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Majorana-Hubbard model on the triangular lattice Tummuru, Tarun Reddy 2019

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Majorana-Hubbard model on thetriangular latticebyTarun Reddy TummuruB.Sc., Shiv Nadar University, 2017A 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 2019c© Tarun Reddy Tummuru 2019Committee PageThe following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Majorana-Hubbard model on the triangular latticesubmitted by Tarun Reddy Tummuru in partial fulfilment of the require-ments for the degree of Master of Science in Physics.Examining Committee:Ian Affleck, Physics and AstronomySupervisorMarcel Franz, Physics and AstronomySupervisory Committee MemberiiAbstractMajorana modes can arise as zero energy bound states in a variety of solidstate systems. A lattice of these quasiparticles, for instance, emerges on thesurface of a topological superconductor with the zero modes localized at thecores of vortices. This thesis is devoted to the study of interactions betweenthe Majorana zero modes when such a lattice has a triangular geometry.We map the phase diagram of this model using a combination of meanfield theory and numerical simulation of ladders through the density-matrixrenormalization-group technique. Our analysis suggests that interactionsdrive a topologically non-trivial gapped phase into a gapless Luttinger liquid.iiiLay SummaryElementary particles that constitute matter have corresponding distinctanti-particles; for an electron this is the positron. It had been proposedthat some solid state systems can realize particles that are their own anti-particles, dubbed Majorana fermions. This has sparked immense researchactivity because of the potential application of these exotic particles in thedevelopment of quantum computation. In this work, we study the physicsthat ensues when Majorana fermions are allowed to interact with each otherin a particular manner.ivPrefaceThis thesis is original, unpublished, independent work by the author.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Majorana zero modes . . . . . . . . . . . . . . . . . . . . . . 11.2 Majorana-Hubbard model . . . . . . . . . . . . . . . . . . . . 22 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Non-interacting limit . . . . . . . . . . . . . . . . . . . . . . 52.2.1 Topological classification . . . . . . . . . . . . . . . . 62.3 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.1 Strong coupling limit . . . . . . . . . . . . . . . . . . 93 Self-consistent mean field theory . . . . . . . . . . . . . . . . 103.1 Mean field approximation . . . . . . . . . . . . . . . . . . . . 113.2 Order parameters . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Self-consistency conditions . . . . . . . . . . . . . . . . . . . 133.4 Brute-force minimization . . . . . . . . . . . . . . . . . . . . 154 2-leg ladder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.1 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Non-interacting limit . . . . . . . . . . . . . . . . . . . . . . 184.3 Low-energy theory . . . . . . . . . . . . . . . . . . . . . . . . 19viTable of Contents4.4 Strong coupling limit . . . . . . . . . . . . . . . . . . . . . . 215 4-leg ladder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.1 Gapless phase . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2 Central charge . . . . . . . . . . . . . . . . . . . . . . . . . . 256 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30AppendicesA Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34A.1 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . 