voltage in Kubo formalism, sothe limit η → 0 should be taken before the dc limit Ω→ 0.) We have assumed εk1 +Ω≡ εp1 where0 ≤ p1 ≤ pi if p1 is real; the poles of the integrand inside the unit circle are then z2 = ei(p1+i0) andz2 = ei(k1+iη). At the band edges, 2t−Ω< εk1 < 2t, and p1 is purely imaginary; we can choose it tohave a positive imaginary part so the above expression remains valid. Similarly∫ pi−pidk12pif (εk2)εk1− εk2 +Ω+11− ei(k1−k2+iη)11− ei(k2−k1+iη)=f (εk2)2it sin p211− ei(−p2−k2)11− ei(p2+k2) +f (εk2)Ω11− e−2η , (B.12b)where εk2−Ω= εp2 , 0≤ p2≤ pi if p2 is real, or p2 =−i |p2| if p2 is purely imaginary. Now combinethe two parts. In the Ω→ 0 limit, p1→ k1 only for p1 real and k1 > 0, and p2→−k2 only for p2real and k2 < 0; the most divergent contribution is therefore137∫ pi−pidk1dk2(2pi)2f (εk1)− f (εk2)εk1− εk2 +Ω+11− ei(k1−k2+i0)11− ei(k2−k1+i0)=∫ 2t−Ω−2tdεk12pii[ f (εk1)− f (εk1 +Ω)]1(2t sink1)(2t sin p1)11− ei(k1−p1)11− ei(p1−k1) +O(1) .(B.13)We have substituted the dummy variables k2→ p1, p2→ k1, and noted that εp1 = εk1+Ω. Expandingvarious parts of the integrand in Ω→ 0 limit, we find∫ pi−pidk1dk2(2pi)2f (εk1)− f (εk2)εk1− εk2 +Ω+11− ei(k1−k2+i0)11− ei(k2−k1+i0)=12piiΩ∫ 2t−Ω−2tdεk1[− f ′ (εk1)]+O(1) . (B.14)The two terms in G′Dj j′,00 (Ω) which are linear in the S-matrix do not contribute any terms ofO(1/Ω) to G′Dj j′ : the difference of Fermi functions is proportional to Ω, but the denominators arealso O(Ω), unlike the case for the δ j j′ terms whose denominators are O(Ω2). This leaves us withthe term quadratic in the S-matrix, which can be similarly evaluated. For the part proportional tof (εk1),∫ pi−pidk22pif (εk1)εk1− εk2 +Ω+S∗j j′ (k1)S j j′ (k2)1[1− ei(k2−k1+i0)]2=f (εk1)2it sin p1S∗j j′ (k1)S j j′ (p1)1[1− ei(p1−k1)]2 + (contribution of poles of S j j′) ; (B.15a)the poles of S j j′ inside the unit circle (on the real axis) may contribute to the contour integral, butthese terms are regular in the Ω→ 0 limit and do not contribute to the dc conductance. Similarly∫ pi−pidk12pif (εk2)εk1− εk2 +Ω+S∗j j′ (k1)S j j′ (k2)1[1− ei(k2−k1+i0)]2=f (εk2)2it sin p2S∗j j′ (p2)S j j′ (k2)1[1− ei(k2−p2)]2 + (contribution of poles of S∗j j′) , (B.15b)Therefore138∫ pi−pidk1dk2(2pi)2f (εk1)− f (εk2)εk1− εk2 +Ω+S∗j j′ (k1)S j j′ (k2)1[1− ei(k2−k1+i0)]2=∫ 2t−Ω−2tdεk12piif (εk1)− f (εk1 +Ω)(2t sink1)(2t sin p1)S∗j j′ (k1)S j j′ (p1)1[1− ei(p1−k1)]2 +O(1)=− 12piiΩ∫ 2t−Ω−2tdεk1[− f ′ (εk1)]S∗j j′ (k1)S j j′ (p1)+O(1) . (B.16)From Eqs. (B.14) and (B.16), we conclude thatG′Dj j′,00 (Ω) =1piiΩ∫ 2t−Ω−2tdεk1[− f ′ (εk1)][δ j j′−S∗j j′ (k1)S j j′ (p1)]+O(1) ; (B.17)taking the Ω→ 0 limit, noting that p1→ k1, we recover the Landauer formula, Eq. (3.47b).B.3 Terms linear in T-matrixWe focus on G′Dj j′,0R+G′Dj j′,0A; the calculation of G′Dj j′,R0+G′Dj j′,A0 is analogous.G′Dj j′,0R (Ω)+G′Dj j′,0A (Ω)= 2∫ pi0dk1(2pi)2dq1dq2(2pi)2∫dωf (ω)− f (εq1)ω− εq1 +Ω+tr{M jk1q1Mj′q1q2(iτψ)× [gRq2 (ω)Vq2GRdd (ω)Vk1gRk1 (ω)−gAq2 (ω)Vq2GAdd (ω)Vk1gAk1 (ω)]} . (B.18)Using Eqs. (3.22), (3.35), (B.5) and (B.6), a huge simplification takes place:G′Dj j′,0R (Ω)= 2∫ pi−pidk1dq1dq2(2pi)4∫dωf (ω)− f (εq1)ω− εq1 +Ω+igRq2 (ω)GRdd (ω)gRk1 (ω)∑r1r2t∗r1tr2×Γr1 j (k1)[δ j j′11− ei(k1−q1+i0) +S j j′ (q1)11− ei(k1+q1+i0)]Γ∗r2 j′ (q2)11− ei(q1−q2+i0) , (B.19a)139G′Dj