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Charge transport in a graphene flake realization of the Sachdev-Ye-Kitaev model Can, Oguzhan 2018

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Charge Transport in a Graphene FlakeRealization of the Sachdev-Ye-KitaevModelbyOguzhan CanB.Sc., The University of Toronto, 2016A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2018c© Oguzhan Can, 2018The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the thesis entitled:Charge Transport in a Graphene Flake Realization of the Sachdev-Ye-Kitaev Modelsubmitted by Oguzhan Can in partial fulfillment of the requirements forthe degree of Master of Sciencein PhysicsExamining Committee:Marcel Franz, PhysicsSupervisorIan Affleck, PhysicsSupervisory Committee MemberSupervisory Committee MemberAdditional ExaminerAdditional Supervisory Committee Members:Supervisory Committee MemberSupervisory Committee MemberiiAbstractWe address the transport properties of a mesoscopic realization of the Sachdev-Ye-Kitaev (SYK) model which is an exactly solvable system of interactingspinless fermions connected to the black hole physics through the holo-graphic principle. Starting with a recent proposal for simulating the SYKmodel in a graphene flake in an external magnetic field and extending it byconsidering leads attached to it, we model a realistic transport experimentand calculate directly measurable quantities featuring non-Fermi liquid sig-natures of the SYK physics. We show that the graphene flake realizationis robust in the presence of leads and that measuring the tunneling currentacross the leads one can experimentally observe a non-Fermi liquid - Fermiliquid transition by tuning the external magnetic field threading the flake.After establishing the transport signatures of the SYK model near equilib-rium using linear response framework, we then derive a formula to extendour results for tunneling current using Keldysh formalism to explore theeffects of finite bias voltage across the leads, going beyond equilibrium.iiiLay SummaryVery strong interactions between electrons in a metal are well known to giverise to rich physics yet such materials are in general difficult to analyze the-oretically. However, a rare exception has emerged recently. In this work, weconsider a new and very popular system called the Sachdev-Ye-Kitaev modelwhich has strong interactions and can be solved exactly. This model is veryattractive because it is not only accessible analytically but also has certainproperties which are mathematically similar to those of black holes due to itsvery random, strongly interacting nature. We start with a recently proposedexperimental realization of this model and study its electrical conductance inthe hopes of understanding implications of very strong interactions betweenelectrons.ivPrefaceAll of the results presented in this thesis have been published on the arXive-print archive. [Oguzhan Can, Emilian M. Nica, and Marcel Franz. Chargetransport in graphene-based mesoscopic realizations of Sachdev-Ye-Kitaevmodels, ArXiv e-prints, arXiv:1808.06584 [cond-mat], Aug 2018 ] At thetime of writing this thesis, this paper has also been submitted to a peer-reviewed journal.The universal jump we observe in section 2.1 has been discovered, andthe finite temperature weak tunneling regime dependencies of the currentpresented in section 2.3 have been analytically calculated by my supervi-sor, Marcel Franz who has also provided the initial idea for this project. Ihave worked out the details of his preliminary analytical calculation for thezero temperature linear response regime using Keldysh path integral formal-ism (see appendix) and numerically confirmed and extended these results tofinite temperature. My collaborator Emilian Nica has later calculated the fi-nite temperature analytical expression for the linear response current, whichis a result we refer to yet do not discuss in this thesis but it can be foundin the aforementioned publication.Keldysh saddle point derivation of the general current expression pre-sented in section 2.4, which is my original contribution, is the main resultin this thesis and its details can be found in the appendix. It has also beenindependently derived by Emilian Nica using diagrammatic techniques.Analytic results in section 2.5 that provide us with a modified form ofthe saddle point equations incorporating the effects of the reservoirs on thesystem are Emilian Nica’s work. I have performed the numerical solution tothese equations which then showed that the low energy SYK physics on theflake is robust in the presence of explicit coupling to the reservoirs.Contributions of my collaborators to the analytical results describedabove, which are critical for presentation and completeness of this work arecited appropriately within the text. I have performed all of the numericalwork using Python and these calculations are completed on the high per-formance computing cluster LISA at the Stewart Blusson Quantum MatterInstitute (SBQMI), UBC.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction to SYK model and the proposal . . . . . . . . . 11.2 Roadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Starting Point - BA model in equilibrium . . . . . . . . . . . 41.3.1 Extending to two symmetric leads . . . . . . . . . . . 61.4 Keldysh Formalism for Beyond Equilibrium . . . . . . . . . . 82 Transport Properties of the Quantum Dot . . . . . . . . . 102.1 Linear Response . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Experimental Considerations . . . . . . . . . . . . . . . . . . 192.3 Weak Tunneling Regime . . . . . . . . . . . . . . . . . . . . 202.4 Beyond Equilibrium . . . . . . . . . . . . . . . . . . . . . . . 232.4.1 I-V characteristics in Weak-Tunneling Regime . . . . 242.4.2 I-V characteristics in Linear Response Regime . . . . 262.4.3 Current conservation . . . . . . . . . . . . . . . . . . 282.5 Coupling to Reservoirs . . . . . . . . . . . . . . . . . . . . . 293 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . 34viTable of ContentsBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36AppendixA Saddle Point Calculations . . . . . . . . . . . . . . . . . . . . 38A.1 Keldysh Action . . . . . . . . . . . . . . . . . . . . . . . . . 38A.2 Linear Response at Saddle Point . . . . . . . . . . . . . . . . 40A.3 Current at Saddle Point . . . . . . . . . . . . . . . . . . . . . 42A.3.1 Evaluation of Gaussian integrals . . . . . . . . . . . . 44A.3.2 Effective action and the large-N limit . . . . . . . . . 46A.3.3 Analytic continuation . . . . . . . . . . . . . . . . . . 48A.3.4 Disorder averaging of the coupling . . . . . . . . . . . 49A.3.5 Proof of the Gaussian average identity (A.16) . . . . 50A.3.6 Proof of Gaussian Identity (A.13) . . . . . . . . . . . 51viiList of Figures1.1 A schematic of the proposed experimental setup . . . . . . . 22.1 Dimensionless DC Conductance in linear response regime atvarious temperatures as a function of p = M/N Gray dashedlines correspond to zero temperature analytic results (eqn.2.12) We defined G0 =√NM e22h . . . . . . . . . . . . . . . . 182.2 Dimensionless DC Conductance in linear response regime attemperatures above T  T ∗ . . . . . . . . . . . . . . . . . . 192.3 DC conductance versus the number of flux quanta per leadmode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4 I-V characteristics in the weak tunneling regime for varioustemperatures. Numerical results are shown in solid curves. Inhigh bias regime eU  kBT we find that the current calcu-lated with (2.22) using numerical solutions of the saddle pointequations matches weak tunneling analytical prediction (2.19)I√T - eU/J characteristics in the weak tunneling regime forvarious temperatures. For low bias regime eU  kBT weobserve a scaling collapse, confirming the predicted eU/√Tdependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5 DC current versus bias voltage eU at steady state transportin arbitrary units for T= 400mK and various values of p =M/N < 0.5 in the NFL regime . . . . . . . . . . . . . . . . . 262.6 DC current at steady state transport in arbitrary units for T=400mK. Solid curves correspond to various bias voltages Uacross two leads. The dashed curve shows the linear responsecurrent projected to eU = 0.006J from the DC conductanceG we have calculated numerically using (2.12) . . . . . . . . . 27viiiList of Figures2.7 Spectral functions of the dot (solid lines) and the left lead(dashed lines) for various values of ρEt2E in units of J in equi-librium, p = 0.1 The low energy behaviour is unaffected (upto ρEt2E ∼ J) by the coupling of the lead endpoints to thesemi-infinite wires coupling the system to the reservoirs. Inconformal regime, dot and the lead spectral functions showω−1/2 and ω1/2 dependence (2.14) respectively. This confor-mal behaviour is outlined by grey dashed lines. . . . . . . . . 302.8 Spectral functions of the dot (solid lines) and the left lead(dashed lines) for various values of ρEt2E in units of J in equi-librium, p = 0.1, in the absence of SYK2 (t = 0) at the leadendpoints. The low energy behaviour is the same (compareto Fig. 2.7), regardless of whether disorder is present at thelead endpoints. . . . . . . . . . . . . . . . . . . . . . . . . . . 312.9 DC conductance as a function of p. We consider the effectsof coupling to the reservoirs tE =√J/2piρE and the presenceof disorder t at the lead endpoints . . . . . . . . . . . . . . . 