34A.2 Reflections and time reversal . . . . . . . . . . . . . . . . . . 34A.3 Rotation by pi/3 . . . . . . . . . . . . . . . . . . . . . . . . . 35B Mean field theory of the square lattice Majorana-Hubbardmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37C 2-leg ladder: field theory etc. . . . . . . . . . . . . . . . . . . 40C.1 Diagonalization and Lorentz invariance . . . . . . . . . . . . 40C.2 Normalization of the field operators . . . . . . . . . . . . . . 41C.3 Jordan-Wigner transform . . . . . . . . . . . . . . . . . . . . 41viiList of Figures2.1 (left) A possible Z2 gauge choice for the triangular lattice.Hopping along (against) the directions indicated incurs a phaseof +i (−i). The two sub-lattices of the Bravais lattice havebeen labeled by α and β. (right) Ordering of MZM operatorsin the three plaquette interaction terms: counter-clockwisedirection with respect to the reference site. . . . . . . . . . . 42.2 The gapped spectrum in units of the hopping amplitude t. . . 72.3 The edge states for flat and zig-zag boundaries. Note thatonly the unshaded region of the BZ is physically meaningful. 83.1 Triangular vortex lattice with a four-site unit cell. . . . . . . 103.2 The self-consistent values of {τj} and {∆j}. As expected,{τj} = t = 1 at g = 0. . . . . . . . . . . . . . . . . . . . . . . 143.3 Values of {τj} that minimize 〈H〉 for different values of thecoupling g. The corresponding {∆j} have been evaluatedusing |ΨMF〉. . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.1 2-leg ladder maps on to a chain with next-nearest-neighborhopping terms. . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 The gapless spectrum in with the energy denoted in units ofthe hopping amplitude t. . . . . . . . . . . . . . . . . . . . . . 195.1 Behavior of energy gaps as a function of the coupling for dif-ferent system geometries under periodic boundary conditions.We have set t = 1. . . . . . . . . . . . . . . . . . . . . . . . . 245.2 Extrapolation of the energy gap for values of g in the twophases. Separate straight lines have been fit for even and oddNx owing to their distinct behaviors (see text). The systemsconsidered vary from N = 30 through N = 50, while keepinga maximum bond dimension of 103 in the DMRG sweeps.The truncation errors are less than 10−8. Note that the gapsfor g = 2 have been rescaled by a factor of 50. . . . . . . . . . 25viiiList of Figures5.3 The fermionic parity of the two lowest energy states for thecase of N = 50. The first excited state switches from even toodd across the transition point; a behavior that is observedfor all system lengths. . . . . . . . . . . . . . . . . . . . . . . 265.4 (top) Entanglement entropy SN (x) for a length x subregionof a periodic system with length N = 50 at g = −4. (bottom)A linear fit through the same data with conformal distanceplotted on the horizontal axis. . . . . . . . . . . . . . . . . . . 275.5 Central charge as a function of the coupling, while keepingup to 103 states and under PBC. We have verified that itcontinues to remain unity for values up to g = −20 in thegapless phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28B.1 A Z2 gauge choice for the square vortex lattice with a four-siteunit cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37B.2 The mean-field parameters {τj} and {∆j} that have beenobtained by solving eq. (B.3) and eq. (B.4) self-consistently. . 