j′,0A (Ω)= 2∫ pi−pidk1dq1dq2(2pi)4∫dωf (ω)− f (εq1)ω− εq1 +Ω+(−i)gAq2 (ω)GAdd (ω)gAk1 (ω)∑r1r2t∗r1tr2×Γr1 j (k1)[δ j j′11− ei(k1−q1+i0) +S j j′ (q1)11− ei(k1+q1+i0)]Γ∗r2 j′ (q2)11− ei(q1−q2+i0) . (B.19b)Writing ω = εk, where 0 ≤ k ≤ pi or k = i |k|, we are now free to do the k1 and q2 integrals. Thepoles of Γ(k1) and Γ∗ (q2) are again not important in the dc limit:G′Dj j′,0R (Ω)= 2∫ pi−pidq1(2pi)2∫dεkf (εk)− f (εq1)εk− εq1 +Ω+i∑r1r2t∗r1tr212it sinkGRdd (εk)12it sink×Γr1 j (k)[δ j j′11− ei(k−q1+i0) +S j j′ (q1)11− ei(k+q1+i0)]Γ∗r2 j′ (−k)11− ei(q1+k+i0) +O(1) ,(B.20a)G′Dj j′,0A (Ω)= 2∫ pi−pidq1(2pi)2∫dεkf (εk)− f (εq1)εk− εq1 +Ω+(−i)∑r1r2t∗r1tr21−2it sinkGAdd (εk)1−2it sink×Γr1 j (−k)[δ j j′11− ei(−k−q1+i0) +S j j′ (q1)11− ei(−k+q1+i0)]Γ∗r2 j′ (k)11− ei(q1−k+i0) +O(1) .(B.20b)Now do the εk and q1 integrals. The δ j j′ terms are regular in the dc limit, so we only need to keep theS j j′ terms. In the 0R term, while the q1 integral in the f (εk) part is straightforward, the εk integral140in the f (εq1) part can be simplified by expanding around k+q1 = 0:G′Dj j′,0R (Ω)= 212pi∫ 2t−Ω−2tdεkf (εk)2it sin pi∑r1r2t∗r1tr212it sinkGRdd (εk)Γr1 j (k)Γ∗r2 j′ (−k)2it sinkS j j′ (p)× 1[1− ei(k+p+i0)]2 −2∫ 0−pidq1(2pi)2(2t sinq1)S j j′ (q1)∫ ∞−∞dεkf (εq1)(εk− εq1 +Ω+)× i∑r1r2t∗r1tr2GRdd (εk)Γr1 j (k)Γ∗r2 j′ (−k)2t sink1(εk− ε−q1 + i0)2+O(1)= O(1) . (B.21a)Here, in the f (εk) part, we have written εk +Ω ≡ εp (0 ≤ p ≤ pi) assuming Ω > 0, integratedover q1 using the complex variable eiq1 , and again neglected O(1) contributions from the polesof S (q1). Because k+ p is always positive and never close to 0, the denominator for the f (εk)part is O(1); thus the f (εk) part is itself O(1). Meanwhile, in the f (εq1) part, we have used(2t sink)(k+q1+ i0) ≈ (2t sinq1)(k+q1+ i0) ≈ εk− ε−q1 + i0 for |k+q1| 1. We then extendthe εk domain of integration back to the entire real axis. Both GRdd (εk) and Γr1 j (k)Γ∗r2 j′ (−k)/(sink)are analytic in the upper εk half plane; thus, closing the εk contour above the real axis, the εk integralin the f (εq1) part sees no pole and vanishes. Similarly, in the 0A term,G′Dj j′,0A (Ω)= 212pii∫ 2t−Ω−2tdεk f (εk)(−i)∑r1r2t∗r1tr2GAdd (εk)Γr1 j (−k)Γ∗r2 j′ (k)2t sinkS j j′ (p)1Ω2−2∫ 2t−2t+Ωdεq12piiS j j′ (q1) f (εq1)(−i)∑r1r2t∗r1tr2GAdd (εp1)Γr1 j (−p1)Γ∗r2 j′ (p1)2t sin p11Ω2+O(1)=−2 1piΩ∫ 2t−Ω−2tdεk[− f ′ (εk)]∑r1r2t∗r1tr2Γr1 j (−k)Γ∗r2 j′ (k)piνkGAdd (εk)S j j′ (p)+O(1) . (B.21b)We have adopted the shorthand εq1 −Ω≡ εp1 (0≤ p1 ≤ pi) and identified k with p1 and p with q1.This result, together with G′Dj j′,R0+G′Dj j′,A0 which yields its complex conjugate, leads to Eqs. (3.47c)and (3.47d).B.4 Terms quadratic in T-matrixWe focus on G′Dj j′,RR (Ω)+G′Dj j′,RA (Ω) first.141G′Dj j′,RR (Ω)+G′Dj j′,RA (Ω)= 2∫ dωdω ′(2pi)2f (ω)− f (ω ′)ω−ω ′+Ω+∫ pi0dk1dk2(2pi)2dq1dq2(2pi)2tr{M jk1k2(iτψ)gRk2(ω ′)Vk2GRdd(ω ′)Vq1gRq1(ω ′)×M j′q1q2(iτψ)[gRq2 (ω)Vq2GRdd (ω)Vk1gRk1 (ω)−gAq2 (ω)Vq2GAdd (ω)Vk1gAk1 (ω)]}. (B.22)Inserting Eqs. (3.22), (3.35) and using Eq. (B.6) again, we findG′Dj j′,RR (Ω)+G′Dj j′,RA (Ω)= 2∫ dωdω ′(2pi)2f (ω)− f (ω ′)ω−ω ′+Ω+∫ pi−pidk1dk2(2pi)2dq1dq2(2pi)2 ∑r1r2r′1r′2t∗r1tr2t∗r′1tr′2Γr1 j (k1)×Γ∗r2 j (k2)11− ei(k1−k2+i0) igRk2(ω ′)GRdd(ω ′)gRq1(ω ′)Γr′1 j′ (q1)Γ∗r′2 j′(q2)× 11− ei(q1−q2+i0) i[gRq2 (ω)GRdd (ω)gRk1 (ω)−gAq2 (ω)GAdd (ω)gAk1 (ω)]. (B.23)We can integrate over all four momenta. Let ω = εk where 0≤ k≤ pi or k= i |k|, and ω ′ = εk′ where0≤ k′ ≤ pi or k′ =−i |k′|; integrating over k2 and q1,G′Dj j′,RR (Ω)+G′Dj j′,RA (Ω)= 2∫ dεkdεk′(2pi)2f (εk)− f (εk′)εk− εk′+Ω+∫ pi−pidk1dq2(2pi)2 ∑r1r2r′1r′2t∗r1tr2t∗r′1tr′2Γr1 j (k1)×Γ∗r2 j(−k′) 11− ei(k1+k′+i0) i12it sink′GRdd (εk′)12it sink′Γr′1 j′(k′)Γ∗r′2 j′ (q2)× 11− ei(k′−q2+i0) i[gRq2 (εk)GRdd (εk)gRk1 (εk)−gAq2 (εk)GAdd (εk)gAk1 (εk)]; (B.24)finally, integrating over k1 and q2, we findG′Dj j′,RR (Ω) = O(1) , (B.25a)142G′Dj j′,RA (Ω)= 2∫ dεkdεk′(2pi)2f (εk)− f (εk′)εk− εk′+Ω+ ∑r1r2r′1r′2t∗r1tr2t∗r′1tr′2Γr1 j (−k)Γ∗r2 j(−k′)Γr′1 j′ (k′)×Γ∗r′2 j′ (k)1(2t sink′)21(2t sink)21[1− ei(k′−k+i0)]2GRdd (εk′)GAdd (εk) . (B.25b)Expanding around k = k′ and integrating over εk and εk′ , assuming Ω> 0, we obtainG′Dj j′,RA (Ω)= 2∫ 2t−Ω−2tdεk2piiΩ[− f ′ (εk)] ∑r1r2r′1r′2t∗r1tr2t∗r′1tr′2Γr1 j (−k)Γ∗r′2 j′ (k)2t sink× Γ∗r2 j (−p)Γr′1 j′ (p)2t sin p1−Ω2GRdd (εp)GAdd (εk)+O(1) , (B.26)where we have written εk +Ω ≡ εp. A similar calculation can be performed on G′Dj j′,AR (Ω) andG′Dj j′,AA (Ω); both are O(1) for the same reason that G′Dj j′,RR (Ω) is O(1). Therefore, G′Dj j′,RA is theonly term quadratic in the T-matrix which contributes to the linear dc conductance. Note that this isnot the case in Ref. [66], where the RA term and the AR term are complex conjugates as a result oftaking the δ -function part in Eq. (3.38). It is easy to see that Eq. (B.26) reproduces Eq. (3.47e).We mention in passing that Eq. (3.47a) can be derived in the wide band limit with essentiallythe same method, although the pole structure is much simpler in that case.143Appendix CDetails of weak-coupling and FLperturbation theoryThis appendix contains technical details related to the perturbation theory calculation[112] in Sec-tion 3.4.C.1 Weak-coupling perturbation theoryIn this subsection, we present the details for the perturbation theory in the weak-coupling regime.These include the calculations of the T-matrix and the connected contribution, both to the third orderin the Kondo coupling J.C.1.1 T-matrixThe object of interest is imaginary time two-point Green’s function of the screening basis. Keepingall terms up to O(V 4)and only the divergent terms at O(V 6), the Fourier transformed Green’sfunction has the form−∫ β0dτeiωnτ〈Tτψk (τ)ψ†k′ (0)〉H= 2piδ(k− k′)gk (iωn)+Kkk′gk (iωn)gk′ (iωn)+∫ dq2pi∫ β0dτeiωnτ∫ β0dτ1dτ2gk (τ− τ1)gq (τ1− τ2)gk′ (τ2)×[KkqKqk′+14∑abJkqJqk′σaσb〈TτSad (τ1)Sbd (τ2)〉]+∫ dq1dq2(2pi)2∫ β0dτeiωnτ∫ β0dτ1dτ2dτ3Jkq1Jq1q2Jq2k′×gk (τ− τ1)gq1 (τ1− τ2)gq2 (τ2− τ3)gk′ (τ3)18∑abcσaσbσ c〈TτSad (τ1)Sbd (τ2)Scd (τ3)〉, (C.1)144where the subscript H stands for Heisenberg picture,gp (iωn) =1iωn− εp (C.2)is the Fourier transform of the free Matsubara Green’s function Eq. (3.53), ωn = (2n+1)pi/β ,and we have used the fact that the free propagators are proportional to identity in spin space andtherefore commute with the Pauli matrices. Recalling Eq. (3.69), the imaginary time integrals inthe third term are trivial after Fourier transform. In the O(J3)term, the τ integral is also easyafter Fourier transform, but integrals over τ1, τ2 and τ3 are best calculated in the time domain. Thetime-ordered product of spins