33ixAcknowledgementsI would like to thank Emilian Nica, Chengshu Li, E´tienne Lantagne-Hurtubiseand Ryan Day for interesting and stimulating discussions. I also thankIlya Elfimov for his support regarding the computing cluster at the SBQMIwhich was extremely helpful in this project. Special thanks extend to MarcelFranz, my supervisor, for providing the initial idea and patiently advisingme throughout this project. Throughout my program, I was supported bythe QuEST initiative at the SBQMI. Finally, I thank my parents SeyhanCan and Abdullah Can for their unconditional support.xFor my parents who have always supported me. Most recently, completion ofthis thesis would not have been possible without the delicious baklavas theymade.xiChapter 1The Model1.1 Introduction to SYK model and the proposalSYK is an exactly solvable quantum mechanical model connected to blackhole physics in AdS2 space-time gravity theories through holographic prin-ciple. [1] Complex fermion variant of the SYK model is given byHSYK =1(2N)3/2∑ijklJijklc†ic†jckclwhere the couplings among N spinless fermionic modes, Jijkl are drawnfrom complex random Gaussian distribution with variance |Jijkl|2 = J2.This model develops an emergent conformal symmetry [2] at low energies,for 1  βJ  N . In this so called conformal regime, SYK model displaysholographic behaviour such as saturating the chaos bound [3] and finite zerotemperature entropy [4] which are also properties of charged black holes. [5]Though SYK is not the first holographic model, it is remarkable dueto its simplicity and the fact that it is exactly solvable. [6] This makesthe model crucial in further understanding of the AdS/CFT correspondencewhich is also called the holographic duality. Holography relates two seem-ingly unrelated physical systems in the sense that the correlators and certainthermodynamic quantities of the two theories show the same functional de-pendence [5] and the symmetries of these two theories match [7]SYK and its variants are popular examples of holographic quantum mat-ter where non-Fermi liquid (NFL) behaviour is observed in the presence ofstrong correlations and strong disorder. In a non-Fermi liquid, elementaryexcitations of the system can not be associated with non interacting elec-tronic excitations through adiabatic continuity arguments. This means thatthe familiar quasiparticle description fails, making theoretical considerationsdifficult. Nevertheless, SYK model is special: despite the strong correlationsit can be solved and certain observable quantities can be analytically ob-tained [6]. However, the distinct non-Fermi liquid behaviour of SYK modelremains to be experimentally observed. Recently, various realizations have11.1. Introduction to SYK model and the proposalbeen proposed involving ultracold atoms [8], Majorana modes on the surfaceof a topological superconductor [9], semiconductor quantum wires attachedto a quantum dot [10] and finally a graphene flake in external magnetic field[11], See a recent review [12] for a comprehensive overview of these differentapproaches. Our focus will be the complex fermion realization of the SYKmodel utilizing the highly degenerate zeroth Landau levels on a grapheneflake in external transverse magnetic field [11]. Though it is shown that sucha model realizes the SYK hamiltonian, an actual experimental situation andthe quantities that would be measured in a laboratory setup are relativelyunexplored.In this work, we model a transport experiment that would probe theconformal behaviour of the SYK model with this graphene flake realizationby attaching two leads that would drive a current through the flake hostingthe SYK model. We address the signatures of the conformal SYK behaviourin a transport experiment, effects of attaching the leads to the original modeland in particular, study whether the SYK behaviour is robust in the presenceof leads. We start with a rather conventional linear response approach nearequilibrium and then extend our results to nonequilibrium phenomena.BFigure 1.1: A schematic of the proposed experimental setupIn order to study the effects of the leads, we consider an interesting exten-sion of SYK model, proposed by Banerjee and Altman [13], which is an SYKmodel of N complex fermions coupled to a set of M auxiliary non interact-ing fermionic degrees of freedom. The authors have shown that by tuningonly the fermion density of the SYK model, one observes a quantum phase21.2. Roadmaptransition from a Fermi liquid to a non-Fermi liquid. A remarkable resultthey obtain is that the phase diagram of this model does not depend on thestrength of the coupling between these two sub-systems but only on the SYKfermion density and the ratio of the number of fermions p = M/N constitutethe two. Just as in the original SYK model, this system is also exactly solv-able in the large-N,M limit. We propose that the extension of SYK modelby Banerjee and Altman [13] can be adapted to the graphene flake realization[11] where the noninteracting auxiliary fermions would model the endpointsof leads coupled to graphene flake which hosts SYK complex fermions. Oncesuch a system is realized in an experimental setting, through a simple trans-port experiment that only requires readily available technologies, we couldobtain the experimental signatures of the SYK model in the large-N limitwhere the numerical techniques run into difficulties. This model would notonly make it possible to probe holographic matter, but also make the ex-perimental study of the predicted Fermi liquid - non-Fermi liquid transition[13] possible, in the hopes of confirming the theoretical predictions.We also demonstrate that the random Gaussian model for the lead end-points is not absolutely necessary and we consider a lead model made ofnoninteracting ballistic chains attached to the SYK graphene flake and showthat the SYK physics persists even in the presence of strong coupling to suchleads.1.2 RoadmapAn outline of this thesis is the following. In chapter 1, we first introducethe Banerjee-Altman model [13] briefly (section 1.3) and generalize it toour experimental setup proposal. In section 1.4 we present the equations ofmotion we solve in Keldysh formalism. These equations are valid beyondequilibrium and can be reduced to the Banerjee-Altman model equations ofmotion in equilibrium.In chapter 2, we study the transport properties of the system. We studythe system near equilibrium in two separate regimes we call “linear responseregime” (section 2.1) and the “weak tunneling regime” (section 2.3). Wethen derive an expression (section 2.4) for the current at saddle point whichis valid in equilibrium and beyond. We then show that the numerical resultsobtained with this formula match weak tunneling results near equilibrium(section 2.4.1). Next, we turn to linear response regime and again show thatthe formula we derived matches the linear response regime results (2.4.2).Finally, we go beyond equilibrium and explore the I-V characteristics for31.3. Starting Point - BA model in equilibriumfinite bias voltage across the two leads. Up to this point, we have assumedthat the lead endpoints are in equilibrium with reservoirs but have not con-sidered their effect on the spectral densities of the leads for simplicity ofthe model. We justify this assumption in section 2.5 and reproduce someof the results we had in earlier sections in the presence of explicit couplingto the reservoirs. We find that the SYK signatures we predict in transportobservables are robust even when the the graphene flake is strongly coupledto the reservoirs.1.3 Starting Point - BA model in equilibriumWe start our discussion by reviewing the variant of the SYK model proposedby Banerjee and Altman [13]. The model couples the original N complexfermion SYK4 model Hc to a set of M auxiliary noninteracting fermionicdegrees of freedom Hψ which are also random disordered (which we will callSYK2). The couplings Hcψ between these two systems are again randomdisordered. The model is given by the following Hamiltonian:Hc =1(2N)3/2∑ijklJijklc†ic†jckcl − µ∑ic†ici (1.1)Hψ =1(M)1/2∑ijtijψ†iψj − µ∑iψ†iψi (1.2)Hcψ =1(NM)1/4∑ijVijc†iψj + V∗ijψ†jci (1.3)where {Jijkl}, {tij} and {Vij} are random Gaussian distributions with vari-ances defined as |Jijkl|2 = J2, |tij |2 = t2 and |Vij |2 = V 2 where antisym-metrization and Hermiticity of Hc impliesJjikl = −Jijkl Jijlk = −Jijkl Jklij = −J∗ijkl (1.4)since Hψ must also be Hermitian,tji = t∗ij (1.5)We shall start with this Banerjee-Altman model (which we shall refer toas BA from now on) which has only one set of auxiliary fermions (modeledby Hψ) and then extend it to two auxiliary flavours of fermions which aremodeled as the endpoints of the leads attached to the SYK4 graphene flake41.3. Starting Point - BA model in equilibriumquantum dot (which will correspond to Hc). In the following we use theterms “graphene flake”, “quantum dot” or “the dot” interchangably.Let us first review their main results that are relevant to our model.BA model displays a quantum phase transition between a non-Fermi liquid(NFL) and a Fermi liquid (FL) which is controlled by only two parametersp = M/N , and the total fermion density n = (N〈nc〉 + M〈nψ〉)/(M + N).At half filling, n = 1/2, the NFL/FL transition occurs are p = pc = 1.Away from half filling, this transition occurs at lower values of p = pc < 1,depending on the value of n. A remarkable feature of this transition is thatpc marking the transition does not depend on any of the parameters t, V orJ but n only. They show that in the NFL phase, though in the presence ofthe coupling to the auxiliary fermions, the model has finite zero temperatureentropy and it saturates the chaos bound [3] as expected of the SYK model.However, once the transition occurs, the zero temperature entropy vanishesand the model is not maximally chaotic anymore.Here we copy the saddle point equations [13] that are obtained in Mat-subara imaginary time formalism after disorder averaging the action. Wewill later derive a similar set of equations in Keldysh formalism since weare also interested in nonequilibrium properties of this model. Going backto the BA model, the saddle point equations in Matsubara frequencies aregiven by:G−1(iωn) = iωn + µ− ΣJ(iωn)− V 2√pG(iωn)G−1(iωn) = iωn + µ− V2√pG(iωn)− t2G(iωn)where G(τ) = −〈Tˆ c(τ)c†(0)〉 and G(τ) = −〈Tˆψ(τ)ψ†(0)〉 are the disorderaveraged imaginary time ordered Green’s functions. The saddle point equa-tions above can be analytically continued to:G−1R (ω) = ω + µ− ΣRJ (ω)− V 2√pGR(ω)G−1R (ω) = ω + µ−V 2√pGR(ω)− t2G(ω)In the conformal limit, analytical solution to above equations can beobtained. For the NFL phase, (at half filling, n = 1/2 or θ = 0 wherep < pc = 1):GR(ω) =Λe−ipi/4√JωGR(ω) = −√p√Jωeipi/4V 2Λ51.3. Starting Point - BA model in equilibriumΣRJ (ω) = −Λ3pieipi/4√JωThis holds when ω  ΣRJ (ω) and ω  V 2√pGR(ω) such that the first equa-tion becomesG−1R (ω) = −ΣRJ (ω)−V 2√pGR(ω) and t2GR(ω) V 2/√pGR(ω)as well as ω  V 2/√pGR(ω) such that G−1R (ω) = −V2√pGR(ω) which can besubstituted back into the previous equation. We then obtain the same formas the original SYK model saddle point equations. The above four inequal-ities define four critical frequencies below which the solutions above arejustified [13]:ωc1 =J2√pi(1− p)3/2 ωc2 = J2√pip2√1− pωc3 =(√piV 42J√1− pp)1/3ωc4 =√piV 4t2J√1− pp(1.6)The conformal solution then holds when all four inequalities hold [13]ω  min(ωc1 + ωc2, ωc3, ωc4)1.3.1 Extending to two symmetric leadsGeneralization to two auxiliary and identical SYK2 models (withM fermionseach) is