39ixAcknowledgementsForemost, I would like to thank my supervisor, Ian Affleck, for the mentor-ship. It has a been my great pleasure to work with him. Coursework atUBC has greatly helped me shape my understanding of physics and for thisI am indebted to the professors, especially Marcel Franz and Mona Berciu.I have also learned from a number of discussions and conversations - I amgrateful to Alberto, Chengshu, E´tienne, Kyle and Samuel. Finally, I thankmy parents. Only with their loving support was it all possible.xChapter 1Introduction1.1 Majorana zero modesThe quest for quantum computation that is topologically protected involvesparticles that have non-trivial exchange statistics. These quantum objectsare called anyons. Of the two broad types of anyons, Abelian and non-Abelian, it is the latter kind that finds use in topological quantum compu-tation.The simplest non-Abelian anyon, the so called Ising anyon, is realizedwhen a quasiparticle or a defect supports a Majorana zero mode (MZM)[1–3]. The zero here means that the state is a zero energy mid-gap excita-tion that is localized in space, while Majorana refers to the real nature ofthe operator representing the mode, drawing parallel to Majorana’s real ver-sion of Dirac’s equation. In their original form, Majorana fermions, as thename suggests, obey Fermi-Dirac statistics. An MZM, however, is specialbecause its presence imbues the quasiparticle or the defect with non-Abelianstatistics.Mathematically, a MZM is represented by self-adjoint fermionic operatorγ that squares to an identity and commutes with the Hamiltonian underconsideration:γ2 = 1 and [H, γ] = 0 (1.1)Any operator that satisfies the first condition is a Majorana fermion oper-ator. The second condition, which sets a MZM apart, also brings about adegeneracy in the groundstate.Suppose that there are only two MZMs γ1 and γ2 in the system. Apair of zero modes constitutes an ordinary spinless complex fermion statec = γ1 + iγ2. Now the Hamiltonian may be block diagonalized with respectto the operator iγ1γ2 = 2c†c− 1. This implies that each state is labeled bya quantum number which is the parity of the state c. Since this fermionicstate occurs at zero energy, there is no cost to occupy this state and, hence,for any given energy level with a given parity there exists another state withthe same energy but opposite parity.11.2. Majorana-Hubbard modelAlong these lines, if there are 2N Majorana modes, γ1, . . . γ2N , satisfying{γi, γj} = 2δij , (1.2)then H can be simultaneously diagonalized by the operators iγ1γ2, iγ3γ4,. . . , iγ2N−1γ2N . The groundstates of H are then labeled by the eigenvalues±1 of these N operators, thereby leading to a 2N fold degeneracy. Notethat this pairing of MZMs is not unique; different pairings correspond todifferent choices of basis that can be used to represent the 2N dimensionalgroundstate manifold. Further, if the total fermion parity of the system isfixed, the degeneracy is 2N−1. Braiding the quasiparticles or the defectshosting MZMs is equivalent to applying a unitary operation on this Hilbertspace. For more details on this aspect of MZMs, the reader is referred to[4].1.2 Majorana-Hubbard modelAs alluded to before, MZMs occur in two different situations. The first isa quasiparticle excitation in a topological phase. A charge e/4 excitationin the ν = 5/2 fractional quantum Hall state is believed to host a MZM[5, 6]. In the second scenario, a defect in an otherwise ordered state, such asa domain wall in 1D [7–9] or a vortex in a superconductor, harbors the zeromode. In the present work, we are interested in the latter. The vortices onthe surface of an intrinsic topological superconductors with p-wave pairingare expected to have MZMs at their cores [10]. An effective p-wave pairingcan also be realized at the interface of a s-wave superconductor and a strongtopological insulator [11].From the perspective of a physical setup, the second requirement