evaluates to〈TτSad (τ1)Sbd (τ2)Scd (τ3)〉=18iεabc [θ (τ1,τ2,τ3)−θ (τ1,τ3,τ2)−θ (τ2,τ1,τ3)+θ (τ2,τ3,τ1)+θ (τ3,τ1,τ2)−θ (τ3,τ2,τ1)]=18iεabc {2 [θ (τ1,τ2,τ3)+θ (τ2,τ3,τ1)+θ (τ3,τ1,τ2)]−1} , (C.3)where εabc is the 3D Levi-Civita symbol and θ (τ1,τ2,τ3) = θ (τ1− τ2)θ (τ2− τ3). The non-trivialtime-ordering is known to produce a logarithmic divergence. Straightforward algebra yields∫ β0dτ1dτ2dτ3eiωnτ1gq1 (τ1− τ2)gq2 (τ2− τ3)gk′ (τ3)(2 [θ (τ1,τ2,τ3)+θ (τ2,τ3,τ1)+θ (τ3,τ1,τ2)]−1)= gk′ (iωn)1εq2− εq1{gq1 (iωn) [ f (εq2)− f (−εq2)]−gq2 (iωn) [ f (εq1)− f (−εq1)]}; (C.4)therefore, using σaσbσ c = iεabc and ∑abc εabcεabc = 6, we can write−∫ β0dτeiωnτ〈Tτψk (τ)ψ†k′ (0)〉H= 2piδ(k− k′)gk (iωn)+gk (iωn)Tkk′ (iωn)gk′ (iωn) , (C.5)whereTkk′ (iωn) = Kkk′+∫ dq2pigq (iωn)(KkqKqk′+316JkqJqk′)− 332∫ dq1dq2(2pi)2Jkq1Jq1q2Jq2k′× 1εq2− εq1{gq1 (iωn) [ f (εq2)− f (−εq2)]−gq2 (iωn) [ f (εq1)− f (−εq1)]}. (C.6)Upon analytic continuation, we find the retarded T-matrix145νTkk′ (Ω) =[νKkk′+ν2∫dεq1Ω+− εq(KkqKqk′+316JkqJqk′)−ν3∫dεqdεq′316JkqJqq′Jq′k′Ω+− εqf(εq′)− f (−εq′)εq′− εq], (C.7)where ν is the Fermi surface density of states. We have used Eqs. (3.25b) and (3.25d), noticing thatjk and κk are essentially independent of k in the Kondo limit; the dummy variables q1 and q2 arethen interchangeable, and are rewritten as q and q′.If we imagine that our Kondo model were defined for dilute impurities rather than a singlemesoscopic device, then translational invariance would be recovered after impurity averaging, andthe imaginary part of the on-shell self-energy would represent the single particle lifetime, which isa physical observable. Therefore, when we reduce the momentum cutoff of the continuum Kondomodel, the imaginary part of the on-shell self-energy per unit impurity concentration, or equivalentlythe imaginary part of the on-shell T-matrix, should be cutoff independent. It is given byImTkk (εk) =(−piνV 2k )V 2k{14κ2+34[j−ν j2∫dεqV 2qf (εq)− f (−εq)εq− εk]2}. (C.8)As the running energy cutoff is reduced from D to D− δD, we recover Eq. (3.27) [or equivalentlyEq. (3.26)], the RG equation for the coupling constant j (or Jkk′).C.1.2 Connected contribution to the conductanceThe second order calculation has been discussed previously so we focus on the third order. Twonon-zero diagrams exist at this order:GC(3)k1k2q1q2 (iωp) =∫ β0dτeiωpτ∫ β0dτ1dτ2dτ318tr(σaσbσ c)∑abc〈TτSad (τ1)Sbd (τ2)Scd (τ3)〉∫ dq2pi× [Jq2qJqk1Jk2q1gq2 (−τ1)gq (τ1− τ2)gk1 (τ2− τ)gk2 (τ− τ3)gq1 (τ3)+Jq2k1Jk2qJqq1gq2 (−τ1)gk1 (τ1− τ)gk2 (τ− τ2)gq (τ2− τ3)gq1 (τ3)]. (C.9)The τ integral becomes trivial in the frequency domain. By Eq. (C.3),146GC(3)k1k2q1q2 (iωp) =−3161β ∑ωn1gk1 (iωn1)gk2 (iωn1 + iωp)∫ dq2pi∫ β0dτ1dτ2dτ3×{2 [θ (τ1,τ2,τ3)+θ (τ2,τ3,τ1)+θ (τ3,τ1,τ2)]−1}×[Jq2qJqk1Jk2q1gq2 (−τ1)gq (τ1− τ2)e−iωn1τ2ei(ωn1+ωp)τ3gq1 (τ3)+Jq2k1Jk2qJqq1gq2 (−τ1)e−iωn1τ1ei(ωn1+ωp)τ2gq (τ2− τ3)gq1 (τ3)]. (C.10)These two terms contribute equally and we show the details for the first term only. Integrate overτ1, τ2 and τ3:∫ β0dτ1dτ2dτ3 {2 [θ (τ1,τ2,τ3)+θ (τ2,τ3,τ1)+θ (τ3,τ1,τ2)]−1}gq2 (−τ1)gq (τ1− τ2)e−iωn1τ2ei(ωn1+ωp)τ3gq1 (τ3)=1εq1− εq2− iωp{f (εq)− f (−εq)εq2− εq1iωn1− εq2− f (εq)− f (−εq)εq1− εq− iωp1iωn1 + iωp− εq1−[f (εq2)− f (−εq2)εq2− εq− f (εq1)− f (−εq1)εq1− εq− iωp]1iωn1− εq}; (C.11)this allows us to do the ωn1 summation, e.g.