straightforward. Now we relabel G auxiliary fermion Green’s func-tions as G(L) for the left lead and introduce G(R) for the right lead whichcouples in the same way as G(L) since we have two identical leads (note thatfor the lead Green’s functions below the subscript R means the retardedGreen’s function while the superscript (L/R) is the site index correspond-ing to left and right leads respectively):G−1R (ω) = ω + µ− ΣRJ (ω)− V 2√p(G(L)R (ω) + G(R)R (ω))(G(L)R )−1(ω) = ω + µ−V 2√pGR(ω)− t2G(L)R (ω)(G(R)R )−1(ω) = ω + µ−V 2√pGR(ω)− t2G(R)R (ω)If we take two symmetric leads, we can then take G(R)R (ω) = G(L)R (ω) =GR(ω). If we also define V˜ = 21/4V and p˜ = 2p, then above equationsreduce toG−1R (ω) = ω + µ− ΣRJ (ω)− V˜ 2√p˜GR(ω)61.3. Starting Point - BA model in equilibriumG−1R (ω) = ω + µ−V˜ 2√p˜GR(ω)− t2GR(ω)But these equations have the same form as the original BA model. Therefore,everything we know about the BA model holds except that V and p arerenormalized. For instance the FL/NFL transition occurs at 2p = p˜ = 1.Therefore we expect to see the FL/NFL transition at p = 0.5 inthis extended model. Then the NFL solutions for the identical lead caseGR(ω) =Λe−ipi/4√Jω(1.7)G(R)R (ω) = G(L)R (ω) = −√p√Jωeipi/4V 2Λ(1.8)(note that we used the ratio√p˜/V˜ 2 =√p/V 2 nevertheless we get the sameform for the solution.) In the Fermi liquid case, (p > pc = 0.5), the Green’sfunctions (two symmetrical leads) are found to beGR(ω) = − it√2V 21√2p− 1 (1.9)G(L)R (ω) = G(R)R (ω) = GR(ω) = −it√2p− 12p(1.10)In this case, the critical frequencies below which we observe conformal regimebecomeωc1 =J2√pi(1− 2p)3/2 (1.11)ωc2 =2J√pip2√1− 2p (1.12)ωc3 =(√piV 4J√1− 2p2p)1/3(1.13)ωc4 =√piV 4t2J√1− 2pp(1.14)note that as we tune p, near p = 0.5, ωc = min(ωc1 + ωc2, ωc3, ωc4) becomesarbitrarily small. Therefore, at the FL/NFL transition we expect toleave conformal regime as we tune p through pc = 0.5 at particle holesymmetry. This also implies that outside the conformal regime, the analyti-cal forms (1.7) will start showing corrections even at arbitrarily low energies.We will see consequences of this in the following sections.71.4. Keldysh Formalism for Beyond Equilibrium1.4 Keldysh Formalism for Beyond EquilibriumIn order to study far from equilibrium transport properties and beyond linearresponse, we have to resort to more advanced Keldysh formalism. Results weobtain here will be more general and should reduce to equilibrium resultsthat we considered in previous section. Following the same formalism asan earlier study [14], we work with Keldysh time contour path integral asopposed to the imaginary time path integrals in Matsubara formalism. Seethe appendix for the details of the saddle point approximation (Section A.1)for Keldysh action. The saddle point equations in real time are given by:Σcss′(t) = ss′J2G2ss′(t)Gs′s(−t)+ss′√pV 2GL,ss′(t)+ss′√pV 2GR,ss′(t) (1.15)ΣψLss′ (t) = ss′t2GL,ss′(t) + ss′(1/√p)V 2Gss′(t) (1.16)ΣψRss′ (t) = ss′t2GR,ss′(t) + ss′(1/√p)V 2Gss′(t) (1.17)along with the Dyson’s equations in frequency spaceGss′(ω) = [σz (ω + µ)− Σc]−1ss′ (1.18)GL,ss′(ω) =[σz (ω + µ)− ΣψL]−1ss′(1.19)GR,ss′(ω) =[σz (ω + µ)− ΣψR]−1ss′(1.20)where G,GL and GR are the time contour Green’s functions of the dot, leftand right leads respectively. They have the matrix form Gss′ which is definedas:G =(GT G<G> GT˜)=(G++ G+−G−+ G−−)where GT and GT˜ are the time ordered and anti-time ordered Green’s func-tions, respectively. G< and G> are the lesser and the greater Green’s func-tions. These four quantities are not independent. By construction, theyare related by GT + GT˜ = G< + G>. Path segment index s (takes val-ues of ±1 in the equations (1.15-1.17) for forward (+) and backward (−)parts of the contour respectively) is the real time path index which appearssince we split the time loop contour into two pieces defined on real timeaxis. In above form we assumed time translational invariance, ignoring thetransient behaviour as the couplings are turned on over the time contourpath. The following expression of fluctuation-dissipation theorem allows us81.4. Keldysh Formalism for Beyond Equilibriumto introduce temperature in equilibrium:GK(ω) = 2i tanh(βω2)ImGR(ω). (1.21)where the Keldysh correlator GK is defined to be GK = G< +G>We solve these equations (1.15-1.20) iteratively while imposing the condi-tion (1.21) for parts of the system which are in equilibrium with a reservoir.9Chapter 2Transport Properties of theQuantum DotWe are interested in evaluating the conductance or current between two leadsattached to the dot. Since we will limit ourselves to steady state transport,it is sufficient that we consider transport between one of the leads and thedot. The current should be the same between the dot and the other leadsince the total charge must be conserved and there is no charge accumulationon the dot in the steady transport state.There are two standard regimes we explore near equilibrium. The firstone is the case where the bias voltage between two leads is smaller than allthe other scales in the system so that we can treat the bias as a perturbation.Then, we define equilibrium as the state where the bias difference betweentwo sites of interest is zero. This condition can also be stated as that thechemical potential is identical among all parts if the system thus the elec-trons have no tendency to jump from one site to the other on average. Wewill call this the “linear response regime”.The other case we can treat near equilibrium is where we start with afinite, arbitrary bias difference between sites but initially these sites, namelythe dot and the lead, are decoupled. We then turn on the perturbation asthe coupling between two sites, allowing electrons to tunnel between twosystems. As long as the coupling parameter is smaller than all the otherscales in system, this approach is justified. Following the convention in theliterature, we will call this regime the “weak-tunneling regime”.What these two approaches have in common is that we start with anequilibrium state and perturb the system slightly. As long as the pertur-bation is small, our approach is valid. However, we run into difficulties ifwe would like to go far from equilibrium. First, linear response approach iscalled ‘linear’ because we truncate the S-matrix expansion to first order inperturbation δH in order to obtain the expression (2.1). If the perturba-tion is large, we would have to take the full S-matrix into account. Second,when we calculate the correlators, standard quantum field theory approachwhere we start with a noninteracting ground state in the distant past and10Chapter 2. Transport Properties of the Quantum Dotturn on and off the interactions adiabatically in the distant past and futurerespectively is not valid since there is the implicit assumption that afterwe turn off the interactions the state we end up with is identical (up to aphase factor) to the ground state we started with. This assumption doesnot hold for a generic time dependent Hamiltonian where the system is notin equilibrium. One way to resolve this issue is that instead of consideringthe evolution of the system from distant past to distant future, one intro-duces a “time loop contour” that starts from a ground state in the distantpast, evolves the system forward in time to the future and then backwardsall the way to ground state we started from in the distant past, defining aclosed loop. In this approach, we have only one noninteracting ground statewe work with, therefore we do not run into the issue we described abovein nonequilibrium situations. An additional problem we run into is specificto disorder averaging which is the crucial step that makes the SYK modelsolvable. Consider the generic correlator〈TˆCc†a(τ1)db(τ2)〉 =∫C D[c, d]ca(τ1)db(τ2)eiS∫C D[c, d]eiSRegardless of the formalism we use, Keldysh or Matsubara, expressions arestructurally the same. If the action S, however, contains random Gaussianvariables that need disorder averaging, note that the same variable wouldappear in both the numerator and denominator. It is not immediately clearhow to proceed with disorder averaging due to the path integral in the de-nominator since the disorder averaging must be carried out simultaneouslyfor both the numerator and the denominator before evaluating the path in-tegrals to obtain an effective action. In Matsubara formalism, this issue canbe avoided by introducing the ‘replica trick’ (as illustrated in [13, 15]) In con-trast, in Keldysh formalism, the denominator is identically unity, thereforeit does not need to be evaluated. [16] Thus, it suffices to disorder averageonly the numerator while completely omitting the denominator. The pricewe pay for this convenience is that we have to promote the saddle pointequations and the Dyson’s equations to matrix equations which we have toresort to regardless since we are interested in nonequilibrium physics. There-fore, Keldysh formalism will be the ideal tool to study the nonequilibriumtransport properties of the SYK model.In the following, our approach will be to study the two aforementionedlimits near equilibrium, and then try to bridge the two in nonequilibriumterritory by deriving a more general formula using Keldysh formalism.112.1. Linear Response2.1 Linear ResponseLinear response formalism allows us to extract transport quantities providedthat we do not perturb the system far from equilibrium. For a genericexpectation value 〈A(t)〉 of an observable operator A, we need to evaluatethe following expression:〈A(t)〉 = 〈A〉0 − i∫ t−∞〈[A(t), δH(t′)]〉H0dt′ (2.1)where the Hamiltonian H = H0 + δH consists of the Hamiltonian H0 ofthe system in the absence of perturbation that leads to response and theperturbation δH, typically coupling to an external system or a gauge fieldwhich leads to response. The correlators one needs to calculate in linearresponse formalism are defined with respect to the equilibrium state (δH =0), where Matsubara formalism can be used at finite temperature.Introducing the minimal coupling in the linear response regimeWe start with a generic Hamiltonian H made of c and ψ operators livingon the quantum dot and the lead respectively. We define the HamiltoniansHc for the dot and Hψ the lead. Assuming [Hc, Hψ] = 0, we further intro-duce the coupling Hcψ between these two systems. Then we can write theHamiltonian for the coupled system:H = Hc +Hψ +HcψwhereHcψ =∑ijVijc†iψj + V∗ijψ†jciWe are interested in the linear response regime where the coupling aboveis fully taken into account to all orders. We introduce a small chemicalpotential difference between two sites and treat this difference as the per-turbation in the linear response regime. This can be achieved by introducingthe minimal coupling in HcdVij =⇒ Vije−i ec∫ ji~A.