of theaforementioned condition (1.1) is too idealized. In the case of superconduct-ing systems supporting MZMs, one usually has[H, γ] ∼ e−x/ξ (1.3)where x may be seen as the separation between two MZMs and ξ is thesuperconducting coherence length. Owing to the superconducting gap ∆,all the mid-gap zero energy states are well separated in energy. Then, forenergies well below ∆, the effective Hamiltonian describing the zero modesis a sum of local terms like iγiγj with exponentially small coefficients ∼e|xi−xj |/ξ (with xi referring to the position of γi) [12, 13]. The coefficientmay be interpreted as the probability amplitude for a fermionic quasiparticle21.2. Majorana-Hubbard modelto tunnel between the two MZMs. In the presence of an Abrikosov vortexlattice of zero modes, this motivates the construction of a tight-bindingmodel for MZMs.Interactions between Majoranas can lead a broad variety of rich physics.A 1D chain of MZMs shows emergent supersymmetry with phase transitionsbelonging to the tri-critical Ising universality class [14–16]. In 2D, studieson the square [17, 18] and honeycomb lattices [19] predict interesting phasediagrams.In this thesis, we study a model for interacting MZMs on the triangularlattice. An earlier work pertaining to the triangular lattice had consideredthe role of disorder [20], but not interactions. Our goal is to map the phasediagram of this model as function of the interaction strength. We try toanalyze a plausible symmetry breaking phase transition from the perspectiveof a self-consistent mean field theory (chapter 3) and then study the 2-legand 4-leg variations of the model using a combination of field theory andthe density matrix renormalization group (chapters 4 and 5). But first, webuild the model and discuss its symmetries.3Chapter 2The modelBased solely on the relation γi = γ†i and the requirement of Hermiticity, alattice hosting MZMs is described by the tight-binding HamiltonianH0 = it∑〈ij〉ηijγiγj . (2.1)Here, the tunneling strength t is real and positive and the anti-symmetricmatrix ηij = ±1 indicates the sign of the phase i acquired in tunnelingalong the bond (i, j). The ambiguity in these signs arises from the fact thatone may redefine γi → −γi without altering the MZM anti-commutationrelations. This Z2 gauge freedom implies that ηij depends on the choice ofgauge, but the product of ηij along a closed loop corresponds to a Z2 gaugeflux and, thus, must be an invariant [21, 22].<latexit sha1_base64="kw9MPoNtO0Emwztg l8A4hZMFP8g=">AAACAnicbVDLSsNAFJ3UV62vqks3g0VwVRIVdFnQhcsK9gFtKDfTSTN2Mhlm JkII3fkBbvUT3Ilbf8Qv8Dectllo64ELh3Pu5d57AsmZNq775ZRWVtfWN8qbla3tnd296v5BWy epIrRFEp6obgCaciZoyzDDaVcqCnHAaScYX0/9ziNVmiXi3mSS+jGMBAsZAWOldh+4jGBQrbl1 dwa8TLyC1FCB5qD63R8mJI2pMISD1j3PlcbPQRlGOJ1U+qmmEsgYRrRnqYCYaj+fXTvBJ1YZ4j BRtoTBM/X3RA6x1lkc2M4YTKQXvan4ryejTDOiF9ab8MrPmZCpoYLMt4cpxybB0zzwkClKDM8s AaKYfQCTCBQQY1Or2GS8xRyWSfus7p3X3buLWuOmyKiMjtAxOkUeukQNdIuaqIUIekDP6AW9Ok /Om/PufMxbS04xc4j+wPn8Aba6mBQ=</latexit><latexit sha1_base64="yLKYmBegHL39jCpE wmgRM+alPnQ=">AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lU0GNBDx4rmFpoQ9lsJ+3SzSbs ToQSevIHeNWf4E28+kv8Bf4Nt20O2vpg4PHeDDPzwlQKg6775ZRWVtfWN8qbla3tnd296v5Byy SZ5uDzRCa6HTIDUijwUaCEdqqBxaGEh3B0PfUfHkEbkah7HKcQxGygRCQ4Qyv53RCQ9ao1t+7O QJeJV5AaKdDsVb+7/YRnMSjkkhnT8dwUg5xpFFzCpNLNDKSMj9gAOpYqFoMJ8tmxE3pilT6NEm 1LIZ2pvydyFhszjkPbGTMcmkVvKv7rpcOxEdwsrMfoKsiFSjMExefbo0xSTOg0DtoXGjjKsSWM a2EfoHzINONoQ6vYZLzFHJZJ66zundfdu4ta46bIqEyOyDE5JR65JA1yS5rEJ5wI8kxeyKvz5L w5787HvLXkFDOH5A+czx/mSJeg</latexit><latexit sha1_base64="TGBl6rC2euDATWb6cy/Ue+aYJg8=">AAACBXicbVDLSgNBEJyNrxhfUY9eBoPgKeyqoMeAHjxGMA9Jlj