− 1β ∑iωn11iωn1− εk11iωn1 + iωp− εk21iωn1− εq=− f (εk2)1εk2− εk1− iωp1εk2− εq− iωp+∫ dω2piif (ω)× 1ω+ iωp− εk2(1ω+− εk11ω+− εq −1ω−− εk11ω−− εq). (C.12)Analytic continuation gives the retarded four-point function,147GC(3)k1k2q1q2 (Ω)=316∫ dq2piJq2qJqk1Jk2q11εq1− εq2−Ω+{f (εq)− f (−εq)εq2− εq[− f (εk2)1εk2− εk1−Ω+1εk2− εq2−Ω++∫ dω2piif (ω)1ω+Ω+− εk2(1ω+− εk11ω+− εq2− 1ω−− εk11ω−− εq2)]− f (εq)− f (−εq)εq1− εq−Ω+[− f (εk1)1εk1 +Ω+− εk21εk1 +Ω+− εq1+∫ dω2piif (ω)1ω−Ω+− εk1(1ω+− εk21ω+− εq1− 1ω−− εk21ω−− εq1)]−[f (εq2)− f (−εq2)εq2− εq− f (εq1)− f (−εq1)εq1− εq−Ω+][− f (εk2)1εk2− εk1−Ω+1εk2− εq−Ω++∫ dω2piif (ω)1ω+Ω+− εk2(1ω+− εk11ω+− εq −1ω−− εk11ω−− εq)]}+ equivalent contribution.(C.13)Substituting Eq. (C.13) into Eq. (3.73), we can perform many of the integrals over k1, k2, q1 andq2 using contour methods. We should be careful how the contours are closed: for instance, thei/(k1− k2+ i0) term in Eq. (3.73) (which derives from the particle number operator in a lead)should be interpreted as∫ ∞0 dxei(k1−k2+i0)x, and forces the k1 contour to close on the upper half planeand the k2 contour to close on the lower half plane. When the smoke clearsG′C(3)j j′ (Ω)→ 1Ω316∫ dk1dq2(2pi)2[− f ′ (εk1)](2ipi)δ (εk1− εq2) ∑r1r2r′1r′2t∗r1tr2t∗r′1tr′2Γ′∗r1, j,k1Γ′r2, j,k1+ ΩvFΓr′1, j′,q2+ ΩvFΓ∗r′2, j′,q2×∫ dq2piV 2q (2 j)3 f (εq)− f (−εq)εq2− εq+ equivalent contribution+O(1)=1ipiΩ∫dεp[− f ′ (εp)] 3pi2ν3Jpp16 Z2, j j′ (εp)∫dεqJpqJqpf (εq)− f (−εq)εp− εq +O(1) . (C.14)Therefore, we can combine the second and the third order results as follows:GCj j′ =−2e2h∫dεp[− f ′ (εp)]T Cj j′ (εp) , (C.15)where the connected transmission probability is148T Cj j′ (εp) =Z2, j j′ (εp)3pi216[νJpp− 12ν2∫dεqJpqJqpf (εq)− f (−εq)εq− εp]2+O(J4). (C.16)This is formally similar to the O(J2)result Eq. (3.78) but with a renormalized coupling constant J;the renormalization is again consistent with Eq. (3.26).C.2 FL perturbation theoryIn this subsection, we discuss the perturbation theory in the FL regime T TK EV (also assumingEV ∼ Econn; see Table 3.1). We first present an alternative derivation of Eq. (3.88), the O(1/T 2K)retarded T-matrix obtained in Ref. [3]. Then we perform an additional consistency check on ourformalism of eliminating the connected contribution to the dc conductance at low temperatures:we directly compute the connected contribution to O(T 2/T 2K), and show that Eq. (3.62) is indeedsatisfied.In momentum space, the leading irrelevant operator Eq. (3.87) takes the formHint =2piv2FTK∫dηH (η)∫ dq1dq2dq3dq4(2pi)4ei(q1−q2+q3−q4)η : ψ˜†q1α ψ˜q2α ψ˜†q3β ψ˜q4β :− v2FTK∫dηH (η)∫ dq1dq2(2pi)2(q1+q2)ei(q1−q2)η : ψ˜†q1α ψ˜q2α : . (C.17)(We measure all momenta relative to the Fermi wavevector kF hereafter.) Here η is the locationof the operator; the weight function H (η) is peaked at the origin and can be approximated as aδ -function above the length scale vF/TK . To lighten notations, we take H (η) = δ (η) whenever itis unambiguous to do so.At O(1/T 2K)both terms in Eq. (C.17) contribute to the T-matrix, but only the first term plays arole in the connected 4-point function.C.2.1 T-matrixTo find the retarded T-matrix of the phase-shifted screening channel ψ˜ , we begin from the imaginarytime 2-point Green’s functionG˜kk′ (τ)≡−〈Tτ ψ˜k (τ) ψ˜†k′ (0)〉H. (C.18)This object is diagonal in spin indices. The three diagrams in Fig. 3.4 panel b) evaluate to149G˜kk′ (τ) = 2piδ(k− k′)gk (τ)− v2FTK (k+ k′)∫dτ1gk (τ− τ1)gk′ (τ1)+(v2FTK)2 ∫ dq2pi(k+q)× (q+ k′)∫ dη1dη2H (η1)H (η2)ei(kη1−k′η2)eiq(η2−η1) ∫ dτ1dτ2gk (τ− τ1)gq (τ1− τ2)×gk′ (τ2)−4(2piv2FTK)2 ∫ dq2dq3dq4(2pi)3∫dη1dη2H (η1)H (η2)ei(kη1−k′η2)× ei(q2−q3+q4)(η2−η1)∫dτ1dτ2gk (τ− τ1)gq2 (τ1− τ2)gq3 (τ2− τ1)gq4 (τ1− τ2)gk′ (τ2) .