~dlhere the gauge field ~A will be related to the potential difference U betweentwo sites. In linear response approximation, we expand the perturbed Hamil-tonian to first order in ~A, and write the perturbation δH due to minimal122.1. Linear Responsecoupling separately:δH = −iec∫~A.~dl∑ijVijc†iψj − V ∗ijψ†jciHere we take ~A = icω~E0eiωt since it must hold that ~E = −1c ∂~A∂t and assumethat ~E0 is constant throughout the intermediate region between two sites.We label the length of this region by ~a. Then, the above expression can berewritten in terms of the potential difference between the lead and the dotU/2 = ~E.~a asδH(t) =eUeiωt2ω∑ijVijc†iψj − V ∗ijψ†jcirecall that the current operator can be written asI = −ie∑ijVijc†iψj − V ∗ijψ†jci (2.2)therefore we obtain:δHˆ(t) =iωU2eiωtIˆ(t)Then we use the linear response expression (2.1) for δH and assuming 〈I〉0 =0 since we do not expect a current in the absence of external field:〈I(t)〉 = U2ω∫ t−∞〈[Iˆ(t), Iˆ(t′)]〉δH=0eiωt′dt′ (2.3)= iU2ω∫ ∞−∞−iθ(t− t′)〈[Iˆ(t), Iˆ(t′)]〉δH=0︸ ︷︷ ︸CRII(t−t′)eiωt′dt′ (2.4)where we defined CRII(t − t′) = −iθ(t − t′)〈[I(t), I(t′)]〉δH=0 is the retardedcurrent current correlator between the lead and the dot.〈I(t)〉 = iU2ω∫ ∞−∞CRII(t− t′)eiωt′dt′ (2.5)= − iU2ωeiωt∫ ∞−∞CRII(t− t′)e−iω(t−t′)d(t− t′)︸ ︷︷ ︸CRII(ω)(2.6)132.1. Linear ResponseWhere we multiplied and divided the expression by eiωt and then changedthe integration variable dt′ → d(t− t′) Therefore we arrive at an expressionfor the current between two sites given the potential difference U betweenis given:〈I(t)〉 = − i2ωCRII(ω)︸ ︷︷ ︸σ(ω)UeiωtThe observed quantity would be the conductance:Re[σ(ω)] =ImCRII(ω)2ωThis current current correlator can be obtained using Keldysh time contourformalism. We need to evaluate the time contour correlatorCII(τ1, τ2) = −i〈TˆCI(τ1)I(τ2)〉at saddle point, this expression is found to be (see appendix A.2 for details)CII(τ1, τ2) = −ie2V 2√NM [G(τ1, τ2)G(τ2, τ1) + G(τ1, τ2)G(τ2, τ1)] (2.7)where G is the time contour Green’s function for c operators living on thedot while G represents the Green’s function for the lead operators d Hereτ1, τ2 are defined on the Keldysh contour. We can analytically continue thisexpression to obtain ImCRII(t1, t2). The retarded correlator CRII in real timeis defined as:CRII(t1, t2) = −iθ(t1 − t2)〈[I(t1), I(t2)]〉 (2.8)= −iθ(t1 − t2)(〈I(t1)I(t2)〉︸ ︷︷ ︸iC<II(t+1 ,t−2 )−〈I(t2)I(t1)︸ ︷︷ ︸iC>II(t−1 ,t+2 )〉) (2.9)note that C<II and C>II can be obtained by anaylytically continuing timecontour correlator (2.7) refer to appendix A.3.3. We then obtainC<II(t+1 , t−2 ) = −ie2V 2√NM[G<(t1, t2)G>(t2, t1) + (G↔ G)]C>II(t−1 , t+2 ) = −ie2V 2√NM[G>(t1, t2)G<(t2, t1) + (G↔ G)]if we plug in these expressions into (2.8) and assuming time translationalinvariance (ignoring transient response) we getCRII(t) = −ie2V 2√NMθ(t)[G<(t)G>(−t) + G<(t)G>(−t)−G>(t)G<(−t)− G>(t)G<(−t)] (2.10)142.1. Linear ResponseFourier transforming,CRII(ω) = e2V 2√NM∫dω12pidω22piG<(ω1)G>(ω2)−G>(ω1)G<(ω2)ω1 − ω2 − ω + iδ +(G↔ G)since we are dealing with equilibrium Green’s functions we also haveG<(ω) =inF (ω)A(ω) and G>(ω) = −i(1 − nF (ω))A(ω) and likewise for G Green’sfunctions. Taking the imaginary part,ImCRII(ω) = −pie2V 2√NM2pi∫dω12piG<(ω1)G>(ω1−ω)−G>(ω1)G<(ω1−ω)+ (G↔ G) (2.11)using the relation between lesser and greater Green’s functions and the spec-tral functions in equilibrium, we arrive at:ImCRII(ω) =pie2V 2√NM4pi2∫dω1[nF (ω1−ω)− nF (ω1)][A(ω1)A(ω1−ω)+A(ω1)A(ω1 − ω)] (2.12)〈I(t)〉 = ImCRII(ω)2ω︸ ︷︷ ︸σ(ω)U =1eImCRII(ω)2ωeUZero temperature solution at particle-hole symmetry (µ = 0) NFLAt zero temperature, Fermi functions reduce to step functions:ImCRII(ω) =pie2V 2√NM4pi2∫ ω0dω1[A(ω1)A(ω1 − ω) +A(ω1)A(ω1 − ω)](2.13)For the NFL phase, the solutions to the saddle point equations are given by(at half filling, n = 1/2 or θ = 0 where p < pc = 0.5). Green’s function Gfor the dot and G for one of the leads:GR(ω) =Λe−ipi/4√JωGR(ω) = −√p√Jωeipi/4V 2Λfrom which we can obtain the spectral functions:A(ω) =√2Λ√J |ω| A(ω) =√2p√J |ω|V 2Λ(2.14)152.1. Linear Responseplugging these into above expression for ImCRII(ω) in equation (2.13) weobtain:ImCRII(ω) =√pe2√NM12pi∫ ω0dω1√ω − ω1ω1+√ω1ω − ω1︸ ︷︷ ︸ωpi=√pe2√NM12ωthe current is given by:〈I(t)〉 = ImCRII(ω)2ω︸ ︷︷ ︸σ(ω)U =√pe2√NM4Uwhere U is the overall bias difference between two leads. Reintroducingh = 2pih¯:〈I(t)〉 = pi2√p√NMe2hUZero temperature solution at particle-hole symmetry (µ = 0) FLFor the FL phase (p > pc = 0.5), the Green’s functions (two symmetricalleads) are found to beGR(ω) = − it√2V 21√2p− 1 GR(ω) = −it√2p− 12pand the respective spectral functionsA(ω) =2t√2V 2√2p− 1 A(ω) =2t√2p− 12pand as we calculate the imaginary part of the correlator using the sameexpression in equation (2.13)ImCRII(ω) =e2√NMpiω1√psimilar to previous result in NFL phase, again reintroducing h = 2pih¯〈I(t)〉 = e2h1√p√NMU162.1. Linear ResponseTo summarize (at zero temperature), we find [17] that〈IDC〉 = limω→0ImC(ω)2ωU ={pi2√p√NMU e2h p < 0.51√p√NMU e2h p > 0.5(2.15)where U is the bias difference between one of the leads and the dot. If wedefine G0 =√NM e22h , we find that p = 0.5 the DC conductance of the entiresystem has a jump pi√pG0 → 2√pG0Numerical Study for Linear Response Regime ConductanceIn previous section we have used the conformal solutions that are obtainedfrom the saddle point equations to evaluate the zero temperature DC con-ductance analytically (equation 2.15).Now, we obtain the DC conductance using the numerical solutions ofthe saddle point equations (1.15) to (1.17) in DC current expression weobtained (equation 2.12). Since this is linear response regime, we also use thefluctuation dissipation relation (2.24) for both the dot and the lead Green’sfunctions. This is where temperature enters the calculation. Strictly zerotemperature is not numerically accessible since we have to use a finite valuefor β = 1/kBT 6= 0. We plot the dimensionless DC conductance G/G0 asa function of p = M/N . We expect to see the NFL behaviour for p < 0.5separated from the FL behaviour for p > 0.5 by a critical point. (See thediscussion in section 1.3.1)While the FL (p > 0.5) regime is relatively insensitive to temperature atleast below 1K, in the NFL regime we see deviations from the zero temper-ature curves near the critical point p = 0.5 as we increase the temperature.These deviations are more pronounced at higher temperatures and closer tothe critical point and they occur due to corrections that become significantas the system starts to leave the conformal regime.Numerical solution has the advantage of accessing the entire spectrum ofenergies while the analytic solution [6] is valid only for low energies, boundby a critical frequency ωc(p)  ω (see equation 1.11) below which we ob-serve conformal regime. This means that numerical solutions will includecorrections to the conformal solution as we cross over to higher energiesnear and above ωc(p). Note the p dependence (1.11) of ωc(p) which vanishes(as√1/2− p) [13] as p → 0.5 where we observe the strongest deviationsfrom the conformal behaviour. In the following, we illustrate that the de-viations from the analytic results in Fig. 2.1 are high energy contributionsoutside the conformal part of the spectral densities. Now let us go back172.1. Linear Response0.0 0.2 0.4 0.6 0.8 1.0p0. mK200.0 mK300.0 mK400.0 mK500.0 mK1000.0 mKp2/ p0.0 0.2 0.4 0.6 0.8 1.0p0. K50.0 K100.0 K200.0 Kp2/ pFigure 2.1: Dimensionless DC Conductance in linear response regime at var-ious temperatures as a function of p = M/N Gray dashed lines correspondto zero temperature analytic results (eqn. 2.12) We defined G0 =√NM e22hto the general expression (2.12) after rearranging and writing in terms ofG0 =√NM e22hG/G0 = limω→0V 22∫dω1nF (ω1 − ω)− nF (ω1)ω[A(ω1)A(ω1 − ω)+A(ω1)A(ω1 − ω)]evaluating the limit,G/G0 = −V 2∫dω1∂ωnF (ω1)A(ω1)A(ω1)which further reduces to the following integral after differentiating the Fermifactor:G/G0 = βV2∫dω114 cosh2 βω12A(ω1)A(ω1)note that (4 cosh2 βω12 )−1 is a distribution function which has the form of apeak centred at ω1 = 0 with FWHM = 2 cosh−1(2) 1β ∼ β−1 ∼ T . Conformalregime, where we have analytic expressions for spectral densities [13] is ob-served only for frequencies ω  ωc. For very low temperatures, the 1/ cosh2distribution vanishes before we leave the conformal regime. However, whenT becomes comparable to critical frequency ωc as we increase temperature,182.2. Experimental Considerationscontributions from the nonconformal part of the spectal densities A andA in the integrand above become important and result in deviations fromthe conformal behaviour of the DC conductance. This departure from con-formal behaviour defines a scale T ∗ which marks a crossover to a differentregime which we explore numerically in Fig. 2.2. Notice the drastic changein the qualitative behaviour in both NFL and FL regimes. Even at veryhigh temperatures flat curves above p > 0.5 can easily be distinguished bytheir counterparts below p < 0.2 0.4 0.6 0.8 1.0p0. mK200.0 mK300.0 mK400.0 mK500.0 K1000.0 mKp2/ p0.0 0.2 0.4 0.6 0.8 1.0p0. K50.0 K100.0 K200.0 Kp2/ pFigure 2.2: Dimensionless DC Conductance in linear response regime attemperatures above T  T ∗2.2 Experimental ConsiderationsIn the proposed experimental setup, the main physical parameters that canbe controlled are the bias voltage applied to the leads and the transversemagnetic field that is applied to the graphene flake. It is estimated that inthe original realization [11] of SYK model on a graphene flake in a magneticfield, the number of SYK fermions N is estimated byN =SBΦ0=ΦΦ0which is simply the number of flux quanta threading the graphene flake. Inorder to connect the results we have so far with the real experimental setup,192.3. Weak Tunneling Regimewe relate the parameter p = MN to Φ/Φ0 by eliminating N :1p=Φ/Φ0Mwhich is the number of flux quanta per lead mode where we have M of them.M can be estimated by the conductance GLead of the lead if we assume thelead has M ballistic transport channels such that GLead = Me2h . Therefore,if we wish to use more experimentally relevant quantities, we should plotG/GLead versusΦ/Φ0M . In this case, we can present the data in Fig. 2.1 withrespect to these new parameters in Fig. 2.3.1 2 3 4 5( / 0)/M1. hM)NFLFL100.0 mK200.0 mK300.0 mK400.0 