A76U2GzOwuM71iWHL2A7zqJ3gTr36HX+BvOEn2oNGChqKqm+6uIJHCoOt+OoWl5ZXVteJ6aWNza3unvLvXNHGqOTR4LGPdDpgBKSJooEAJ7UQDU4GEVjC6nPqte9BGxNEtjhPwFRtEIhScoZXuuggPmNW9Sa9ccavuDPQv8XJSITnqvfJXtx/zVEGEXDJjOp6boJ8xjYJL mJS6qYGE8REbQMfSiCkwfjY7eEKPrNKnYaxtRUhn6s+JjCljxiqwnYrh0Cx6U/FfLxmOjeBmYT2GF34moiRFiPh8e5hKijGdRkL7QgNHObaEcS3sA5QPmWYcbXAlm4y3mMNf0jypeqdV9+asUrvKMyqSA3JIjolHzkmNXJM6aRBOFHkiz+TFeXRenTfnfd5acPKZffILzsc 3IWaZag==</latexit><latexit sha1_base64="n6ePtpEqTfTSO8QDF0fT7mqPdTQ=">AAACBXicbVDLSgNBEJyNrxhfUY9eBoPgKexGQY8BPXiMYB6SLG F2MpsMmZldZnrFZcnZD/Cqn+BNvPodfoG/4STZgyYWNBRV3XR3BbHgBlz3yymsrK6tbxQ3S1vbO7t75f2DlokSTVmTRiLSnYAYJrhiTeAgWCfWjMhAsHYwvpr67QemDY/UHaQx8yUZKh5ySsBK9z1gj5A1apN+ueJW3RnwMvFyUkE5Gv3yd28Q0UQyBVQQY7qeG4OfEQ2c CjYp9RLDYkLHZMi6lioimfGz2cETfGKVAQ4jbUsBnqm/JzIijUllYDslgZFZ9Kbiv148Sg2nZmE9hJd+xlWcAFN0vj1MBIYITyPBA64ZBZFaQqjm9gFMR0QTCja4kk3GW8xhmbRqVe+s6t6eV+rXeUZFdISO0Sny0AWqoxvUQE1EkUTP6AW9Ok/Om/PufMxbC04+c4j+wPn 8ASMAmWs=</latexit><latexit sha1_base64="O0SEgD4bQl3ZVibPn7YkPNYmUXQ=">AAACBXicbVDLSgNBEJyNrxhfUY9eBoPgKewaQY8BPXiMYB6SLG F2MpsMmZldZnrFZcnZD/Cqn+BNvPodfoG/4STZgyYWNBRV3XR3BbHgBlz3yymsrK6tbxQ3S1vbO7t75f2DlokSTVmTRiLSnYAYJrhiTeAgWCfWjMhAsHYwvpr67QemDY/UHaQx8yUZKh5ySsBK9z1gj5A1apN+ueJW3RnwMvFyUkE5Gv3yd28Q0UQyBVQQY7qeG4OfEQ2c CjYp9RLDYkLHZMi6lioimfGz2cETfGKVAQ4jbUsBnqm/JzIijUllYDslgZFZ9Kbiv148Sg2nZmE9hJd+xlWcAFN0vj1MBIYITyPBA64ZBZFaQqjm9gFMR0QTCja4kk3GW8xhmbTOql6t6t6eV+rXeUZFdISO0Sny0AWqoxvUQE1EkUTP6AW9Ok/Om/PufMxbC04+c4j+wPn 8ASSamWw=</latexit>Figure 2.1: (left) A possible Z2 gauge choice for the triangular lattice. Hop-ping along (against) the directions indicated incurs a phase of +i (−i).The two sub-lattices of the Bravais lattice have been labeled by α and β.(right) Ordering of MZM operators in the three plaquette interaction terms:counter-clockwise direction with respect to the reference site.A Z2 gauge choice for the triangular lattice that conforms to the re-42.1. Symmetriesquirement of having a pi/2 flux through each elementary triangle is shownin fig. 2.1. Note that this gauge fixing imposes a two-site unit cell and arectangular Bravais lattice. Choosing an alternate gauge would modify thesign convention, but would not change the number of the sub-lattice degreesof freedom. With this Hamiltonian as the starting point, we will now use itssymmetries to constrain the form of the interactions possible in this model.2.1 SymmetriesThe Z2 gauge implies that H0 is not explicitly invariant under direct latticetransformations. For instance, translation by a site along the direction c(see fig. 2.1 for the three directions referred to in the following) modifies theHamiltonian. Instead, the symmetries are represented projectively, that is,the conventional transformations need to be supplemented with additionalphase factors for the system to remain unaltered.While the Bravais lattice is rectangular, the symmetries of eq. (2.1) aredictated by the fact that the underlying vortex lattice is triangular. Inaddition to discrete translations, Tµ along the directions µ = a,b, c1, a pi/3rotation about any α site leaves H0 invariant. While the anti-unitary timereversal Θ and reflections Rx/y about the Cartesian axes are not symmetriesby themselves, their product Rx/yΘ is a symmetry. The transformationsunder these symmetry operations are outlined explicitly in Appendix A.2.2 Non-interacting limitLet us now look at the groundstate properties of the system in the absenceof interactions. We label the MZMs on the two inequivalent sites belongingto a unitcell i by αi and βi, which are self-adjoint and obey{αi, αj} = 2δij {βi, βj} = 2δij {αi, βj} = 0. (2.2)The Bravais lattice vectors of the rectangular lattice are d1 = (a, 0)and