(C.19)Going to the Fourier space, we identify the imaginary time T-matrix asT˜kk′ (iωn) =− v2FTK(k+ k′)+(v2FTK)2 ∫ dq2pi(k+q)(q+ k′)∫dη1dη2H (η1)H (η2)ei(kη1−k′η2)× eiq(η2−η1)gq (iωn)−4(2piv2FTK)2 ∫ dq2dq3dq4(2pi)3∫dη1dη2H (η1)H (η2)ei(kη1−k′η2)× ei(q2−q3+q4)(η2−η1) 1β ∑ωn21β ∑ωn4gq2 (iωn2)gq3 (iωn2 + iωn4− iωn)gq4 (iωn4) . (C.20)where all Matsubara frequencies are fermionic, e.g. ωn = (2n+1)pi/β . Both frequency summa-tions are standard,[74] and analytic continuation iωn→ ω+ yieldsT˜kk′ (ω) =− v2FTK(k+ k′)+(v2FTK)2 ∫ dq2pi(k+q)(q+ k′)∫dη1dη2H (η1)H (η2)ei(kη1−k′η2)× eiq(η2−η1)ω+− εq −4(2piv2FTK)2 ∫dη1dη2H (η1)H (η2)ei(kη1−k′η2)∫ dq2dq3dq4(2pi)3× ei(q2−q3+q4)(η2−η1) [− fB (εq3− εq4)− f (εq2)] [ f (εq4)− f (εq3)]ω++ εq3− εq4− εq2, (C.21)where fB (ω) = 1/(eβω −1) is the Bose function.In the q integral we close the contour in the upper half plane for η2 > η1, and in the lower halfplane for η2 < η1; this leads to∫ dq2pi(k+q)(q+ k′) eiq(η2−η1)ω+− εq =−ivF(k+ωvF)(ωvF+ k′)eiω+vF(η2−η1)H (η2−η1) . (C.22)150For the on-shell T-matrix T˜pp (εp), the phase factors involving η1 and η2 cancel, and the η integralsbecome∫dη1dη2H (η1)H (η2)H (η2−η1) = 1/2. We can simplify the triple integral over q2, q3and q4 by the contour method in a similar fashion, before using the following identity,∫ ∞−∞dεq2dεq3dεq4 [ fB (εq3− εq4)+ f (εq2)] [ f (εq4)− f (εq3)]δ (ω+ εq3− εq4− εq2)=12(pi2T 2+ω2),(C.23)which has been given in Ref. [89] in the context of an inelastic scattering collision integral. Collect-ing all three terms, we recover Eq. (3.88).C.2.2 Connected contribution to the conductanceInserting Eqs. (3.72) and (3.85) into the 4-point function Eq. (3.73), and performing the k1, k2, q1and q2 integrals, we obtainG ′Cj j′ (iωp) =∫ dp1dp2dp3dp4(2pi)4G˜Cp1p2p3p4 (iωp)∑j1 j2∑j′1 j′2U1, j1U∗1, j2U1, j′1U∗1, j′2×(δ j j1δ j j2ip1− p2+ i0 +S∗j j1S j j2ip2− p1+ i0)×(δ j′ j′1δ j′ j′2ip3− p4+ i0 +S∗j′ j′1S j′ j′2ip4− p3+ i0); (C.24)we have ignored the momentum dependence ofU and S in the Fermi liquid regime (which is justifiedat TK Econn). Here the δP-independent connected four-point correlation function for ψ˜ is definedasG˜Cp1p2p3p4 (iωp)≡−∫ β0dτeiωpτ ∑σσ ′〈Tτ ψ˜†p1σ (τ) ψ˜p2σ (τ) ψ˜†p3σ ′ (0) ψ˜p4σ ′ (0)〉C. (C.25)We observe that δψψ drops out of G ′Cj j′ completely, which reflects the inelastic nature of the connectedcontribution.To O(1/T 2K), there are three diagrams resulting in nonzero connected contributions to the lineardc conductance, depicted in Fig. C.1. The corresponding 4-point functions readG˜C,BCSp1p2p3p4 (iωp) =−4(2piv2FTK)2 ∫ β0dτeiωpτ∫ β0dτ1dτ2∑σσ ′δσσ¯ ′∫ dq1dq3(2pi)2×gp1 (τ1− τ)gp2 (τ− τ2)gp3 (τ1)gp4 (−τ2)gq1 (τ2− τ1)gq3 (τ2− τ1) , (C.26a)151Figure C.1: The three connected diagrams at O(T 2/T 2K)contributing to the conductance. ZS,ZS’ and BCS label only the topology of the diagrams and