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Tricritical Ising edge modes in a Majorana-Ising ladder Li, Chengshu 2017

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Tricritical Ising Edge Modes in aMajorana-Ising LadderbyChengshu LiB.Sc., Tsinghua University, 2015A 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)July 2017c© Chengshu Li 2017AbstractWhile Majorana fermions remain at large as fundamental particles, theyemerge in condensed matter systems with peculiar properties. Grover et al.[1] proposed a Majorana-Ising chain model, or the GSV model, where thesystem undergoes a tricritical Ising transition by tuning just one parameter.In this work, we generalize this model to a ladder with inter-chain Majoranacouplings. From a mean field analysis, we argue that the tricritical Isingtransition will also occur with inter-chain couplings that allow the systemto be gapless in the non-interacting case. More crucially, based on analysis ofthe interacting chain model and the non-interacting ladder model, we expectthe tricritical Ising modes to appear on the edges, a feature that mightpersist when going to 2d. We carry out extensive DMRG calculations toverify the theory in the ladder model. Finally, we discuss possible numericalprobes of a 2d model.iiLay SummaryIn solid state systems, particles that do not exist in vacuum can emerge,with interesting properties and possible applications to new materials anddevices. Furthermore, one can assembly these particles in a sophisticatedway that allows even more bizarre behaviors. In this work we propose such amodel, which has a configuration of a ladder, and its properties are studiedboth analytically and numerically.iiiPrefaceThis thesis is original, unpublished, independent work by the author, Cheng-shu Li.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The GSV model . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The TCI edge mode and the ladder/2d model . . . . . . . . 21.3 Possible experimental realizations . . . . . . . . . . . . . . . 22 The GSV model . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 A detailed discussion of the model . . . . . . . . . . . . . . . 32.2 Conformal field theory in 1 + 1D . . . . . . . . . . . . . . . . 62.3 DMRG algorithm . . . . . . . . . . . . . . . . . . . . . . . . 72.4 DMRG results . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Majorana-Ising ladder . . . . . . . . . . . . . . . . . . . . . . . 123.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 From a chain to a ladder: non-interacting model . . . . . . . 133.2.1 The m = 0 case . . . . . . . . . . . . . . . . . . . . . 153.2.2 The m 6= 0 case . . . . . . . . . . . . . . . . . . . . . 153.2.3 Low-energy field theory . . . . . . . . . . . . . . . . . 163.3 From a chain to a ladder: interacting model . . . . . . . . . 183.4 DMRG results . . . . . . . . . . . . . . . . . . . . . . . . . . 20vTable of Contents4 From a ladder to 2d . . . . . . . . . . . . . . . . . . . . . . . . 244.1 The 2d model . . . . . . . . . . . . . . . . . . . . . . . . . . 244.1.1 The non-interacting model . . . . . . . . . . . . . . . 244.1.2 The interacting model . . . . . . . . . . . . . . . . . . 284.2 Entanglement spectrum, edge modes and infinite DMRG . . 294.3 The non-interacting model revisited . . . . . . . . . . . . . . 305 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35AppendixA Majorana fermions . . . . . . . . . . . . . . . . . . . . . . . . . 37viList of Tables2.1 A dictionary for the central charges of some common CFTs. . 9viiList of Figures1.1 Majorana-Ising chain model, or the GSV model. . . . . . . . 11.2 The phase diagram of the GSV model. . . . . . . . . . . . . . 22.1 Phase diagram of the t1-t2 model. In the Majorana repre-sentation (up), t1 = t2 is the gapless point. In the spinrepresentation (down), t1 < t2 and t1 > t2 corresponds toferromagnetic and paramagnetic phases, respectively. . . . . . 52.2 Phase diagram from MFT analysis. To make contact withthe numerical results in Section 2.4, we have label the threephases with the central charge c, also see Table 2.1. . . . . . . 62.3 Conformal invariance and phase transitions. (a)(b) and (a)(c)show two conformal transformations z → w1 = 2z and z →w2 = z2. (d)(e)(f) Under these transformations the config-uration of a critical Ising model is “invariant”, the definitemeaning of which is clarified in CFT. . . . . . . . . . . . . . . 62.4 The pure spin model after Jordan-Wigner transformation. . . 82.5 A typical fitting of the central charge with Eq. 2.17. We omitthe points lA ∼ 1 because the CFT prediction works at longdistances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 DMRG result of the Majorana-Ising chain model. Here wehave L = 80, g = 1, J = 0.3. . . . . . . . . . . . . . . . . . . 113.1 Braids and anyons. We consider five kinds of exchanges ofidentical particles, whose worldlines are shown in (a) ∼ (e).In d ≥ 3, (a), (d), (e) and (b), (c) are two kinds of topo-logically equivalent exchanges. In 2d, all five exchanges, orbraids, are different, and the representation of the braid groupdetermines the type of the anyons. . . . . . . . . . . . . . . . 133.2 Majorana ladder model. The arrows denote the sign of eachcoupling, in agreement with the Grosfeld-Stern rule. . . . . . 143.3 Phase diagram of the m = 0 Majorana ladder model. Thegapless region is the line t1 = 2t2. . . . . . . . . . . . . . . . . 15viiiList of Figures3.4 Phase diagram of the Majorana ladder model. The gaplessregion is shifted as one turns on m. . . . . . . . . . . . . . . . 163.5 Majorana-Ising ladder model. The sign convention is thesame as in Fig. 3.2 and we suppress the arrows for clarity. . . 183.6 From a chain to a ladder. We are interested in the chiral TCICFT in the red box. A comparison between the first and thethird row implies a TCI CFT, while one between the secondand the third column suggests that it is chiral. This argumentis verified by the DMRG calculations. . . . . . . . . . . . . . 193.7 The spin model after Jordan-Wigner transformation. . . . . . 203.8 Decoupled Majorana-Ising ladder. The central charge is twicethat of a chain. We take L = 24, g = 1, J = 0.3 here. . . . . . 