d2 = (0,√3a), with a being the inter-vortex spacing. We denote thenumber of unit cells along the x and y directions by Nx and Ny and totalnumber of unit cells by N = NxNy. Then, periodic boundary conditionsgive kx = pii/(Nxa) and ky = pij/(√3Nya), where i and j run over Nx andNy integer values respectively.1Note that the three translations are not independent of each; only two are.52.2. Non-interacting limitA Fourier transform of the real space MZM operators reads(αjβj)=√2N∑keikrj(αkβk)(2.3)where the normalization has been chosen so as to conform to eq. (2.2). Thenew operators in the momentum space obey{αk, αk′} = {βk, βk′} = δk,−k′ . (2.4)Importantly, the real nature of the MZMs manifests itself as the relation(α−k, β−k) = (α†k, β†k). This means that the operators at k and −k are notindependent. To remedy this, we only focus on one-half of the Brillouin zone(BZ), which makes αk and βk the ordinary complex fermion operators.Using the above arguments, one arrives at the Fourier transformed Hamil-tonian H0 =∑′Ψ†k h0(k) Ψk, where the prime denotes sum over half of theBZ, Ψk = (αk βk)T andh0(k) = 2t(−2 sin(k.d1) D(k)D(k)∗ 2 sin(k.d1))(2.5)with D(k) = i[1 + e−ik.d1 − e−ik.d2 + e−ik.(d1+d2)]. Diagonalizing this BlochHamiltonian is straight forward and we get the dispersion relationE±k = ±2√2t[3− cos(2kxa)− 2 sin(kxa) sin(√3kya)]1/2. (2.6)This band structure is gapped and is shown in fig. 2.2. It is importantto note that the groundstate of the system corresponds to filling up allthe negative energy states. When written in terms of complex fermions,the Hamiltonian consists of pair creation and annihilation operators andcan be diagonalized via the Bogoliubov-de Gennes transformation. As aconsequence, the excitations are strictly positive in energy and are at a costto the condensate energy.2.2.1 Topological classificationOne consequence of having a non-zero Z2 gauge flux through any elementaryclosed loop is that a gapped band structure of a periodic vortex lattice istopologically nontrivial. A fermionic quasiparticle tunneling between twovortices necessarily acquires a Z2 phase of ±i, as evident from eq. 2.1. Thisbackground gauge field in which the MZMs move is reminiscent of gauge62.3. InteractionsFigure 2.2: The gapped spectrum in units of the hopping amplitude t.field in the Haldane model [23]. One can, therefore, expect the bands tohave non-zero Chern number and, hence, gapless edge modes in presence ofboundaries.We verify these observations explicitly. With the Bloch Hamiltonian(2.5) written in the basis of Pauli matrices, h0 = ~d(k).~σ, the Chern numberis given byC = 14pi∫BZ1‖~d(k)‖3[∂ ~d(k)∂kx× ∂~d(k)∂ky]· ~d(k) = sgn(t). (2.7)We indeed find topologically protected edge states that connect the bulkbands for the two types of edge geometries (flat and zig-zag) of the lattice;see fig. 2.3.2.3 InteractionsThe leading order, interactions in a vortex lattice involve four neighboringMZMs that constitute a plaquette. In a square lattice, for instance, these arethe Majoranas at the corners of an elementary square. It is easy to see thatin case of a triangular geometry, three different orientations of rhomboidalplaquettes are possible, with each group tessellating the entire lattice once.That is,HI = g∑P1 − P2 + P3 with Pν = γiγjγkγl (2.8)and the ordering of MZM operators in each of these terms is in the counter-clockwise direction, as shown in fig. 2.1.72.3. Interactions2 0 2kx3210123E kx3 2 302 3 3ky3210123E kyFigure 2.3: The edge states for flat and zig-zag boundaries. Note that onlythe unshaded region of the BZ is physically meaningful.82.3. InteractionsWe note that the negative sign accompanying P2 is a