not necessarily the physics.G˜C,ZSp1p2p3p4 (iωp) =−4(2piv2FTK)2 ∫ β0dτeiωpτ∫ β0dτ1dτ2∑σσ ′δσσ ′∫ dq3dq4(2pi)2×gp1 (τ1− τ)gp2 (τ− τ2)gp3 (τ2)gp4 (−τ1)gq3 (τ2− τ1)gq4 (τ1− τ2) , (C.26b)G˜C,ZS’p1p2p3p4 (iωp) =−4(2piv2FTK)2 ∫ β0dτeiωpτ∫ β0dτ1dτ2∑σσ ′δσσ¯ ′∫ dq1dq4(2pi)2×gp1 (τ1− τ)gp2 (τ− τ2)gp3 (τ2)gp4 (−τ1)gq1 (τ2− τ1)gq4 (τ1− τ2) . (C.26c)Here the terminology of BCS, ZS and ZS′ is borrowed from Ref. [109] and refers only to thetopology of the diagrams.We illustrate the calculation with the BCS diagram; ZS and ZS’ again turn out to be completelyanalogous. Going to the Fourier space,G˜C,BCSp1p2p3p4 (iωp)=−(2piv2FTK)28∫ dq1dq3(2pi)21β ∑ωn11β ∑ωn31β ∑ωn5gp1 (iωn1)gp2 (iωn1 + iωp)×gp3 (iωn3)gp4 (iωn3− iωp)gq1 (iωn5)gq3 (iωn1 + iωn3− iωn5) ; (C.27)152theωn5 summation is standard, whereas theωn1 andωn3 summations require the following identities:1β ∑ωn31iωn3− εp31iωn3− iωp− εp41iωn1 + iωn3− εq1− εq3= f (εp3)1εp3− iωp− εp41iωn1 + εp3− εq1− εq3+ f (εp4)1iωp+ εp4− εp31iωn1 + iωp+ εp4− εq1− εq3− fB (εq1 + εq3)1εq1 + εq3− iωn1− εp31εq1 + εq3− iωn1− iωp− εp4, (C.28a)and− 1β ∑ωn11iωn1− εp11iωn1 + iωp− εp21iωn1 + iωp+ εp4− εq1− εq3=∫ ∞−∞dε2piif (ε)1ε− iωp− εp1(1ε+− εp21ε++ εp4− εq1− εq3− 1ε−− εp21ε−+ εp4− εq1− εq3)− f (εp1)1εp1 + iωp− εp21εp1 + iωp+ εp4− εq1− εq3, (C.28b)where ε± ≡ ε± i0+. The second identity can be derived by allowing the complex plane contour towrap around the line Imz= ωp[74].After applying the identities above, performing analytic continuation iωp→Ω+, and performingall p integrals that are approachable by the contour method in Eq. (C.24), we findG′C,BCSj j′ (Ω)=−(2piv2FTK)28∑j1 j2U1, j1U∗1, j2U1, j′U∗1, j′S∗j j1S j j21Ω∫ dq1dq3(2pi)2[ f (−εq1)− f (εq3)]×{∫ dp1dp4(2pi)2[ f (εp4)+ fB (εq1 + εq3)]f (εp1 +Ω)− f (εp1)Ω1εp1 +Ω++ εp4− εq1− εq3−∫ dp2dp3(2pi)2[ f (εp3)+ fB (εq1 + εq3)]f (εp2−Ω)− f (εp2)−Ω1εp2−Ω++ εp3− εq1− εq3}. (C.29)In the dc limit, the principal value parts of the integrands cancel to O(1/Ω), while the δ -functionparts remain:153G′C,BCSj j′ (Ω) =−(2pi)2T 2K8∑j1 j2U1, j1U∗1, j2U1, j′U∗1, j′S∗j j1S j j21Ω2ipi(2pi)4∫dεp1[− f ′ (εp1)]×∫dεp4dεq1dεq3 [ f (εp4)+ fB (εq1 + εq3)] [ f (−εq1)− f (εq3)]δ (εp1 + εp4− εq1− εq3)+O(1)=− 4T 2K∑j1 j2U1, j1U∗1, j2U1, j′U∗1, j′S∗j j1S j j21Ωi2pi∫dεp1[− f ′ (εp1)](pi2T 2+ ε2p1)+O(1) . (C.30)In the second step we have again invoked Eq. (C.23).Each of the ZS and ZS’ contributions ends up being the opposite of the BCS contribution,G′C,ZSj j′ (Ω) = G′C,ZS’j j′ (Ω) =−G′C,BCSj j′ (Ω) . (C.31)therefore, using Eq. (3.22) for the U matrix elements, we can express the total connected contribu-tion to the conductance to O(1/T 2K)asGCj j′ =e2hlimΩ→0(2piiΩ)G′C,BCSj j′ (Ω) =−2e2h∫dω[− f ′ (ω)]T Cj j′ (ω) , (C.32)whereT Cj j′ (ω) =−2V 4kF[S (kF)Γ† (kF)λΓ(kF)S† (kF)]j j[Γ† (kF)λΓ(kF)]j′ j′pi2T 2+ω2T 2K. (C.33)We have reintroduced the coupling matrix Eq. (3.20). The ω integral can be done explicitly:GCj j′ =2e2h83V 4kF∣∣∣[S (kF)Γ† (kF)λΓ(kF)] j j′∣∣∣2(piTTK)2, (C.34)i.e. the lowest order connected contribution to the conductance is O(T 2/T 2K), characteristic of a FL.Eq. (C.33) is in explicit agreement with Eq. (3.62). We can also check its consistency with theEq. (3.48) and single-particle T-matrix inelasticity. Recall that, by virtue of Eq. (3.65), we shouldhave the following approximate identity for ω ≈ 0 in the FL regime:T Cj j′ (ω) =−1V 2kF[S (kF)Γ† (kF)λΓ(kF)S† (kF)]j j∑j′′T Dj′′ j′ (ω) . (C.35)On the other hand, Eqs. (3.86) and (3.88) yield for the on-shell T-matrixIm [−piνT(ω)]−|−piνT(ω)|2 = pi2T 2+ω22T 2K; (C.36)therefore, plugging Eq. (C.36) into Eq. (3.48), we find that for ω ≈ 0,154∑j′′T Dj′′ j′ (ω) =pi2T 2+ω22T 2K4V 2kF[Γ† (kF)λΓ(kF)]j′ j′ . (C.37)Eqs. (C.35) and (C.37) are fully consistent with Eq. (C.33).155