213.9 Majorana-Ising ladder with g = 1, J = 0.3, t1 = 2t2 =2 (blue), 1 (red), 0.4 (green). All three cases exhibit a TCItransition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.10 Majorana-Ising ladder with L = 60, g = 1, J = 0.3, t1 =1, t2 = 0.8. There is a Ising transition between two gappedphases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.1 The 2d non-interacting model. Note that we have use a dif-ferent convention than in the previous chapter to have a min-imun unit cell. . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 The phase diagram of the 2d non-interacting model. A non-zero Chern number n signifies a topological phase and a chiraledge mode. Note that since we “double” the Hilbert space ina Majorana system, the Chern number we get is twice thereal value. Also, the values of t and t1 are irrelevent as longas they remain positive. . . . . . . . . . . . . . . . . . . . . . 264.3 The edge modes from the low-energy theory. While the bulkmodes are gapped out, the edge modes are left gapless. . . . . 284.4 The DMRG paths for the non-interacting model with a cylin-der geometry, with the size of (a)∞× 4 and (b)∞× 8 in theMajorana basis, or (a) ∞× 2 and (b) ∞× 4 in the spin basis. 304.5 The CFT towers and the entanglement spectrum. The linesare given by the Ising CFT which contains three conformaltowers. The dots are the entanglement spectrum from infiniteDMRG upon a shift and rescaling. We see an almost perfectmatch here. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31ixList of Figures4.6 The twisted transfer matrix (TTM) and the transverse mo-mentum. The twisted transfer matrix is constructed by mul-tiplicating the MPS with a unitry transformation U whichrotates the cylinder by a lattice constant. The fixed point Ris diagonal and the diagonal elements are eika. . . . . . . . . 324.7 The entanglement spectrum of the ∞× 8 case. Here to tellapart different conformal towers the transverse momentum ismeasured and translated by a multiple of 2pi for clarity. Wecan see that the pattern “110111112” holds for every tower. . 33A.1 The Kitaev chain model can be written in Dirac fermion op-erators (a) or Majorana fermion operators (b)(c). (b) Thetrivial phase can be continuously connected to the isolatedatom limit without closing the gap. (c)(d) In the topologicalphase, there is a topologically protected zero energy state. . . 39xAcknowledgementsFirst of all, I would like to extend my thanks to my supervisor, Prof. MarcelFranz, who has patiently helped me throughout my research. I have beendeeply impressed and influenced by his taste and insights in physics. I wouldlike to thank Dr. Miles Stoudenmire for developing the DMRG library ITen-sor. I would like to thank Dr. Miles Stoudenmire and Prof. Lukasz Cinciofor providing timely advice on various technical details. I would also liketo thank Dr. Dmitry Pikulin, Dr. Armin Rahmani, Xiaoyu Zhu, E´tienneLantagne-Hurtubise, Og˘uzhan Can, Tianyu Liu, Anffany Chen, EnriqueColome´s and Dr. Emilian Nica for helpful discussions. Finally, I wouldlike to thank my family for their support.xiChapter 1IntroductionIn this chapter we discuss the background and main concerns of the thesis.We first introduce the Majorana-Ising chain model, or the GSV model, whichmotivates our work. A more technical discussion is detailed in Chapter 2.Then we discuss the features of the model that motivates a generalization toa ladder/2d model. We also briefly discuss possible experimental realizationsof the models.1.1 The GSV modelGrover, Sheng and Vishwanath[1] came up with a Majorana-Ising chainmodel in 2014. For a background discussion on Majorana fermions, see Ap-pendix A. We postpone the detailed analysis of the model till Chapter 2 andonly sketch the physics in this section. The model is shown in Fig. 1.1. Thenearest neighbor couplings between the Majorana fermions are modulatedby an Ising spin field, which in turn has nearest neighbor longitudinal anti-ferromagnetic Ising couplings and experiences a transverse magnetic field h.αj αj+1µj µj+1Figure 1.1: Majorana-Ising chain model, or the GSV model.In this model, if one fixes all other parameters and tunes h, there willbe three different phases. For small h, the system is gapped, while for largeh the system is gapless and in a critical Ising phase. For a particular valueof h, a tricritical Ising (TCI) transition will occur. The phase diagram isshown in Fig. 1.2.Compared with other known models, this is the first model where a TCIis obtained by tuning only one parameter. Moreover, it is known that the11.2. The TCI edge mode and the ladder/2d modelTCI is supersymmetric in an algebraic sense[2], which means that super-symmetry, a long-expected but elusive ingredient of various grand unifiedtheories, is realized in a condensed matter system.hgapped TCI IsingFigure 1.2: The phase diagram of the GSV model.1.2 The TCI edge mode and the ladder/2d modelNow we focus on the TCI point. From the low energy field theoretic pointof view, there is a left-moving mode and a right-moving mode. A naturalquestion is, can we separate the left- and right- movers into different edges?Behaviour of this kind is familiar in various quantum Hall effects and giverise to interesting phenomena, e.g. the quantized conductance. A separationof left- and right- movers will keep them from being gapped out by localcouplings, and will in turn stabilize the edge modes. Thus we expect astable TCI phase when we go to 2d.As a first step to constructing a 2d model, we consider coupling twochains to form a ladder model. We resort to the non-interacting model for acoupling scheme, see Section 3.2 for a thorough discussion. It is known thatif one introduces a direct coupling t1 and a diagonal coupling t2 between twochains, as in Fig. 3.2, we will have edge modes when t1 = 2t2. We introducethis coupling scheme to the Majorana-Ising model, and expect that a TCIedge mode will emerge with appropriate parameters. In this work we showthat this indeed is the case by both analytical and numerical argument.A logical generalization is then a 2d model. However, the numericaltechnique used in the chain and ladder model, the density matrix renormal-ization group (DMRG), is not applicable in 2d. In the last part of this thesis,we discuss a possible numerical approach