consequence ofrequiring that the interactions obey the symmetries of H0. Under the actionof ΘRx, for instance, P1 → −P2, P2 → −P1 and P3 → P3 and, hence,HI → HI .2.3.1 Strong coupling limitIn Fu and Kane’s proposal to realize the Majorana zero modes in a vor-tex lattice, fine tuning of the topological insulator’s chemical potential canlead to the so called neutrality point [12, 24]. At neutrality, the interfacesuperconductor exhibits an emergent chiral symmetry that prevents the Ma-jorana modes from hybridizing. Physically, this means that quadratic termsare now prohibited in the Hamiltonian and the system is inherently stronglyinteracting. A feature of the triangular lattice in this limit is the equivalenceof the attractive and repulsive interactions. This can be seen by noting thatchanging the sign of zero mode operators on every alternate α site effec-tively results in each of the interaction terms picking up a negative sign,while preserving the anti-commutation relation between the operators.Also, with t = 0, the unit cell is no longer bipartite. Since we nowhave an odd number of Majorana sites per unit cell, periodic boundaryconditions and translation symmetry dictate that the groundstate is at leasttwo-fold degenerate [25]. It is important to note that this degeneracy isintrinsically related to the system size. In case of systems with one oddlength (only either Nx or Ny is odd), the two degenerate states belong todifferent fermionic parity sectors and this degeneracy can be attributed tothe system’s underlying supersymmetry. With two even lengths (both Nxor Ny are even), however, the degeneracy is a consequence of mutual anti-commutation of the translation operators along the two spatial directions.9Chapter 3Self-consistent mean fieldtheoryFor a system of non-interacting Majoranas, we had a two-site unit cell. Tak-ing cues from the 1D chain and 2D square Majorana-Hubbard model studies[15, 17], we wish to account for the possibility of zero modes interacting morestrongly with some neighbors than the rest. To that extent, we enlarge theunit cell to include four sites. The MZM operators are denoted by γνr , whereν = p, q, r, s as shown in fig. 3.1. The resulting Bravais lattice is triangular,whose primitive vectors are given by d1 = (2a, 0) and d2 = (a,√3a).Figure 3.1: Triangular vortex lattice with a four-site unit cell.The system is then fully described by H = H0 +HI , whereH0 =∑rit[(γprγqr − γsrγrr) + (γqrγpr+d1 − γrrγsr+d1) + (γprγsr + γqrγrr)− (γsrγpr+d2 + γrrγqr+d2)− (γqrγsr + γpr+d1γrr)− (γsr+d1γqr+d2+ γrrγpr+d2)](3.1)103.1. Mean field approximationHI = g∑r[γprγqrγrrγsr + γqrγpr+d1γsr+d1γrr + γrrγsr+d1γpr+d1+d2γqr+d2+ γsrγrrγqr+d2γpr+d2 − (γpr+d1γrrγsrγqr + γqr+d1γsr+d1γrrγpr+d1+ γrr+d1γpr+d1+d2γqr+d2γsr+d1 + γsr+d1γqr+d2γpr+d2γrr) + γpr+d2γsrγqrγrr+ γqr+d2γrrγpr+d1γsr+d1 + γrr+d2γqr+d2γsr+d1γpr+d1+d2+ γsr+d2γpr+d2γrrγqr+d2].(3.2)We now proceed to study these interactions approximately, by ignoring thecorrelations, in a mean field setting.3.1 Mean field approximationTo begin, we assume that the groundstate of H can be well approximatedby an effective non-interacting counterpart, which we define asHMF = i∑rτa(γprγqr − γsrγrr) + τa¯(γqrγpr+d1 − γrrγsr+d1)+τc(γprγsr + γqrγrr) + τc¯(γsrγpr+d2+ γrrγqr+d2)−τb(γqrγsr + γpr+d2γrr)− τb¯(γsr+d1γqr+d2+ γrrγpr+d2). (3.3)The physical meaning of the new parameters τj (with j = a, b, c, a¯, b¯ or c¯ andj¯ = j) is as follows. In the absence of interactions, HMF and H0 coincide.The interactions, however, renormalize t to give rise to the modified hoppingamplitudes τj . Note that, for this reason, HMF obeys the same Z2 gauge.Since the system is invariant