mK500.0 mK1000.0 mKFigure 2.3: DC conductance versus the number of flux quanta per lead mode.2.3 Weak Tunneling RegimeIn this section we consider the “weak tunneling” regime where the biasvoltage across the two leads can be finite. However, the coupling V betweenthe leads and the dot must be the smallest scale in the system such thatwe can treat it as the perturbation with respect to the equilibrium. Forsimplicity, we consider the following setup: The quantum dot is at particle-hole symmetry, µD = 0 and the lead bias voltages are symmetrically shifted202.3. Weak Tunneling Regimeby ± = U/2 such that µL = eU/2 and µR = −eU/2. We then turn on thecoupling (2.21) provided that V is sufficiently small. (We copy the followinganalysis from [17])We can use the standard tunneling conductance result ([18], Pg. 566) afterGaussian averaging the couplings Vij . We can then obtain the followingtunneling current formula for the weak tunneling regime:〈I(t)〉 = 2pieV 2√NM∫ ∞−∞ρψ(+ eU)ρc()[nF ()− nF (+ eU)]d (2.16)The spectral densities we use in this formula are calculated when the leadsare decoupled from the dot. The retarded Green’s function of the SYK4model [5] at particle-hole symmetry for finite temperatures is given by:GR =−iC√2piTΓ(1/4− iβω/2pi)Γ(3/4− iβω/2pi)We can then obtain the spectral density ρc = − 1pi ImGRρc ∝ 1√T|Γ(1/4 + iβω/2pi)|2 cosh(βω2)and the Green’s function for the SYK2 model on the leads can be obtainedfrom the retarded Green’s function by setting V = 0 in saddle point equa-tions and solving for the lead Green’s functions:ρψ =1pitRe√1−( ω2t)2plugging in these expressions into equation (2.16):〈I(t)〉 ∝ eV2t√NM1√T×∫ ∞−∞|Γ(1/4 + iβ/2pi)|2 cosh(β2)[nF ()− nF (+ eU)]dwhere we assumed that the lead spectral density ρψ ≈ 1pit is flat - consideringcontributions only for  t at low energies, effectively introducing a cutofffor the integral bounds which we send to infinity, ignoring further highenergy contributions. We estimate this integral in two limits:212.3. Weak Tunneling RegimeeU  kBT limit: Fermi factors reduce to the derivative of Fermi factorlimβeU→0[nF ()− nF (+ eU) = βeU4 cosh2 (β/2)the integral above becomes:〈IWT )〉 ∝ eV2t√NM1√TeU∫ ∞−∞|Γ(1/4 + iβ/2pi)|24 cosh (β/2)d(β)Note that the integral reduces to a dimensionless constant. Then, from thisexpression we can easily extract the dependence to the external parametersof the model.〈IWT 〉 ∝ eU√T(eU  kBT ) (2.17)eU  kBT limit: Fermi factors introduce limits to the integral this time,as they are effectively step functions at temperatures much smaller than biasvoltage:〈I(t)〉 ∝ eV2t√NM1√T1β∫ 0−βeU|Γ(1/4 + iβ/2pi)|2 cosh(β2)d(β)for β 1 the integrand can be estimated as the following:|Γ(1/4 + iβ/2pi)|2 cosh(β2)∼ 1√|β|then the integral can be estimated as〈IWT 〉 ∝ eV2t√NM1√βT√eUfrom which we can extract the dependencies to external parameters:〈I(t)〉 ∝√eU (eU  kBT ) (2.18)It is important to recall that we assumed ω  t above, which that the resultsthat we have should be valid only when temperature and the bias is muchsmaller than t. To summarize our results for the weak tunneling regime, wefound that the weak tunneling current IWT is given by〈IWT 〉 ∝{eU/√T (eU  kBT )√eU (eU  kBT )(2.19)222.4. Beyond EquilibriumWe can also read off the tunneling conductance from these results:G(U) ∝{1/√T (eU  kBT )1/√U (eU  kBT )(2.20)Note the non-linear behaviour at temperatures much lower than the biasvoltage across the leads.In next section, we will derive a more general formula (2.22) for thecurrent which is valid even when we are far from equilibrium. After obtainingthe general formula, we show that the numerical results we obtain with itreduces to the dependencies (2.19) near equilibrium.2.4 Beyond EquilibriumIn this section we obtain a general expression for the current between twosites where the coupling between the two is given byHcψ =1(NM)1/4∑ijVijc†iψj + V∗ijψ†jci (2.21)where the couplings Vij are drawn from a random Gaussian distribution withthe variance |Vij |2 = V 2. We have used the Keldysh formalism to addressnonequilibrium transport where the bias voltage between two sites can befinite. The derivation given in detail can be found in the appendix A.3. Thegeneral expression for net current from the left lead (ψL operators) to thequantum dot (c operators) is given by〈ILD〉 = ie√NMV 2∫dω{G<(ω)AL(ω)− G<L (ω)A(ω)}(2.22)Where G and GL are the correlators associated with c and ψL respectively.The Green’s functions enter this formula must be calculated in the presenceof couplings between sites. This formula is valid in large-N,M saddle pointapproximation. Similarly, we could define the current from the quantum dotto the right lead (ψR operators)〈IDR〉 = ie√NMV 2∫dω{G<R (ω)A(ω)−G<(ω)AR(ω)} (2.23)In steady state, where there is no charge accumulation on the quantum dot,the net current 〈I〉 through the entire junction must be 〈I〉 = 〈ILD〉 = 〈IDR〉.232.4. Beyond EquilibriumThis is the statement of current conservation which we will address shortlyin section 2.4.3 We assume that the ψL and ψR degrees of freedom, fermionicmodes at the endpoints of the left and right leads respectively, which arethe SYK2 models of the tips of the leads are in equilibrium with reservoirs.However, we do not assume equilibrium for the graphene flake quantumdot, which we take to be at zero chemical potential. We symmetricallyshift the chemical potentials of the leads by ±µ/2 where µ = eU whereU is the potential difference between two leads, across the entire system.In the following analysis we ignore the effect of the reservoirs on the leadsexcept that the reservoirs determine the thermal distribution functions forthe leads. In section 2.5, we will justify this assumption. Since the leadsare in equilibrium as they are in contact with reservoirs, we can impose theFDT condition on the leads Green’s functions. We assume that the left leadhas the higher potential:GKL/R(ω) = 2i tanh(β(ω ± µ/2)2)ImGRL/R(ω). (2.24)With these assumptions, we compute the current (2.22) as a function ofapplied bias in the next section for linear response regime and weak tunnelingregime.2.4.1 I-V characteristics in Weak-Tunneling RegimeIn weak tunneling regime which we introduced in section 2.3, we studied thethe tunneling current IWT in two different limits. While it is proportionalto eU/√T for eU  kBT , in eU  kBT limit it shows√eU dependence. Inthis section, we will match these near equilibrium weak-tunneling results tothe current computed with the formula (2.22) using the numerical solutionsof the saddle point equations (1.15-1.17) in the weak tunneling regime wherewe take V to be sufficiently small. Here we take t = J/2, V = 0.025Jand p = 0.3. First we consider the high bias regime (eU  kBT ). (SeeFig. 2.4(a)) Note that in high bias regime I-V curves do not depend ontemperature and agree with the analytical prediction I ∝√eU/J (2.19) upto eU ∼ 0.1J . At low biases, we observe a temperature dependent behaviouryet it is linear in applied bias. In Fig. 2.4(b) we plot I√T versus eU toobserve the scaling collapse that occurs for I ∝ eU/√T at low bias regime(eU  kBT )242.4. Beyond Equilibrium10 3 10 2 10 1 10010 410 310 2I - [a.u](a)T = 200.0 mKT = 300.0 mKT = 400.0 mKT = 500.0 mKT = 600.0 mKT = 700.0 mKT = 800.0 mKeU/J10 3 10 2 10 1 100eU/J10 310 210 1IT - [a.u](b)T = 200.0 mKT = 300.0 mKT = 400.0 mKT = 500.0 mKT = 600.0 mKT = 700.0 mKT = 800.0 mKeU/JFigure 2.4: I-V characteristics in the weak tunneling regime for various tem-peratures. Numerical results are shown in solid curves. In high bias regimeeU  kBT we find that the current calculated with (2.22) using numericalsolutions of the saddle point equations matches weak tunneling analyticalprediction (2.19) I√T - eU/J characteristics in the weak tunneling regimefor various temperatures. For low bias regime eU  kBT we observe ascaling collapse, confirming the predicted eU/√T dependence252.4. Beyond Equilibrium2.4.2 I-V characteristics in Linear Response RegimeUsing the numerical solutions of the saddle point equations with the as-sumptions described above, we compute the current as a function of appliedbias. We plot the I−V characteristics of the system in Fig. 2.5. Notice thelinear behaviour at very low values of bias eU/J . The current curves arenormalized by a 1/√p factor and for sufficiently low values of bias all curvestend to collapse to a single line, suggesting that current has√p dependence,which is in agreement with the linear response regime results (2.15). As the0.00 0.01 0.02 0.03 0.04 0.05eU/J0. - [a.u]p = 0.2p = 0.24p = 0.28p = 0.32p = 0.36p = 0.4p = 0.44Figure 2.5: DC current versus bias voltage eU at steady state transport inarbitrary units for T= 400mK and various values of p = M/N < 0.5 in theNFL regimebias eU is increased, the curves depart from the linear response regime. Theonset of nonlinear regime begins relatively earlier as we get closer to thephase transition p→ 0.5 whereas deep in the NFL phase (near p = 0.2) wesee linear I − V characteristics up to roughly eU = 0.025JWe plot the current in Fig 2.6 as a function of p for various bias valuesup to eU ∼ J such that we can observe the how the current evolves aswe depart from the linear response regime as a function of p. The lowestbias that we show is eU = 0.006J . For the same bias value, we project262.4. Beyond Equilibrium0.2 0.3 0.4 0.5 1.0p10 210 1100I - [a.u.]Linear Response10 310 210 1100eU/JFigure 2.6: DC current at steady state transport in arbitrary units forT= 400mK. Solid curves correspond to various bias voltages U across twoleads. The dashed curve shows the linear response current projected toeU = 0.006J from the DC conductance G we have calculated numericallyusing (2.12)272.4. Beyond Equilibriumthe linear response current (dashed curve) from the DC conductance wehave computed numerically in section 2.1. This shows that the formula(2.22) captures the linear response regime. We can now increase bias eUand study the departures from the linear response regime. The lowest threecurves (the third one corresponds to eU = 0.04J) show that while the FLpart (p > 0.5) is unaffected, the range of p values below which we see the√p dependence shrinks until we crossover to a different regime at higherbiases at order eU ∼ J where the p dependence of current is very similarto high temperature linear response behaviour (Fig. 2.2) we obtained nearequilibrium.2.4.3 Current conservationIn steady state transport, there should not be any charge accumulation onthe dot. This means that the current between the left lead and the dot ILDis equal to the