and apply it to the non-interactingsystem.1.3 Possible experimental realizationsIn the original GSV model, Grover et al. proposed a potential realizationat the surface of a slab of He3-B, where Majorana modes are expected to21.3. Possible experimental realizationsemerge. The phase transition is driven by a magnetic field parallel to thesurface. We expect that our model can also be realized in this setup.3Chapter 2The GSV modelIn this chapter we discuss the model and relevant technical background. Wefirst give a detailed discussion of the GSV model in Section 2.1. Then weintroduce in Section 2.2 the conformal field theory, a powerful and versatiletool that we rely on heavily throughout the work. We briefly review the nu-merical technique, DMRG, in Section 2.3. Finally, we present the numericalresults of the GSV model in Section 2.4.2.1 A detailed discussion of the modelThe Hamiltonian of the GSV model isH = HM +HIsing +HgHM = it∑jαjαj+1HIsing = J∑jµzjµzj+1 − h∑jµxjHg = −igt∑jαjαj+1µzj(2.1)We examine each term in detail. HM is a tight binding model for Majoranafermions, where the i guarantees hermiticity. For reasons that will becomeclear shortly, we consider a slightly more general HamiltonianH ′M = it1∑jαjβj + it2∑jβjαj+1 (2.2)42.1. A detailed discussion of the modelwhere t1, t2 > 0, and H′M = HM when t1 = t2 = t. The bulk spectrum isreadily obtained by going to momentum spaceαk =√12N∑jeikjαjαj =√2N∑ke−ikjαkβk =√12N∑jeikjβjβj =√2N∑ke−ikjβk(2.3)where N is the size for α and β (thus the total size is 2N). We have set thelattice constant to be 1 for simplicity and chosen a normalization so that{α†k, αk′} = {β†k, βk′} = δkk′ (2.4)In momentum space the Hamiltonian isH ′M =∑ki(t1 − eikt2)α†kβk − i(t1 − e−ikt2)β†kαk (2.5)We obtain the spectrum upon diagonalizationE =√t21 + t22 − 2t1t2 cos k =√(t1 − t2)2 + 4t1t2 sin2 k2(2.6)The spectrum is gapped when t1 6= t2, and becomes gapless at t1 = t2. Inter-estingly, the gaplessness is related to the phase transition of the transversefield Ising model. To see that we perform a Jordan-Wigner transformationαj = σxj∏k<jσzkβj = σyj∏k<jσzk(2.7)and the Hamiltonian now readsH ′M = −t1∑jσzj − t2∑jσxj σxj+1 (2.8)52.1. A detailed discussion of the modelMajoranaSpint1t2gapped gapless gappedferro. para.Figure 2.1: Phase diagram of the t1-t2 model. In the Majorana representa-tion (up), t1 = t2 is the gapless point. In the spin representation (down),t1 < t2 and t1 > t2 corresponds to ferromagnetic and paramagnetic phases,respectively.We can interpret t1 as the transverse magnetic field and t2 as the ferromag-netic Ising coupling. Now the two phases of t1 > t2 and t1 < t2 correspond tothe paramagnetic and the ferromagnetic phase, respectively, and the phasetransition is equivalent to an Ising one.HIsing is a transverse field antiferromagnetic Ising model. We can mapit into the familiar ferromagnetic one by rotating the spins on the even sitesby pi around the x-axisU =∏even jexp(ipi2µxj)U−1HIsingU = −J∑jµzjµzj+1 − h∑jµxj(2.9)Thus the phase transition occurs at J = h.Finally, Hg couples the spin and Majorana degrees of freedom. Thisterm breaks the Z2 symmetry of HIsing.The interaction term Hg thwarts an exact solution, and as a first steptoward understanding we resort to a mean field theory (MFT) analysis.Fortunately, much of the physics is already captured from this very simpleapproach.The assumption we make is that the spin degree of freedom in Hg, µz, isnot dynamical, but enters only as a parameter, which is in turn determinedby HIsing. In other word, we make the MFT substitutionHg = −igt∑jαjαj+1µzj → HMFT = −igt∑jαjαj+1〈µzj 〉 (2.10)Now we go to the two limiting cases of h→∞ and h→ 0. When h→∞,all the spins align in the x-direction, leaving 〈µzj 〉 = 0. Thus the interactingterm vanishes, and HM is in a gapless phase of the Ising class.When h → 0, the spins are antiferromagnetically ordered, with 〈µzj 〉 =(−1)j . Then HM + HMFT is equivalent to the t1-t2 model discussed in theprevious section, and the spectrum is gapped.62.2. Conformal field theory in 1 + 1Dhc = 0 c = 710 c =12Figure 2.2: Phase diagram from MFT analysis. To make contact with thenumerical results in Section 2.4, we have label the three phases with thecentral charge c, also see Table 2.1.When tuning h from 0 to∞, the system evolves from a gapped phase toan Ising phase, thus we expect a TCI phase transition in between. Numericalresults are shown in Section 2.4.2.2 Conformal field theory in 1 + 1D(a) (b) (c)(d) (e) (f)Figure 2.3: Conformal invariance and phase transitions. (a)(b) and (a)(c)show two conformal transformations z → w1 = 2z and z → w2 = z2.(d)(e)(f) Under these transformations the configuration of a critical Isingmodel is “invariant”, the definite meaning of which is clarified in CFT.At second-order phase transitions, the correlation length diverges, andan emergent scaling invariance is realized. In the critical regime, one canfurther assume conformal invariance, an even larger symmetry to be defined72.3. DMRG algorithmshortly[3][2][4]. Conformal field theory (CFT) is developed to fully exploitthe conformal invariance in the formulation of field theory. In 1 + 1D or2 + 0D in particular, conformal invariance is so restrictive that quite a lotof results can be deduced based on only a few assumptions. We utilize only1 + 1D CFT in this work.Conformal transformation is defined bygµν(x)→ g′µν(x′) = Ω(x)gµν(x) (2.11)where gµν(x) is the metric tensor and Ω(x) is a scalar field. In 1 + 1D, thiscorresponds to all analytic coordinate transformations z → f(z), where wehave gone to complex coordinates z = x+iy and we measure space and timecoordinates in the same unit by setting v = 1. A system can be describedby CFT if there exist fields called primary fields which transform accordingtoΦ(z, z¯)→(∂f∂z)h(∂f¯∂z¯)h¯Φ(f(z), f¯(z¯))(2.12)where h and h¯ are real numbers known as conformal weights.An important parameter in CFT is the central charge c, defined throughthe radially ordered operator product expansion (OPE)R (T (z)T (w)) =c/2(z − w)4 +2(z − w)2T (w) +1z − w∂T (w) (2.13)Here, T (z) is the stress-energy tensor, and R denotes radial ordering. Impos-ing a unitary condition on the theory, one can show that only a few discretevalues of c are allowed, each can be