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Conductance of junctions of multiple interacting quantum wires and long Aharonov-Bohm-Kondo rings Shi, Zheng 2017
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Title | Conductance of junctions of multiple interacting quantum wires and long Aharonov-Bohm-Kondo rings |
Creator |
Shi, Zheng |
Publisher | University of British Columbia |
Date Issued | 2017 |
Description | In this thesis, we calculate the linear dc conductance of two types of multi-terminal interacting systems: junctions of interacting quantum wires attached to Tomonaga-Luttinger liquid (TLL) leads, and closed and open long Aharonov-Bohm-Kondo (ABK) rings. In both cases, we obtain corrections to the non-interacting Landauer formula, arising from interactions in the TLL leads and the quantum dot in the Kondo regime respectively. In junctions of interacting quantum wires, if the wires are attached to Fermi liquid (FL) leads, the conductance is formally given by the Landauer formula with renormalized single-particle S-matrix elements. If, however, the wires are attached to TLL leads, i.e. the interaction does not vanish even in the leads, the conductance has an additional contribution dependent on the interaction strength in the leads. We calculate this additional contribution both at the first order in interaction and in the random phase approximation, and heuristically relate the FL conductance to the TLL conductance through a "contact resistance" between an FL lead and a TLL wire. In long ABK rings, where the interaction is due to spin-flip scattering at a quantum dot in the Kondo regime, the linear dc conductance consists of two parts: a disconnected part of the Landauer form, and a connected part that can be approximately eliminated at low temperatures. For a closed long ABK ring, where the electric current is conserved in the ring, the high-temperature conductance has qualitatively different behaviors for temperatures greater than and lower than the characteristic energy scale v_F /L, where v_F is the Fermi velocity and L is the ring circumference. Meanwhile, for an open long ABK ring where electrons may leak into the side leads coupled to ring arms, as long as the ring arms have both small transmission and small reflection, the ring behaves as a two-path Aharonov-Bohm interferometer, and we predict the observation of a π/2 phase shift due to scattering off the Kondo singlet formed at low energies around the impurity spin. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2017-07-12 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0348803 |
URI | http://hdl.handle.net/2429/62224 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2017-09 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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