current between the dot and the right lead IDR. The currentconservationILD = IDRthen implies when we incorporate the equation (2.22) into this equality:∫dω{G<(ω)AL(ω)− G<L (ω)A(ω)}=∫dω{G<R (ω)A(ω)−G<(ω)AR(ω)}which can be rearranged as∫dω{G<(ω)(AL(ω) +AR(ω))− (G<L (ω) + G<R (ω))A(ω)}= 0 (2.25)This is the statement of current conservation and does not depend on thespecifics of the system apart from the random Gaussian form of coupling(2.21) we have defined in the large-N,M limit. It can be shown, for ourmodel, that in the saddle point approximation a stronger statement couldbe made than equation (2.25). Combining the saddle point equations (1.15-1.17) with the Keldysh equation G< = GRΣ<GA which follows from theDyson’s equation in Keldysh formalism, it can be shown that not only theequation (2.25) holds but also the integrand is zero for all frequencies [17].Then we obtain the following form for the G<:G<(ω)(AL(ω) +AR(ω)) = (G<L (ω) + G<R (ω))A(ω) (2.26)since we assume that the leads are in equilibrium with reservoirs, they followthe distributions:fL(ω) =1eβω+eU/2 + 1and fR(ω) =1eβω−eU/2 + 1(2.27)282.5. Coupling to Reservoirsat half filling µ = 0. And in equilibrium for the leads, we can combineG<L/R = ifL/R(ω)AL/R with equation (2.26) to obtain a distribution formfor the dot green’s function G which is not in equilibrium:G<(ω) = ifL(ω)AL(ω) + fR(ω)AR(ω)AL(ω) +AR(ω) A(ω) (2.28)This is the statement of current conservation. We use this condition on thedot Green’s function G when we solve the saddle point equations (1.15-1.17)numerically.2.5 Coupling to ReservoirsIn our analysis we have assumed that the lead endpoints are in equilibriumwith reservoirs but we have so far ignored the effect of the reservoirs onthe leads even though the lead endpoints must be strongly coupled to thereservoirs so that we can assume equilibrium forms (2.27) for the lead end-points. When the extended leads are ignored we have M fermionic modeswhich constitute the SYK2 models on the lead endpoints which are coupledto the dot via equation (2.21). To introduce the effect of the reservoirs,one can couple the reservoirs to the lead endpoints (where the SYK2 lives)with semi-infinite noninteracting 1D ballistic chains. (See SM for reference[17]). This coupling can be taken into account by renormalizing the barelead propagators [17] G0L and G0R (this is shown in equilibrium, thereforeone can work in Matsubara formalism which can then be incorporated intoKeldysh equations in equilibrium):G0L/R(iωn)→ G0L/R(iωn)− ΣE(L/R)(iωn)where, after analytic continuation, retarded self energy ΣRE(L/R) is given by:ΣRE(L/R) = ρEt2E ln∣∣∣∣ω +Dω −D∣∣∣∣− ipiρEt2Ewhere tE is the coupling of the 1D chains to the lead endpoints, ρE is localdensity of states at the end of leads and D  J is a cutoff of the order ofthe lead bandwidths. Once we have the retarded self energy ΣRE(L/R) forboth leads, we can write it a s in Keldysh basis and update the bare leadendpoint propagators in Dyson equations (1.18-1.20) as following:σz (ω + µ)→ σz (ω + µ)− ΣwL/R(ω)292.5. Coupling to Reservoirstherefore, taking the corrections due to coupling to the extended leads intoaccount. Now we can solve the updated saddle point equations numericallyand compare the solutions where we considered the coupling to reservoirswith the earlier solutions where we have ignored it. In Fig. 2.7 we considerthe spectral functions of the dot and the left lead in the same plot for variousvalues of ρEt2E , in equilibrium.10 2 10 1 100/J10 210 1100101102103Spectral Densities0.0J0.1J0.2J0.3J0.4J0.5J0.6J0.7J0.8J0.9J1.0JFigure 2.7: Spectral functions of the dot (solid lines) and the left lead(dashed lines) for various values of ρEt2E in units of J in equilibrium, p = 0.1The low energy behaviour is unaffected (up to ρEt2E ∼ J) by the couplingof the lead endpoints to the semi-infinite wires coupling the system to thereservoirs. In conformal regime, dot and the lead spectral functions showω−1/2 and ω1/2 dependence (2.14) respectively. This conformal behaviour isoutlined by grey dashed lines.We tune the coupling tE from 0 to tE ∼√J/ρE . (t = V = J/2, in equi-librium) Notice that the low energy spectrum where we see the conformalbehaviour is robust in the presence of coupling to semi infinite wires. Sincethe transport quantities we have considered so far depend on low energyfeatures of the spectral functions at low temperatures, we therefore expect302.5. Coupling to Reservoirsto see similar transport behaviour when we actually couple the system tothe reservoirs.Now consider the case where we turn off the disorder, completely ignoringthe SYK2 part at the endpoints of the leads and still couple the SYK4 dot tothe semi-infinite wires with random couplings (2.21). This can be achievedby setting V = J/2 as before, but t = 0 then turn on tE which is the couplingof the lead endpoints to the rest of the semi-infinite wire (Fig. 2.8) Even10 2 10 1 100/J10 210 1100101102103Spectral Densities0.1J0.2J0.3J0.4J0.5J0.6J0.7J0.8J0.9J1.0JFigure 2.8: Spectral functions of the dot (solid lines) and the left lead(dashed lines) for various values of ρEt2E in units of J in equilibrium, p = 0.1,in the absence of SYK2 (t = 0) at the lead endpoints. The low energy be-haviour is the same (compare to Fig. 2.7), regardless of whether disorder ispresent at the lead endpoints.if we completely ignore disorder at the lead endpoints, we find that theconformal low energy behaviour is the same as BA [13] model. We canalso consider the linear response conductance results which we computedearlier in Fig. 2.1. We compare three cases where the system is completelydecoupled from the reservoirs (Fig. 2.9a), coupled to reservoirs (Fig. 2.9b)and finally, the case where there is no disorder at the lead endpoints (t = 0)312.5. Coupling to Reservoirsbut the system is coupled to the reservoirs (Fig. 2.9c) We use the same modelparamters as above, but tE =√J/2piρE when the system is coupled to thereservoirs. See Fig.2.9 for comparison of DC conductances as functions ofp in these three cases we described above. We observe the same qualitativebehaviour in all three cases away from the critical point pc = 0.5, justifyingour assumption of ignoring the explicit coupling to the reservoirs. Notethat the deviations from the conformal behaviour near the critical point isweaker in the presence of coupling to the reservoirs (Figs. 2.9b and 2.9c) incomparison to the isolated model (Fig. 2.9a) we started with.322.5. Coupling to Reservoirs0.2 0.4 0.6 0.8 = 100.0 mKT = 300.0 mKT = 500.0 mKT = 700.0 mKT = 900.0 mKT = 1000.0 mK(a) Decoupled from reservoirs, (t = V = J/2)0.2 0.4 0.6 0.8 Coupled to reservoirs, (t = V = J/2)0.2 0.4 0.6 0.8 Coupled to reservoirs, no disorder at the lead endpoints (V = J/2, t = 0)Figure 2.9: DC conductance as a function of p. We consider the effects ofcoupling to the reservoirs tE =√J/2piρE and the presence of disorder t atthe lead endpoints33Chapter 3Summary and ConclusionWe have explored the charge transport signatures of the SYK model.Startingwith the “linear response” (section 2.1) and the “weak tunneling” (section2.3) regimes near equilibrium, we then bridged these two approaches with amore general formula (A.34) we have derived which is valid in and beyondequilibrium at finite bias voltage across the leads. Although we have usedthis formula for the specific model we studied in this work, the derivationof the current expression for large-N at saddle point approximation did notdepend on the specific details of the model except for the form of coupling(2.21) between two sites between which we consider the current. The formulais valid as long as the two sites we consider for transport admit large-N“classical” saddle point solutions as SYK model does. For instance, thesame formula can be used to study nonequilibrium transport properties ofa chain of SYK models which are coupled via the form (2.21).In linear response regime, at low temperatures, we observed a jump (seeFigs. 2.1 and 2.3) in tunneling conductance as we tune the magnetic fieldthreading the flake. This jump corresponds to an experimental signature ofthe NFL-FL transition proposed in the Banerjee-Altman model [13]. As weincrease the temperature, we find that the jump disappears and we see acrossover to a high temperature (Fig. 2.2) regime where the remnants of theNFL-FL transition at zero temperature can still be seen.Weak tunneling regime allows us to study the current-bias characteristicsat finite biases within the linear response framework. We have found [17](2.20) that in the NFL regime at temperatures much lower than the biasvoltage, tunneling conductance G ∝ U−1/2 is highly nonlinear and does notdepend on temperature. At higher temperatures, conductance G ∝ T−1/2becomes ohmic and exhibits temperature dependence.Using the general current formula (A.34), we have computed the current-bias curves (Fig. 2.6) for finite bias voltage values up to order J . We findthat the jump that we have seen in linear response regime disappears andwe observe a crossover to a high bias regime where the I-V characteristicsare very similar to the high temperature regime I-V characteristics of thelinear response regime at infinitesimal bias across the two leads.34Chapter 3. Summary and ConclusionWe have also considered the effect of the reservoirs on the transportsignatures of the SYK model and shown numerically that the low energybehaviour is unaffected (Fig. 2.7) by explicit coupling of the lead endpointsto featureless extended wires which are modeled as the reservoirs, justifyingour inital assumption of considering only the lead endpoints (assumed to bedecoupled from, yet in equilibrium with the reservoirs) which are genericallymodified by explicit coupling to reservoirs.To summarize, we have proposed a relatively simple transport experi-ment and computed directly measurable quantities displaying signatures ofSYK physics. We believe that the graphene flake realization [11], throughsuch an experiment, would open up more possibilities for further explorationof the SYK model in a laboratory setting, confirm theoretical predictionsand possibly help us overcome the difficulties we encounter using numericaland theoretical techniques.35Bibliography[1] Alexei Kitaev. A simple model of quantum holography. talk given atKITP, Apr 2015.[2] Juan Maldacena and Douglas Stanford. Remarks on the sachdev-ye-kitaev model. Phys. Rev. D, 94:106002, Nov 2016.[3] Juan Maldacena, Stephen H. Shenker, and Douglas Stanford. A boundon chaos. Journal of High Energy Physics, 2016(8):106, Aug 2016.