identified with some phase transitionclasses. Thus, c provides an easy probe for the identification of a phasetransition. In this work we focus on the tricritical Ising transition (TCI),with central charge c = 710 .2.3 DMRG algorithmInteracting systems are notoriously hard to solve numerically, mainly due tothe fact that the dimension of the Hilbert space grows exponentially withthe system size. As a classical example, the Hilbert space of the Hubbardmodel is 4N and is as large as 4800 ' 10480 for a moderate system size of20×20×20. As a result, only systems of relatively small sizes can be solvedby brute force.Fortunately, for (quasi-)1D system, a good ansatz known as the matrixproduct state (MPS)[5] turns out to give an almost accurate ground state82.4. DMRG resultswavefunction. The basic idea is to rewrite the wavefunction in terms ofproducts of matrices by singular value decomposition (SVD)ψi1,i2,··· ,in =∑j1,j2,··· ,jn−1Aj1i1Aj1,j2i2· · ·Ajn−2,jn−1in−1 Ajn−1in(2.14)Within each step, we truncate the matrix by keeping only the dominantsingular values. From the modern view, the density matrix renormalizationgroup (DMRG) is a variational method based on MPS. The computationalcost is only O(Lm3), where L is the system size and m is the bond dimensionof MPS, typically 102 ∼ 103. The efficiency comes from the fact that in1D the entanglement between two subsystems grows only logarithmically, aspecial case of the area law. In this work we carry out extensive DMRGcalculations to obtain the ground state with a cutoff error as small as 10−6.2.4 DMRG resultsWe apply finite DMRG algorithm to calculate the ground state wavefunctionof the full Hamiltonian with periodic boundary condition using ITensor[6].We have again performed Jordan-Wigner transformation so that the Hamil-tonian is spin-12 only, which is easier for numerical simulations. The Hamil-tonian isH = −t∑j(σxj σxj+1 − σzj ) + gt∑j(σxj σxj+1µzj,a − σzjµzj,b)+ J∑j(µzj,aµzj,b + µzj,bµzj+1,a)− h∑j(µxj,a + µxj,b)(2.15)Figure 2.4: The pure spin model after Jordan-Wigner transformation.As discussed in Section 2.2, the main weapon at our disposal is the centralcharge c. The central charges of some common CFTs are shown in Table 2.1.In order to extract the central charge, we measure the entanglement entropy92.4. DMRG resultsCFT central charge cvacuum 0free fermions/Ising 12free bosons 1tricritical Ising 710Table 2.1: A dictionary for the central charges of some common CFTs.SA of subsystems of various sizes from the ground state wavefunction. SAis defined from the reduced density matrix ρASA = Tr(ρA ln ρA) (2.16)where ρA = TrBρ. SA is related to central charge c by the famous formula[7]SA =c3ln[Lpiasin(pilAaL)]+ S0 (2.17)where lA and L are lengths of the subsystem and the whole system, respec-tively, a is the lattice constant, and S0 is a constant independent of lA. Atypical fitting is shown in Fig. 2.5.Our main results are shown in Fig. 2.6, in agreement with Grover et al.’sresults.102.4. DMRG resultsFigure 2.5: A typical fitting of the central charge with Eq. 2.17. We omitthe points lA ∼ 1 because the CFT prediction works at long distances.112.4. DMRG resultsFigure 2.6: DMRG result of the Majorana-Ising chain model. Here we haveL = 80, g = 1, J = 0.3.12Chapter 3Majorana-Ising ladderIn this chapter the main results of this research project are presented. Weprovide some general remarks in Section 3.1, motivating the generalizationto a ladder and eventually to 2d. Then in Section 3.2 we discuss the non-interacting Majorana ladder model, which, while serving as the MFT ap-proximation of the fully interacting model, is of great interest on its own.The fully interacting model is introduced and analyzed in Section 3.3, wherewe argue the existence of the TCI chiral edge modes, the key result of thiswork. Finally, the numerics we perform to support the analytical argumentare summarized in Section 3.4.3.1 General remarksBased on the discussions so far, we want to generalize the model to 2d,where richer physics is expected. We briefly discuss two aspects therein.Edge physics and edge-bulk correspondence. The edge-bulk corre-spondence is an echoing feature in topological phases. It appears alreadyin 1d systems such as the Su-Schrieffer-Heeger model[8], where the edgecharge is determined by the bulk phase. In 2d an archetypal example is thequantum Hall effect (QHE), one of the first and most thoroughly studiedtopological phases. In QHE the experimentally observed quantized conduc-tance boils down to the chiral edge modes, or equivalently a topologicalinvariance of the bulk states, a surprising connection bridged by Thouless,Kohmoto, Nightingale and Den Nijs[9]. While in QHE the bulk is exactlysolvable, in the current situation the edge is more easily probed thanks toCFT. We thus expect an investigation of the edge physics interesting andinformative.Anyons in 2d. In 1d, particles can not pass through each other, and thereis no difference between bosons and fermions. In d ≥ 3, there is only onetopologically (more precisely, homotopically) equivalent way of exchanging133.2. From a chain to a ladder: non-interacting modeltwo identical particles, and thus the unitary transformation of exchanging Nparticles is given by the representation of the permutation group SN . Indeed,bosons correspond to the trivial representation and fermions correspond toanother 1-d representation. In 2d, however, when particles exchange theworldlines can braid in an infinite number of different ways. The symmetrygroup for N identical particles is therefore the braid group BN , and anyonsemerge as the representations of the group.