[4] Subir Sachdev. Holographic metals and the fractionalized fermi liquid.Phys. Rev. Lett., 105:151602, Oct 2010.[5] Subir Sachdev. Bekenstein-hawking entropy and strange metals. Phys.Rev. X, 5:041025, Nov 2015.[6] Subir Sachdev and Jinwu Ye. Gapless spin-fluid ground state in arandom quantum heisenberg magnet. Phys. Rev. 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Landauer formula for the currentthrough an interacting electron region. Phys. Rev. Lett., 68:2512–2515,Apr 1992.[20] Hartmut J. W. Haug and Anti-Pekka Jauho. Quantum Kinetics inTransport and Optics of Semiconductors. Springer-Verlag, Berlin, 2008.37Appendix ASaddle Point CalculationsA.1 Keldysh ActionFollowing Ref. [14], we write down the path integral for equations 1.1-1.3,ignoring the extended leads, and obtain an effective action after disorderaveragingZ =∫D [ψ,ψ, c, c] eiS , (A.1)where the Grassmann fields ψ, c correspond to a lead endpoint and the dot,respectively. We work with only one lead for simplicity since both leads areidentical. The other lead will be introduced at the end of the calculation.The real-time action is defined on the Keldysh contour and can be writtenas a sum of contributions from the lead, dot, and coupling between them:S = SL + SD + SLD (A.2)SL =∑s∑α∫dt{ψαs(t)s [i∂t + µ]ψαs(t)}−∑ss′∫ ∫dtdt′ss′ it22M(∑αψαs(t)ψαs′(t′))∑βψβs′(t′)ψβs(t)SD =∑s∑i∫dt {cis(t)s [i∂t + µ] cis(t)}+∑ss′∫ ∫dtdt′ss′ iJ24N3(∑icis(t)cis′(t′))2∑jcjs′(t′)cjs(t)2(A.3)38A.1. Keldysh ActionSLD = −∑ss′∫ ∫dtdt′{ss′iV 2√NM(∑icis(t)cis′(t′))(∑αψαs′(t′)ψαs(t))}.(A.4)The integrals run from −∞ to ∞ and the index s = ±1 labels the forwardand backward direction on the Keldysh contour. We introduce the fields Gand G together with the Lagrange multipliers Σc,ψ:∫D[G,Σc]eN∑ss′∫ ∫dtdt′Σcss′ (t,t′)[Gs′s(t′,t)− iN∑i cis(t)cis′ (t′)] = 1∫D[G,Σψ]eM∑ss′∫ ∫dtdt′Σψss′ (t,t′)[Gs′s(t′,t)− iM∑i ψis(t)ψis′ (t′)] = 1.The resulting action isSL =∑ss′∑α∫ ∫dtdt′{ψαs(t)[σzss′δtt′ (i∂t + µ)− Σψss′(t, t′)]ψαs′(t′)}+∑ss′∫ ∫dtdt′{iss′Mt22Gs′s(t′, t)Gss′(t, t′)− iMΣψss′(t, t′)Gs′s(t′, t)}SD =∑ss′∑i∫ ∫dtdt′{cis(t)[σzss′δtt′ (i∂t + µ)− Σcss′(t, t′)]cis′(t′)}+∑ss′∫ ∫dtdt′{iss′NJ24G2s′s(t′, t)G2ss′(t, t′)− iNΣcss′(t, t′)Gs′s(t′, t)}SLD =∑ss′∫ ∫dtdt′{iss′√NMV 2Gs′s(t′, t)Gss′(t, t′)}After integrating out fermions, we find the saddle point of the actionδSδGss′(t, t′)= 0,δSδGss′(t, t′) = 0 (A.5)δSδΣcss′(t, t′)= 0,δSδΣψss′(t, t′)= 0 (A.6)39A.2. Linear Response at Saddle PointDropping the dependence on two time indices, we obtain the saddle-pointequations which follow from equations (A.5)Σcss′(t) =ss′J2G2ss′(t)Gs′s(−t) + ss′√pV 2Gss′(t) (A.7)Σψss′(t) =ss′t2Gss′(t) + ss′ V2√pGss′(t), (A.8)where p = M/N . These are supplemented by the (matrix) Dyson equationfor the frequency-dependent Green’s functions which we obtain from (A.6)Gss′(ω) = [σz (ω + µ)− Σc]−1ss′ (A.9)Gss′(ω) =[σz (ω + µ)− Σψ]−1ss′(A.10)These equations can easily be generalized to the case where there are twoseparate flavours of ψ operators corresponding to left and right leads. Weneed to add a third term to (A.7) similar to its second term (to introducethe other lead) and write down another self energy equation similar to (A.8)as well as a separate Dyson’s equaiton for this new fermion flavour we in-troduced. The final result is given by (1.15-1.17)A.2 Linear Response at Saddle PointWe need to evaluate the time contour correlatorCII(τ1, τ2) = −i〈TˆCI(τ1)I(τ2)〉which can then be analytically continued to the retarded correlation functionCRII(ω). Plugging in the expression for current operator in equation (2.2), weobtain the following (we suppress the denominator in Vij = Vij/(NM)1/4)CII(τ1, τ2) = ie2∑ijklVijVkl〈Tˆ c†i (τ1)dj(τ1)c†k(τ2)dl(τ2)〉− VijV ∗kl〈Tˆ c†i (τ1)dj(τ1)d†l (τ2)ck(τ2)〉− V ∗ijVkl〈Tˆ d†j(τ1)ci(τ1)c†k(τ2)dl(τ2)〉+ V ∗ijV∗kl〈Tˆ d†j(τ1)ci(τ1)d†l (τ2)ck(τ2)〉 (A.11)We can evaluate these correlators in path integral formalism (Keldysh con-tour) at the saddle point. Let us compute one of these terms explicitly as40A.2. Linear Response at Saddle Pointthe others will follow similarly. For instance, consider the third term in theabove expression. In time-contour path integral formalism, we can write:− ie2∑ijklV ∗ijVkl〈Tˆ d†j(τ1)ci(τ1)c†k(τ2)dl(τ2)〉= −ie2∑ijkl∫D[c, d]eiSc+iSddjτ1ciτ1ckτ2dlτ2V ∗ijVkle∑ij φijVij+φ∗ijV∗ij (A.12)where Sc and Sd are the Keldysh actions for c and d fermions. We wrotethe coupling term separately, with the shorthands φij = −i∑τ ciτdjτ andφ∗ij = −i∑τ djτciτ . In order to obtain the Gaussian average of this quantityover the distribution {Vij} we need the following intermediate result (seeAppendix A.3.6 for a proof):V ∗a Vbe∑m Vmφm+V∗mφ∗m ={(V 2 + V 4φaφ∗a)e∑φφ∗ a = bV 4φaφ∗be∑φφ∗ a 6= b (A.13)this then allows us to writeV ∗ijVkle∑ij φijVij+φ∗ijV∗ij = δikδjl[V 2 + V 4φijφ∗ij]+ (1− δikδjl)V 4φijφ∗klIn large N,M limit, only the V 2 term survives to leading order inside thepath integral. Only second and third terms include terms of order V 2 andwe only keep these terms in the large N,M limit. We then end up withCII(τ1, τ2) = −ie2∫D[c, d]eiSc+iSd∑ijV 2[djτ1ciτ1ciτ2djτ2+ ciτ1djτ1djτ2ciτ2 ]e∑φφ∗Rearranging this expression yieldsCII(τ1, τ2) = iV2e2NM∫D[c, d]eiSc+iSd+∑φφ∗ (1N∑iciτ1ciτ2)︸ ︷︷ ︸iG(τ1,τ2)× ( 1M∑jdjτ2djτ1)︸ ︷︷ ︸iG(τ2,τ1)+ (1N∑iciτ2ciτ1)︸ ︷︷ ︸iG(τ2,τ1)(1M∑jdjτ1djτ2)︸ ︷︷ ︸iG(τ1,τ2)(A.14)where the expressions in brackets above yield the Green’s functions G and Gin large-N saddle point after disorder averaging and Hubbard-Stratanovich41A.3. Current at Saddle Pointdecoupling. Recall we had suppressed the denominator in Vij = Vij/(NM)1/4Therefore we should also have V 2 → V 2/√NM . We then obtain the ex-pression:CII(τ1, τ2) = −ie2V 2√NM [G(τ1, τ2)G(τ2, τ1) + G(τ1, τ2)G(τ2, τ1)]Here τ1, τ2 are defined on the Keldysh contour. We can analytically continuethis expression to obtain ImCRII(t1, t2).A.3 Current at Saddle PointIn the following, we derive a formula for the current between two islandsusing the Keldysh formalism. Our approach is similar to the work of Meirand Wingreen [19]. We start with a current operator and then evaluateits expectation value while considering coupling between the two sites toall orders. The formula we derive applies when the couplins between thetwo sites are Gaussian random of the form (2.21) and is valid as long asthe individual Hamiltonians at each island admit “classical” large-N saddlepoint solutions. For instance, the formula is valid even if both islands areSYK4 like. The current operator (rate of change of the charge on the lead)is given byI = −ie∑abVabc†aψb − V ∗abψ†bcaNote that we suppress the denominator in Vij = Vij/(NM)1/4 To evaluatethe expectation value of the current, we need an expression for the time(contour) ordered operator under disorder average:Vab〈Tˆ c†a(τ1)ψb(τ2)〉and similarly V ∗ab〈Tˆψ†b(τ1)ca(τ2)〉 which appears due to the second term inthe current expression we have above. The above correlator (before disorderaverage) can be written in the path integral formalism as following:∑abVab(i〈Tˆ c†a(τ1)ψb(τ2)〉)= i∑abVab∫C D[c, ψ]ca(τ1)ψb(τ2) exp (iS)∫C D[c, ψ] exp (iS)where the Keldysh action S is given byS = Sc + Sψ −∫dτHcψ(τ)42A.3. Current at Saddle PointHere we split the action for c and ψ fermions from the term in the Hamilto-nian that couples the two flavours which we label by Hcψ. Keldysh formalismis especially convenient for performing disorder averages since the denomi-nator Z = D[c, ψ] exp (iS) for the above correlator expression is genericallyunity. [16] Therefore, for the following discussion we can ignorethe denominator and proceed with the averaging. We consider thecoupling term to be of the formHcψ =∑ijVijc†iψj + V∗ijψ†jcito simplify the notation, we write the time (contour) integrals as sums∫ →∑τ and change the time index notation on fermionic Grassman numbersas the following: ca(τ)→ caτ . Now we can rewrite the expression (A.3) as:∑abVabi〈Tˆ c†a(τ1)ψb(τ2)〉 = i∑ab∫CD[c, ψ]caτ1ψbτ2eiSc+iSψ× Vab exp−i∑ijτVijciτψjτ exp−i∑ijτV ∗ijψjτciτ (A.15)Now we would like to compute the average of this quantity over complexGaussian distributions Vij with variance V2. Here we assume that amongall i, j labels, Vij distributions are independent. We are interested in theidentity:Vabe(∑ij Vijφij+∑ij V∗ijφ∗ij) = V 2φ∗ab × e(∑ij Vijφij+∑ij V∗ijφ∗ij) (A.16)where we defined the shorthands φij = −i∑τ ciτψjτ and φ∗ij = −i∑τ ψjτciτ .Note that star (*) here does not mean complex conjugate. Proof of thisidentity is given in appendix A.3.5. The average on the RHS above can beevaluated by completing Gaussian integrals to square (see appendix A.3.4)Replacing φijs back, we obtain (the expression under the bar in equation(A.15)):−iV 2∫dτ ′′ψbτ ′′caτ ′′ expV 2 ∫ dτdτ ′(∑iciτ ciτ ′)∑jψjτ ′ψjτ43A.3. Current at Saddle Pointwhere we replaced sums with integrals over time contours back ∑τ →∫:finally we plug this expression into (A.15):Vabi〈Tˆ c†a(τ1)ψb(τ2)〉 = −V 2∫Cdτ ′′∫D[c, ψ]∑acaτ1caτ ′′∑bψbτ ′′ψbτ2× eiSc+iSψeV 2∫dτdτ ′(∑i ciτ ciτ ′)(∑j ψjτ ′ψjτ)Now let us decouple the exponential by introducing a bosonic unity:∫D[P,Q] exp(−∫dτdτ ′Pττ ′Qτ ′τ)= 1 (A.17)the exponential in the path integral above can we written as:eV2∫dτdτ ′... =∫D[P,Q]e∫dτdτ ′[V 2∑i ciτ ciτ ′∑j ψjτ ′ψjτ−Pττ ′Qτ ′τ ]now we do the following change of variables Pττ ′ → Pττ ′ + V∑i ciτ ciτ ′ andQτ ′τ → Qτ ′τ + V∑j ψjτ ′ψjτ cancelling the quadratic fermion cross term:eV2∫dτdτ ′... =∫D[P,Q]e∫dτdτ ′[−Pττ ′Qτ ′τ+V Pττ ′∑j ψjτ ′ψjτ+V Qτ ′τ∑i ciτ ciτ ′ ]But this completely decouples two different flavours of fermions, allowing usto rewrite the combined path integral as a product:Vab〈iTˆ c†a(τ1)ψb(τ2)〉 = −V 2∫Cdτ ′′∫D[P,Q]e−∫ττ ′ Pττ ′Qτ ′τ×∫D[c]∑acaτ1caτ ′′eiSceV (∫ττ ′ Qτ ′τ∑i ciτ ciτ ′)×∫D[ψ]∑bψbτ ′′ψbτ2eiSdeV (∫ττ ′ Pττ ′∑j ψjτ ′ψjτ) (A.18)A.3.1 Evaluation of Gaussian integralsWe would like to evaluate the expression (A.18) above. If fermionic degreesof freedom are already noninteracting (bilinear), their respective actionscan be Gaussian integrated. Let us consider the following part of the aboveaction:∫D[c]∑acaτ1caτ ′′eiSceV (∫ττ ′ Qτ ′τ∑i ciτ ciτ ′)=1VδδQτ ′′τ1∫D[c]eiSceV (∫ττ ′ Qτ ′τ∑i ciτ ciτ ′) (A.19)44A.3. Current at Saddle PointIn RHS above, we simplified the expression by writing it with a functionalderivative. Note that the path integral on the RHS must be a Gaussianintegral or it can be brought to a Gaussian form by further decoupling withHS transformations. This will introduce more bosonic fields which appearas terms like Qτ ′τ in the exponential as well as overall bosonic path integralsfor the correlator