(a) (b) (c) (d) (e)Figure 3.1: Braids and anyons. We consider five kinds of exchanges ofidentical particles, whose worldlines are shown in (a) ∼ (e). In d ≥ 3, (a),(d), (e) and (b), (c) are two kinds of topologically equivalent exchanges. In2d, all five exchanges, or braids, are different, and the representation of thebraid group determines the type of the anyons.3.2 From a chain to a ladder: non-interactingmodelAs a first step towards 2d, we couple two chains and consider a ladder model.Again, before plunging into the full interacting model, we first examine anon-interacting one. We note that for a 2d Majorana system there is aconsistency condition on the coupling parameters, known as the Grosfeld-Stern rule[10], which requires that the accumulated phase φ along a closedloop is related to the number of vertices n byφ =pi2(n− 2) (3.1)The sign convention we choose is shown in Fig. 3.2, and the Hamiltonian143.2. From a chain to a ladder: non-interacting modelαj αj+1βj βj+1Figure 3.2: Majorana ladder model. The arrows denote the sign of eachcoupling, in agreement with the Grosfeld-Stern rule.isH = i∑j(t+m(−1)j) (αjαj+1 − βjβj+1) + it1∑jβjαj+ it2∑j(αjβj+1 − βjαj+1)(3.2)where t ±m are the intra-chain couplings and t1,2 are the inter-chain cou-plings. Note that t1,2 have a different meaning in the previous chapter, andtheir role is played by m here. We go to the momentum space using Eq. 2.3H =∑kΨ†kHkΨkΨk = (αk,even, βk,even, αk,odd, βk,odd)THk = i0 0 0 0t1 0 0 0−(t+m) + (t−m)e−ik t2(1 + e−ik) 0 0−t2(1 + e−ik) (t+m)− (t−m)e−ik t1 0+ h.c.(3.3)In general, diagonalizing a 4 × 4 hermitian matrix is quite laborious, anduseful information is hard to extract. (It is always possible since the problemreduces to a quartic equation, whose root formula is known.) Fortunately,for certain matrices there is a nice trick. A hermitian matrix can always beexpanded by (I,σ) ⊗ (I, τ ), where σ and τ are Pauli matrices. If there arenot too many terms, upon squaring and rearranging the matrix a few timesit will become proportional to unit matrix, with the cross terms cancelledout by the anticommutation relations. In the current situation, we haveHk = t1σy − t2(1 + cos k)σyτx + t2 sin k σyτy+ ((t−m) cos k − (t+m))σzτy+ (t−m) sin k σzτx(3.4)153.2. From a chain to a ladder: non-interacting modeland the spectrum isE =(t21 + t22(1 + cos k)2 + t22 sin2 k +((t−m) cos k − (t+m))2+(t−m)2 sin2 k ± 2t2(t21((1 + cos k)2 + sin2 k)+((1 + cos k) ((t−m) cos k − (t+m)) + sin2 k(t−m))2 ) 12) 12(3.5)This result is rather complicated and we discuss the two cases m = 0 andm 6= 0 separately. We also formulate a low-energy field theory to obtainmore insights.3.2.1 The m = 0 caseNow the energy spectrum can be further simplified toE =√4t2 sin2k2+(t1 ± 2t2 cos k2)2(3.6)The system is gapped for general t, t1 and t2, but becomes gapless whent1 = 2t2. To see this, we note that both terms under the square root mustbe 0 in order that the system is gapless. The first term is 0 only whensin k2 = 0, and thus t1 ± 2t2 = 0. The phase diagram is shown in Fig. 3.3.t1t2gaplessgappedFigure 3.3: Phase diagram of the m = 0 Majorana ladder model. Thegapless region is the line t1 = 2t2.3.2.2 The m 6= 0 caseThe spectrum is more complicated. To proceed, we restrict out attention tonot too large m, so that the spectrum does not change too much. A corollary163.2. From a chain to a ladder: non-interacting modelis that the E is small only around k = 0. We also note that ∂E∂k∣∣k=0= 0,since only cos k and sin2 k are present in E. Combining this with the factthat E ≥ 0 and E is smooth, we expect E = 0 can only occur at k = 0E|k=0 =(t21 + 4t22 + 4m2 ± 4t2(t21 + 4m2)12) 12=∣∣∣∣2t2 ±√t21 + 4m2∣∣∣∣ (3.7)and the gapless condition followst2 =√(t12)2+m2 (3.8)The upshot is that the gapless region is not removed, but shifted. Whilethis does not pose a problem in the ladder model, when we go to 2d thingsbecome tricky. The discussion is postponed to Chapter 4.2101 2 3 t1t2m = 0m = 1m = 2Figure 3.4: Phase diagram of the Majorana ladder model. The gaplessregion is shifted as one turns on m.3.2.3 Low-energy field theoryTo gain more insight into the relation between the ladder and the chain,we now adopt a low-energy field theoretic view. To do so we assume the173.2. From a chain to a ladder: non-interacting modeloperators on even/odd sites change slowly so that∑j→∫dxα2j → α1(x)α2j+2 → α1(x) + 2∂α1∂xα2j−1 → α2(x)α2j+1 → α2(x) + 2∂α2∂x(3.9)and similar for β’s. The low-energy Hamiltonian readsH = 2i∫dx(mα1α2 + (t+m)α1α′2 − (α↔ β))+ it1∫dx (β1α1 + β2α2)+ it2∫dx(α1β2 + α2β1 + 2α1β′2 − (α↔ β))(3.10)We further defineαR/L = (α1 ± α2)/2βR/L = (β1 ∓ β2)/2(3.11)and in terms of these operators the Hamiltonian finally becomesH = 2i∫dx(2mαLαR + (t+m)(αRα′R − αLα′L) + (α↔ β))+ 2it1∫dx (βRαL + βLαR)+ 4it2∫dx (αRβL − αLβR − αRβ′R + αLβ′L)(3.12)where we have ignored the total derivative or boundary terms. In the iso-lated chain limit, t1 = t2 = 0, the Hamiltonian reduces to the sum of twochains, of which the physical meaning is clear: within each chain there is aright-moving mode and a left-moving mode, with velocity ±2t. A nonzerom couples both modes, just like a mass term in the high-energy context,hence the nomenclature. For m = 0, the two modes contribute to the lowenergy states and explains the gaplessness of each chain.When m = 0, a general inter-chain coupling t1 and t2 couples all the fourmodes and thus gap out the spectrum. However, from Eq. 3.12 it is easy183.3. From a chain to a ladder: interacting modelto see that if one chooses t1 = 2t2, only two out of four modes are coupled,and we are left with two chiral gapless modes on the edge. This analysisis readily generalizable to a 2d model, where a similar coupling scheme willgap out all the bulk modes but leave two edge modes untouched, as we willdiscuss in detail in Chapter 4.3.3 From a chain to a ladder: interacting modelEquipped with the insights from the non-interacting model, we now explorethe full interacting model, Fig. 3.5, with intra-chain spin-spin and spin-Majorana couplings. The Hamiltonian isH = it∑j(αjαj+1 − βjβj+1)− igt∑j(αjαj+1µzj,a − βjβj+1µzj,b)+ it1∑jβjαj + it2∑j(αjβj+1 − βjαj+1)+ J∑j(µzj,aµzj+1,a + µzj,bµzj+1,b)− h∑j(µxj,a + µxj,b)(3.13)αjβj βj+1αj+1µj,aµj,bFigure 3.5: Majorana-Ising ladder model. The sign convention is the sameas in Fig. 3.2 and we suppress the arrows for clarity.We expect the success of MFT in the chain model will persist in the193.3. From a chain to a ladder: interacting modelladder model, with the MFT HamiltonianHMFT = it∑j(αjαj+1 − βjβj+1)− igt∑j(αjαj+1〈µzj,a〉 − βjβj+1〈µzj,b〉)+ it1∑jβjαj + it2∑j(αjβj+1 − βjαj+1)+ J∑j(µzj,aµzj+1,a + µzj,bµzj+1,b)− h∑j(µxj,a + µxj,b)(3.14)By the same argument as in the chain model, the limiting cases h → 0,∞correspond to m 6= 0 and m = 0 in the non-interacting model, respectively.If we fix t1 = 2t2, they further correspond to a gapped phase and