expression. Let us assume without loss of generality thatSc =∫τ∑i ciτ [G−10 δττ ′+Bττ ′ ]ciτ ′ has a Gaussian form where Bττ ′ representthe possible the extra terms we just mentioned above. In the presence ofsuch fields, overall path integrals∫ DB over these degrees of freedom areimplied. (G−10 = i∂τ) The Gaussian integral above then can be written as:∫D[c] exp(∫ττ ′∑iciτ[iG−10 δττ ′ +Bττ ′ + V Qτ ′τ]ciτ ′)=∏i∫D[ci] exp−∑ττ ′ciτ (−2)[iG−10 δττ ′ +Bττ ′ + V Qτ ′τ]︸ ︷︷ ︸Ci(τ,τ ′)ciτ ′=∏idet(Ci) = exp(∑iTr[log(Ci)]) = exp(NTr[log(C)]) (A.20)The last equality holds since Ci = C is the same for all i in the sum. Noticethat C(τ, τ ′) = −i2G−1(τ, τ ′) where G is the Green’s function renormalizeddue to V couplings (as well as extra B fields). We read off this relationfrom the above form of the Gaussian integral. If we go back to the aboveexpression (A.19) to evaluate the path integral,∫D[c]∑acaτ1caτ ′′eiSceV (∫ττ ′ Qτ ′τ∑i ciτ ciτ ′) =1VδδQτ ′′τ1eNTr[log(C)]= NeNTr log(C)(−2)C(τ ′′, τ1)−1 = −iNG(τ ′′, τ1)eNTr[log(C)] (A.21)where in the last step we used the relation between C and G that we foundabove. Note that the above expression did not depend on the pres-ence or form of additional B bosonic degrees of freedom. Theywill vanish under the functional derivative with respect to Q. Theexpression for the the other path integral similarly follows. To summarizethese results, we have:∫D[c]∑acaτ1caτ ′′eiSceV (∫ττ ′ Qτ ′τ∑i ciτ ciτ ′) = −iNG(τ ′′, τ1)eNTr[log(C)](A.22)45A.3. Current at Saddle Point∫D[ψ]∑bψbτ ′′dψbτ2eiSψeV (∫ττ ′ Pττ ′∑j ψjτ ′djτ) = −iMG(τ2, τ ′′)eMTr[log(D)](A.23)whereCαβ = (−2)[iG−10 δαβ +Bαβ + V Qβα]= −i2G−1αβ (A.24)Dαβ = (−2)[iG−10 δαβ +B′αβ + V Pβα]= −i2G−1αβ (A.25)The expression for the ψ fermions is very similar - note that we introducedB′ similar to the B we had for c fermions. Now we use equations (A.22)and (A.23) in the original path integral (A.18).∑abVabi〈Tˆ c†a(τ1)db(τ2)〉= MNV 2∫Cdτ ′′∫D[P,Q,B,B′]G(τ2, τ ′′)G(τ ′′, τ1)× e−∫B,B′...e−∫ττ ′ Pττ ′Qτ ′τ+NTr[log(C)]+MTr[log(D)] (A.26)A.3.2 Effective action and the large-N limitTo compute the effective action at saddle point we would have followed theexact same steps to obtain the disorder averaged expression for Z = 1 exceptthat we would not take the functional derivatives to bring down the Green’sfunctions in the fermionic path integrals as we did in (A.21)Z =∫D[P,Q,B,B′]e−∫B,B′...e−∫ττ ′ Pττ ′Qτ ′τ+NTr[log(C)]+MTr[log(D)](A.27)we represent the additional bosonic fields by B and B′ which appear af-ter decoupling of interacting fermionic degrees of freedom. The expressione−∫B,B′... is a shorthand for these additional terms in the action. We canrewrite the above expression asZ =∫D[P,Q,B,B′]e−Seff [P,Q,B,B′]whereSeff [P,Q,B,B′] =∫B,B′...+∫ττ ′Pττ ′Qτ ′τ −NTr[logC(B,Q)]−MTr[logD(B′, Q)] (A.28)46A.3. Current at Saddle Pointwe can obtain the saddle point equations with:δSeffδPab= 0δSeffδQab= 0δSeffδB= 0 ...For a general formula, however, we do not need to obtain these equationsexplicitly. Fermionic degrees of freedom are integrated out but they dependon the bosonic fields. Next step is to evaluate the bosonic path integrals. Wecan approximate these path integrals with the saddle point approximation.The crucial observation is that as long as the fluctuations with respect tothe saddle point vanish in the large-N limit, the path integral will be givenby the classical action S0eff evaluated at the saddle point. In large-N limit,saddle point approximation then reads:Z =∫D[P,Q,B,B′]e−Seff [P,Q,B,B′] ≈ e−Seff [P 0,Q0,B0,B′0] = 1since in Keldysh formalism Z = 1. Therefore we obtain:e−Seff [P0,Q0,B0,B′0] = 1where the superscript means the saddle point values of the bosonic fields.Now we go back to the numerator we obtained above (equation (A.26)).Note that we can do the saddle point approximation that we used for Z asit is essentially the same expression except that it also contains the Green’sfunctions before the exponent. Note that the Green’s functions G(B,Q)and G(B′, P ) depend on the fields P,Q,B,B′... (eqns A.24, A.25) and theycan not be taken outside the integral immediately. But in the large-N limit,their fluctuations with respect to the saddle point vanish and we can replacethem with their values at the saddle point and we can take them outsidethe path integral. Therefore, in large-N limit, we can rewrite the numerator(A.26) as:∑abVabi〈Tˆ c†a(τ1)ψb(τ2)〉 = MNV 2∫Cdτ ′′ G(τ2, τ ′′)G(τ ′′, τ1)︸ ︷︷ ︸saddle-point values×∫D[P,Q,B,B′]e−∫B,B′...e−∫ττ ′ Pττ ′Qτ ′τ+NTr[log(C)]+MTr[log(D)]︸ ︷︷ ︸e−Seff [P0,Q0,B0,B′0]=1(A.29)but note that the rest of the integral is nothing but Z = e−Seff [P 0,Q0,B0,B′0] =1. Therefore, we arrive at the disorder averaged saddle-point current for-mula:∑abVabi〈Tˆ c†a(τ1)ψb(τ2)〉 = MNV 2∫Cdτ ′′G(τ2, τ ′′)G(τ ′′, τ1) (A.30)47A.3. Current at Saddle Point∑abV ∗abi〈Tˆψ†b(τ1)ca(τ2)〉 = MNV 2∫Cdτ ′′G(τ2, τ ′′)G(τ ′′, τ1) (A.31)where G and G are the saddle point Green’s functions evaluated in thepresence of all interactions.A.3.3 Analytic continuationWe are interested in the real-time quantitites:C<1 (t1, t2) =∑abVabi〈c†a(t2)ψb(t1)〉 C<2 (t1, t2) =∑abV ∗abi〈ψ†b(t2)ca(t1)〉with these we can write the current expectation value as:〈I〉 = limt1,t2→t−e (C<1 (t1, t2)− C<2 (t1, t2))We can obtain these lesser Green’s functions by using Langareth’s rules [20].The expressionC(τ1, τ2) =∫Cdτ ′A(τ1, τ ′)B(τ ′, τ2)on time ordered contour can be analytically continued to:C<(t1, t2) =∫ ∞−∞dt′Ar(t1, t′)B<(t′, t2) +A<(t1, t′)Ba(t′, t2)using this expression for equations (A.30) and (A.31) The current expressionthen becomes:〈I〉 = −eNMV 2∫dt′Gr(t, t′)G<(t′, t) + G<(t, t′)Ga(t′, t)−Gr(t, t′)G<(t′, t)−G<(t, t′)Ga(t′, t)Fourier transforming (assume time translational invariance), we arrive atthe final formula:〈I〉 = −eNMV 2∫dω{G<(ω) [Gr(ω)− Ga(ω)]− G<(ω) [Gr(ω)−Ga(ω)]}(A.32)48A.3. Current at Saddle Pointwe can simplify this further by using the relation: GR − GA = 2iImGR =−iA(ω)〈I〉 = ieNMV 2∫dω{G<(ω)A(ω)− G<(ω)A(ω)} (A.33)Recall we had suppressed the denominator in Vij = Vij/(NM)1/4 Thereforewe should also have V 2 → V 2/√NM .〈I〉 = ie√NMV 2∫dω{G<(ω)A(ω)− G<(ω)A(ω)} (A.34)A.3.4 Disorder averaging of the couplingWe would like to evaluateexp∑ijVijφij +∑ijV ∗ijφ∗ijlet us suppress the indices ij → i for simplicity. We can rewrite this expres-sion as:exp(∑iViφi + V ∗i φ∗i)=∏iexp (Viφi + V ∗i φ∗i )since all Vi are independent random variables, it suffices that we evaluatethe average (for Γ = V 2 where V is the varianceexp (Viφi + V ∗i φ∗i ) =∫dze−zz∗/ΓpiΓe(zφ+z∗φ∗)If we switch to real variables z = x+ iy:exp (Viφi + V ∗i φ∗i ) =∫dxdypiΓexp(−x2 + y2Γ) + (x+ iy)φ+ (x− iy)φ∗if we split the x and y integrals and complete them to square, we are leftwith:=1piΓ∫dxdye−(x2+y2)/Γ︸ ︷︷ ︸=1eΓφφ∗for the last step we assumed that φ and φ∗ commute. This is true in ourcase since they are Grassman bilinears. Therefore, we find that:exp∑ijVijφij + V ∗ijφ∗ij = expV 2∑ijφijφ∗ij49A.3. Current at Saddle PointA.3.5 Proof of the Gaussian average identity (A.16)We define the complex Gaussian distribution as:p(z) =e−zz∗/ΓpiΓNow let us further suppress the pair labels i, j → i and a, b→ a without lossof generality. Then if we evaluate the variance σ we find 〈zz∗〉 = Γ = σ2.Now we split the sums in the exponential as following:VaeVaφaeV∗a φ∗a exp∑i 6=aViφi exp∑i 6=aV ∗i φ∗i= VaeVaφaeV∗a φ∗a × exp∑i 6=aViφi exp∑i 6=aV ∗i φ∗i (A.35)since the distribution Va is independent of Vi for all i 6= a, we can factor theaverages as above. Now we focus on the first factor and the average over Vaas we expand the exponentials:VaeVaφaeV∗a φ∗a =∑mnVa(V ∗a φ∗a)nn!(Vaφa)mm!=∑mnVa(V ∗a )m(Va)n(φ∗a)mm!(φa)nn!(A.36)Now we use a version of Wick’s theorem for complex Gaussian integrals.This theorem tells us that the average of a product of Gaussian distributedvariables can we factored into averages of all pairings. For example:〈z∗i zjz∗kzl〉 = 〈z∗i zj〉〈z∗kzl〉+ 〈z∗i zl〉〈z∗kzj〉Only pairings of type 〈z∗z〉 = σ2 survive. It can be shown that 〈zz〉 =〈z∗z∗〉 = 0. Therefore we have〈VijV ∗kl〉 = V 2δikδjlNow go back to the average we wish to evaluate:Va(V ∗a )m(Va)nAccording to Wick’s theorem, we can factor this product into all possiblepairings of V s and V ∗s. Let us focus on the first Va that occurs in the50A.3. Current at Saddle Pointproduct. For this expression to survive the average, this first Va must bepaired with a V ∗a . There are m different ways of doing this and they are allidentical:Va(V ∗a )m(Va)n = m× VaV ∗a ×(all pairings of (V ∗a )m−1(Va)n)= m× VaV ∗a︸ ︷︷ ︸V 2×(V ∗a )m−1(Va)nif we use this result in above expression (A.36), we obtainVaeVaφaeV∗a φ∗a = V 2φ∗a∑mn(V ∗a )m−1(Va)n(φ∗a)m−1(m− 1)!(φa)nn!= V 2φ∗a∑mn(V ∗a φ∗a)m−1(m− 1)!(Vaφa)nn!but now we see the averaged expression in the above line is nothing but theproduct of exponentials. Plugging this result back into (A.35)Va exp(∑iViφi)exp(∑iV ∗i φ∗i)= V 2φ∗aeVaφaeV∗a φ∗a×e∑i 6=a Viφie∑i 6=a V∗i φ∗i(A.37)but again, the product of these two averages can be combined because aver-ages of Vi are independent. Then we finish the proof state our intermediateresult about V averages here:Vab exp∑ijVijφij +∑ijV ∗ijφ∗ij = V 2φ∗ab × exp∑ijVijφij +∑ijV ∗ijφ∗ij(A.38)A.3.6 Proof of Gaussian Identity (A.13)In this section, we prove the following identity:V ∗a Vbe∑m Vmφm+V∗mφ∗m ={(V 2 + V 4φaφ∗a)e∑φφ∗ a = bV 4φaφ∗be∑φφ∗ a 6= b51A.3. Current at Saddle PointCase 1, a = bThe above expression reduces to:V ∗a Vb exp(∑mVmφm + V ∗mφ∗m)= V ∗a Va exp (Vaφa + V ∗a φ∗a)× exp∑m6=aVmφm + V ∗mφ∗m (A.39)The first factor V ∗a Va exp (Vaφa + V ∗a φ∗a) can be written as∑m,nV ∗a VaV ma φmam!(V ∗a )n(φ∗a)nn!= V ∗a Va∑m,nV ma φmam!(V ∗a )n(φ∗a)nn!+∑m,nm× V ∗a Vam!n× VaV ∗an!V m−1a (V ∗a )n−1φma (φ∗a)nwhere used a variant of the Wick’s theorem we introduced in the previoussection. While the first term on the RHS is the pairing of Va and V∗a thatwas outside the exponential initially, the second term involves the pairingsof these two factors with the exponential series. These two terms can againbe written up as exponentials:= (V 2 + V 4φaφ∗a)exp (Vaφa + V∗a φ∗a)completing the proof, where we used 〈VaV ∗b 〉 = V 2δab.Case 2, a 6= bIn this case we can split the averages as following:V ∗a Vb exp(∑mVmφm + V ∗mφ∗m)= V ∗a exp (Vaφa + V ∗a φ∗a)× Vb exp (Vaφa + V ∗a φ∗a)× e(∑m 6=a,b Vmφm+V ∗mφ∗m)we can use the result (A.37) from the previous section to simplify this ex-pression= V 2φaexp (Vaφa + V ∗a φ∗a)× V 2φ∗bexp(Vbφb + V∗b φ∗b)× e(∑m 6=a,b Vmφm+V ∗mφ∗m)= V 4φaφ∗be(∑m Vmφm+V∗mφ∗m)completing the proof.52


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