an Isingphase. Crucially, there is only one copy of Ising CFT in the latter. Thus, aswe tune h from 0 to ∞, we expect a TCI transition in between, where thenumber of TCI CFT is also one. Furthermore, as the Ising CFT is chiral,so should be the TCI CFT. We illustrate the argument in Fig. 3.6.gappedc = 0TCIc = 710gappedc = 0gappedc = 0single chaintwo decoupled chainsladder with t1 = 2t2h = 0 h = hc h =∞Isingc = 122×TCIc = 752×Isingc = 1chiral TCIc = 710chiral Isingc = 12Figure 3.6: From a chain to a ladder. We are interested in the chiral TCICFT in the red box. A comparison between the first and the third rowimplies a TCI CFT, while one between the second and the third columnsuggests that it is chiral. This argument is verified by the DMRG calcula-tions.203.4. DMRG results3.4 DMRG resultsWe perform extensive DMRG simulations to confirm the analytical analysisin the previous section. For bookkeeping we copy here the spin-versionHamiltonianH = −t∑j(σxj σyj+1 + σyj σxj+1) + gt∑j(σyj σxj+1µzj,a + σxj σyj+1µzj,b)− t1∑jσzj + t2∑j(σxj σxj+1 + σyj σyj+1)+ J∑j(µzj,aµzj+1,a + µzj,bµzj+1,b)− h∑j(µxj,a + µxj,b)(3.15)where αj and βj become σj under the Jordan-Wigner transformation, asshown in Fig. 3.7.µj,aµj,bαjβjαj+1βj+1σj σj+1µj,a µj,bFigure 3.7: The spin model after Jordan-Wigner transformation.First, we take t1 = t2 = 0, i.e. the case of two isolated chains. Weexpect the central charge to be twice of a single chain. The result is shownin Fig. 3.8.Next, we take t1 = 2t2 = 2, 1, 0.4. We expect that only one pair of chiraledge modes survive. The result is shown in Fig. 3.9.We also explore the case t1 6= 2t2. We take t1 = 1 and t2 = 0.8. Sincethe Majorana degrees of freedom are gapped out, we expect the systemis dominated by the spin ladder, and an Ising transition lies between twogapped phases. The result is shown in Fig. 3.10.213.4. DMRG resultsFigure 3.8: Decoupled Majorana-Ising ladder. The central charge is twicethat of a chain. We take L = 24, g = 1, J = 0.3 here.223.4. DMRG resultsFigure 3.9: Majorana-Ising ladder with g = 1, J = 0.3, t1 = 2t2 =2 (blue), 1 (red), 0.4 (green). All three cases exhibit a TCI transition.233.4. DMRG resultsFigure 3.10: Majorana-Ising ladder with L = 60, g = 1, J = 0.3, t1 =1, t2 = 0.8. There is a Ising transition between two gapped phases.24Chapter 4From a ladder to 2dAs motivated in Chapter 3, the ultimate goal is to probe a 2d model. Unfor-tunately, DMRG is impeded by the area law and a direct simulation is outof the question. In this chapter we first analytically discuss the 2d modelfrom various aspects in Section 4.1. Then we introduce a useful concept,known as the entanglement spectrum, and discuss its relevance to the edgegapless modes in Section 4.2. We also discuss the numerical techniques.An application of the approach to the non-interacting model is shown inSection 4.3.4.1 The 2d model4.1.1 The non-interacting modelThe Hamiltonian of the 2d non-interacting Majorana model isH = i∑ij((−1)it−m) γi,jγi+1,j − it1∑ij(−1)iγi,jγi,j+1+ it2∑ij(γi,jγi−1,j+1 − γi,jγi+1,j+1)(4.1)The advantage of studying a ladder model is that we have direct accessto the edge states in the analytical solution, which is unfortunately not thecase for a 2d model. Here, we can only solve the bulk spectrum. As usual254.1. The 2d modeltt1 t2Figure 4.1: The 2d non-interacting model. Note that we have use a differentconvention than in the previous chapter to have a minimun unit cell.we go to the momentum spaceαkx,ky =√12N∑ijeikx2i+ikyjγ2i,jβkx,ky =√12N∑ijeikx(2i+1)+ikyjγ2i+1,jγ2i,j =√2N∑kxkye−ikx2i−ikyjαkx,kyγ2i+1,j =√2N∑kxkye−ikx(2i+1)−ikyjβkx,ky(4.2)In momentum space the Hamiltonian readsH =∑kΨ†kHkΨkΨk = (αk , βk)T(4.3)withHk = 2( −t1 sin ky it cos kx − (m+ 2t2 cos ky) sin kx−it cos kx − (m+ 2t2 cos ky) sin kx t1 sin ky)= −2 sin kx(m+ 2t2 cos ky)σx − 2t cos kxσy − 2t1 sin kyσz(4.4)264.1. The 2d modeland we immediately get the bulk spectrumE = 2√sin2 kx(m+ 2t2 cos ky)2 + t2 cos2 kx + t21 sin2 ky (4.5)It is straightforward to see when the bulk spectrum is gapless. The threeterms are all zero only whencos kx = sin ky = m+ 2t2 cos ky = 0⇒ kx = ±pi2,{ m = 2t2ky = pior{ m = −2t2ky = 0(4.6)We can calculate the Chern number of the system from the low energy pointsin the Brillouin zone[11], where the Hamiltonian isH(−pi2 ,0) = 2(m+ 2t2)σx − 2tqxσy − 2t1qyσzH(−pi2 ,pi) = 2(m− 2t2)σx − 2tqxσy + 2t1qyσzH(pi2 ,0)= −2(m+ 2t2)σx + 2tqxσy − 2t1qyσzH(pi2 ,pi)= −2(m− 2t2)σx + 2tqxσy + 2t1qyσz(4.7)and the Chern number followsn =12∑sign(vxvy∆) ={ 0 if |m| > 2t22 if |m| < 2t2 (4.8)m−2t2 2t2n = 2 n = 0n = 0Figure 4.2: The phase diagram of the 2d non-interacting model. A non-zero Chern number n signifies a topological phase and a chiral edge mode.Note that since we “double” the Hilbert space in a Majorana system, theChern number we get is twice the real value. Also, the values of t and t1 areirrelevent as long as they remain positive.The upshot of this calculation is that, the edge of the 2d system issimilar to the ladder case, with a gapless phase and a gapped phase, whichis determined by m. However, there are two noteworthy differences betweenthe 2d and the ladder case. In the latter, an infinitesimal m will gap out thespectrum, while in the former the transition happens at m = 2t2, a finite274.1. The 2d modelvalue. Moreover, in 2d the bulk spectrum becomes gapless at the transitionpoint, a fact obscured in the ladder case, as there is not a well-defined “bulk”for a ladder.The edge mode is best understood from the low-energy theory. We recallthat when m = 0 there is a left-moving mode and a right-moving modewithin each chain, and the t1 and t2 terms gap out these modes exceptwhen t1 = 2t2, in which case only a pair of modes are gapped out andanother pair is left untouched. As we introduce more chains, the new t1 andt2 terms will successively gap out the bulk modes, and leave a pair of edgemodes gapless. As the edge modes become macroscopically separated, wecan relax the constrain of t1 = 2t2 and the gapless modes will persist, asseen from the Chern number calculation.We now convert the analysis into equations. We perform the substitution∑i→ 12∫dxγ2i,j → (−1)iα1jγ2i+1,j → (−1)iα2j(4.9)and defineαR/Lj = (α1j ± α2j )/2 (4.10)Then the Hamiltonian becomesH → 2i∑j∫dx(−mα1jα2j + (t+m)α2jα1′j )− it1∑j∫dx (α1jα1j+1 − α2jα2j+1)+ 2it2∑j∫dx(− α1jα2j+1 + α2jα1j+1 + α1jα2′j+1 + α2jα1′j+1)= 2i∑j∫dx(2mαRj αLj + (t+m)(αRj αR′j − αLj αL′j ))− 2it1∑j∫dx (αRj αLj+1 + αLj αRj+1)+ 4it2∑j∫dx (αRj αLj+1 − αLj αRj+1 + αRj αR′j+1 − αLj αL′j+1)(4.11)When m = 0 and t1 = 2t2, the αRj αLj+1 terms vanish while the αLj αRj+1 terms284.1. The 2d modelgap out all the gapless modes except αR1 and αLN . Thus this pair of edgemodes remains gapless.Figure 4.3: The edge modes from the low-energy theory. While the bulkmodes are gapped out, the edge modes are left gapless.4.1.2 The interacting modelThe Hamiltonian of interacting model isH = it∑ij(−1)i (1− gµzij) γi,jγi+1,j − it1∑ij(−1)iγi,jγi,j+1+ it2∑ij(γi,jγi−1,j+1 − γi,jγi+1,j+1)+ J∑ijµzijµzi,j+1 − h∑ijµxij(4.12)and the corresponding MFT one is, as before,H = it∑ij(−1)i (1− g〈µzij〉) γi,jγi+1,j − it1∑ij(−1)iγi,jγi,j+1+ it2∑ij(γi,jγi−1,j+1 − γi,jγi+1,j+1)+ J∑ijµzijµzi,j+1 − h∑ijµxij(4.13)At h = ∞, we again obtain a non-interacting, m = 0 Majorana model.We expect the edge modes to be an Ising CFT, a story we are familiar up to294.2. Entanglement spectrum, edge modes and infinite DMRGnow. As h → 0, however, there are two possible scenarios. If g is not largeenough and thus m is not large enough either, the system will stay in theoriginal phase. For a large g, a large m will gap out the edge modes. Thus,we can still expect a TCI point in between the two limits of h.Based on the ladder model calculation, we can also formulate a “coupled-chain” analysis. As we couple two TCI chains with appropriate couplingparameters, we have seen that a pair of right and left movers are gappedout and the other pair left untouched. If we further introduce more TCIchains, we expect the bulk movers will be consecutively gapped out withthe edge modes gapless. As we have seen from the previous section, thisanalysis works well for the non-interacting model with Ising CFT.To sum up, we expect a TCI edge mode to occur as we tune the h from0 to ∞, while, on the other hand, the bulk is gapped, as hinted from theMFT and the coupled-chain analysis. There is another possibility that isnot so interesting, though. As h = 0 and h =∞ correspond to topologicallydifferent phases, there might be just a “common” phase transition in be-tween, where the bulk is gapless and edge modes are ill defined. Due to lackof numerical probes, we can not assert which one is correct at this point.4.2 Entanglement spectrum, edge modes andinfinite DMRGIn order to numerically explore the nature of the edge modes, new techniquesare necessary. For our present purpose, a measurement of the entanglementspectrum with infinite DMRG seems to be most promising.Back in Section 2.4, we defined the entanglement entropy from a bipar-tition of a systemSA = Tr(ρA ln ρA) (4.14)Apparently, much information stored in ρA is lost in this quantity, and thismotivates the definition of the entanglement spectrum, first proposed by Liand Haldane[12]Eα = − log ρA,α (4.15)where ρA,α’s are the eigenvalues of ρA. People then realize that, generally,the entanglement spectrum resembles the edge states spectrum of a topo-logical phase, up to a shift and rescaling [13].The infinite DMRG algorithm turns out to be a powerful method ofextracting the entanglement spectrum[14]. Instead of the truly 2d modelthat is beyond the capacity of DMRG, one considers a cylinder geometry,304.3. The non-interacting model revisitedwith Lx . 10 and Ly = ∞. By choosing an appropriate DMRG path, theentanglement spectrum is easily obtained from the MPS. We will elaboratethis idea in the next section using a concrete example.4.3 The non-interacting model revisitedWe go back to the non-interacting model with a cylinder geometry andmeasure the entanglement spectrum. We consider the ∞ × 4 and ∞ × 8cases. Upon a Jordan-Wigner transformation, the spin models have sizes of∞× 2 and ∞× 4, respectively. The DMRG paths are shown in Fig. 4.4.AAABBBCCCDDD· · ·· · ·AAABBB· · ·· · ·(a) (b)Figure 4.4: The DMRG paths for the non-interacting model with a cylindergeometry, with the size of (a) ∞× 4 and (b) ∞× 8 in the Majorana basis,or (a) ∞× 2 and (b) ∞× 4 in the spin basis.For the ∞ × 4 case, the entanglement spectrum is readily identifiedwith the Ising CFT result, see Fig. 4.5. The spectrum follows a patternof “1101111122223”. The ∞ × 8 case is more complicated. Now there ismore than one copy of Ising CFT on the edge, and we need to calculate thetransverse lattice momentum to tell them apart and identify the pattern.To do so we follow the methods in [14] and find the (diagonal) fixed pointof the “twisted transfer matrix” (TTM), whose diagonal elements are thecorresponding spacial translation phases eika, where the momentum k canbe extracted. The construction of the TTM is shown diagrammatically inFig. 4.6 We shift the k’s by a multiple of 2pi so that the pattern is clearlyvisible. The result is shown in Fig. 4.7.314.3. The non-interacting model revisitedEh = 0 h = 12 h =116Figure 4.5: The CFT towers and the entanglement spectrum. The lines aregiven by the Ising CFT which contains three conformal towers. The dots arethe entanglement spectrum from infinite DMRG upon a shift and rescaling.We see an almost perfect match here.Finally, we briefly discuss the difficulties of applying this technique to theinteracting model. As can be already seen from the∞×8 case, the conformaltowers are mixed together, and resolving each tower becomes even harderfor a TCI CFT where there are five possible towers. In [14] and other worksadopting this method, there exists an additive conserved quantum number,e.g. total spin in z direction Sz or total particle number n, that helps todistinguish between different towers, a feature we unfortunately lack in ourmodel. This drawback might be compensated for by more sophisticatedconsiderations of CFT, a direction worth exploring.324.3. The non-interacting model revisitedRA B C DA B C DTTMU R=A BA BUC DC DFigure 4.6: The twisted transfer matrix (TTM) and the transverse momen-tum. The twisted transfer matrix is constructed by multiplicating the MPSwith a unitry transformation U which rotates the cylinder by a lattice con-stant. The fixed point R is diagonal and the diagonal elements are eika.334.3. The non-interacting model revisitedETransverse momentum k/2piFigure 4.7: The entanglement spectrum of the ∞ × 8 case. Here to tellapart different conformal towers the transverse momentum is measured andtranslated by a multiple of 2pi for clarity. We can see that the pattern“110111112” holds for every tower.34Chapter 5ConclusionIn this thesis, the main result is the generalization of the GSV model to aladder model and the identification of the TCI edge mode.In the GSV model, a TCI transition is obtained by tuning only oneparameter. By examining the chain model, we establish a mean field theorythat agrees with the limiting cases of the full interacting model, which hintsa TCI transition as we tune the parameter. The mean field theory is readilygeneralized to a ladder and further to 2d.Based on the results from the interacting chain model and the MFTladder/2d model, we expect a TCI edge mode in the interacting ladder/2dmodel. We carry out extensive DMRG calculation of the interacting laddermodel and the results agree with the analytical argument. To the best ofour knowledge, this is the first work where a TCI edge mode is identified.The DMRG calculations are not applicable in 2d models. Thus so farwe can not assert whether the analysis in the interacting ladder model stillholds in the interacting 2d model. We discuss a possible numerical probeof the latter using infinite DMRG which has access to the entanglementspectrum, which might give some interesting results in future research.35Bibliography[1] Tarun Grover, DN Sheng, and Ashvin Vishwanath. Emergent space-time supersymmetry at the boundary of a topological phase. 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Unpaired majorana fermions in quantum wires. Physics-Uspekhi, 44(10S):131, 2001.37Appendix AMajorana fermionsMajorana fermions were first proposed by E. Majorana in 1937[15]. Herealized, as stated here in a modern way[16][17], that it is possible to definea charge conjugation symmetric solution of the Dirac equation in the sensethatC−1ΨC = Ψ (A.1)where C denotes the charge conjugation operator and Ψ is the second-quantized field operator. By choosing a representation known as Majo-rana representation where the charge conjugation matrix C = 1 and thusC−1ΨC = CΨ∗ = Ψ∗, the condition translates toΨa† = Ψa (A.2)and Ψa’s, with the fermionic condition {Ψai ,Ψbj} = 2δabδij satisfied, areknown as Majorana fermion operators. From now on we will use the let-ter γ(a) instead to denote Majorana fermions to emphasize the differencebetween Majorana and Dirac. Indeed, while the equation of motion is thesame, i.e. the Dirac equation, the Lagrangians are different. The relation be-tween the two is analogous to the real and complex scalar fields in the bosoniccase[16]. In nonrelativistic systems, we are free from the spin-statistics the-orem, and it is legitimate to regard γ’s as spinless Majorana operators withno spin indices. Below we will mainly focus on spinless operators.A few comments are in order. We first note that the Majorana operatorscan be related to Dirac fermion operators by defining pairwiseγ1 = c† + c, γ2 = i(c† − c) (A.3)Using this transformation, we can always map a fermionic Hamiltonian intoa Majorana one, and vice versa. Thus, paired Majorana fermions are abun-dant in electronic systems, giving an equivalent, though not interesting,representation. What we are interested in, as a result, are only the unpairedones. Another scenario where Majorana fermions appear “trivially” is inordinary superconductors. In the BdG mean field approach, Cooper pairingare accounted for by a bilinear pairing term ∆c†c†+ h.c. A sage and almost38Appendix A. Majorana fermionsunique choice of field operator for Hamiltonians of this kind is the Nambuspinor, defined as Ψ =(c↑, c↓, c†↓,−c†↑)T. The fact that excitations in thissystem are Majorana fermions follows from CΨ∗ = Ψ, where C = τyσy andσ and τ are Pauli matrices in spin and Nambu spaces, respectively. Whilethis is conceptually interesting, the Majorana excitations are not local andfurther manipulations are hindered.Recently people realize[17] that, unlike the two cases discussed above,zero-energy Majorana excitations typically bound with edges or defects intopological phases, or Majorana zero modes (MZM), have a few nice prop-erties. Apart from being localized in space, they also behave as non-Abeliananyons[18] when braided against each other, allowing potential applicationsin topological quantum computation[19]. An prototypical realization ofMZM is the Kitaev chain model[20]. The Hamiltonian isH =∑j(−tc†jcj+1 + ∆c†jc†j+1 + h.c.− µc†jcj)(A.4)The ∆ = 0 case is the familiar tight binding model, and with an appropriateµ it is a topologically trivial insulator. This can be seen from the fact thatthe phase is connected to the isolated atom limit without closing the gap bycontinuously decreasing t to 0. Interesting physics appears if we take ∆ = tand µ = 0. In this case, we haveH = itN−1∑j=1γj,2γj+1,1 (A.5)where we have used Eq. A.3 for each site to rewrite the Hamiltonian usingMajorana operators. Now the bulk Hamiltonian is in a gapped phase, as canbe seen by recombining γj,2 and γj+1,1 into Dirac operators. What is crucial,however, is that γ1,1 and γN,2 do not enter the Hamiltonian, and thus rep-resent two MZMs. Combining these two operators into f = (γ1,1 + iγN,2)/2,we can further identify the degenerate ground states |0〉 and |1〉 by f |0〉 = 0and |1〉 = f † |0〉. Since there is only one zero energy state, it is robust againstperturbation as long as the particle-hole symmetry is preserved. Note thatthe Majorana operators are localized at the two edges of the chain. Whileimmune from decoherence induced by local perturbation, the MZMs canbe braided by, e.g., introducing a T-junction, allowing potential quantumcomputational operations.39Appendix A. Majorana fermionsEnergy0Trivial phase Topological phaseMZM(a)(b)(c)(d)Figure A.1: The Kitaev chain model can be written in Dirac fermion oper-ators (a) or Majorana fermion operators (b)(c). (b) The trivial phase canbe continuously connected to the isolated atom limit without closing thegap. (c)(d) In the topological phase, there is a topologically protected zeroenergy state.40

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