under translations, we work in the momen-tum space and, again, as a consequence of the relation γν−k = γν†k , only halfof the Brillouin zone is physically meaningful. This leads toHMF =∑kx>0X (γpk†γqk − γrk†γsk) + Y (γpk†γsk + γqk†γrk)+ (V γqk†γsk +Wγpk†γrk) + h.c. (3.4)with the coefficients defined asV = −i (τb − τb¯eik·(d1−d2))W = i (τbe−ik·d2 − τb¯e−ik·d1)X = i (τa − τa¯e−ik·d1)Y = i (τc + τc¯e−ik·d2). (3.5)113.2. Order parametersWith the groundstate wavefunction |ΨMF〉 of this Hamiltonian as a vari-ational ansatz, we now intend to minimize the energy 〈ΨMF|H |ΨMF〉. Atthis point one can minimize 〈H〉 as a function of {τj} either by brute-forceor in a self-consistent (SC) manner2. Here we focus on the latter and thenbriefly comment on the former. But first, let us define the order parametersof the system.3.2 Order parametersBased on the previously studied Majorana-Hubbard models, we expect theonset of a phase transition to be marked by unequal expectation values ofneighboring bonds, along either a, b or c directions. While the onset ofthis order would correspond to broken translation symmetry in the originallattice, in the model with an enlarged unit cell it does not. The transitioncan be studied by looking at the difference in the expectation values of thenearest-neighbor bonds.To proceed, we first consider the pairwise MZM operators∆a = 〈iγprγqr〉 = −〈iγsrγrr〉∆a¯ =〈iγqrγpr+x〉= − 〈iγrrγsr+x〉∆c = 〈iγprγsr〉 = 〈iγqrγrr〉∆c¯ =−〈iγsrγpr+y〉= − 〈iγrrγqr+y〉∆b =− 〈iγqrγsr〉 = −〈iγpr+xγrr〉∆b¯ =−〈iγsr+xγqr+y〉= − 〈iγrrγpr+y〉 . (3.6)The aforementioned difference in the expectation values of neighboring bondsmay then be measured by looking at ∆j −∆j¯ , which quantify the extent ofdimerization along a, b and c. Suppose that we define∆a = A+B ∆a¯ = A−B∆c = C +D ∆c¯ = C −D∆b = E + F ∆b¯ = E − F. (3.7)Then, B,D or F serve as the three mean field order parameters and asymmetry broken phase is marked when either of these is non-zero.2In order to ascertain that this SC mean-field theory is actually suitable for the problemat hand, we apply it to the square lattice Majorana-Hubbard model in appendix B.123.3. Self-consistency conditions3.3 Self-consistency conditionsAs mentioned in the above, we are interested in minimizing the total energywith respect to the non-interacting mean-field wavefunction. Firstly, 〈H〉can be evaluated as〈H〉 = 2N∑jt∆j − g(∆2j + ∆j∆j¯) (3.8)where, for simplicity, we have assumed that the second neighbor hoppingsgenerated by the interactions do not play a role in the kind of symmetrybreaking under consideration here3. We then wish to find {τj} that satisfy∂〈H〉∂τi= 0, or equivalently∑j[t− 2g(∆j + ∆j¯)]∂∆j∂τi= 0. (3.9)In order to connect this with the mean field Hamiltonian, note that with{∆j} as defined eq. (3.6), HMF motivates an alternate definition∆j =12N〈∂HMF∂τj〉(3.10)where we have made use of the fact that the system is translationally invari-ant. Further, {∆j} can be related to EMF via the the Hellmann-Feynmantheorem4 to give∆j =12N∂EMF∂τj. (3.11)But, from the definition (3.3), we knowEMF = 〈ΨMF|HMF |ΨMF〉 = 2N∑jτj∆j . (3.12)Therefore,12N∂EMF∂τi= ∆i +∑jτj∂∆j∂τior∑jτj∂∆j∂τi= 0. (3.13)3If next neighbor bonds are denoted by κ, the terms generated by the interactions areof the form gκ∆j . Including these terms merely shifts the t→ t− gκ and does not changethe results qualitatively.4It relates the expectation value of a derivative of an operator with the derivative ofthe expectation value of the operator133.3. Self-consistency conditionsComparing (3.9) with (3.13), we finally have the self-consistency relationsτj = t− 2g(∆j + ∆j¯) (3.14)that can be evaluated iteratively for self-consistency. The solutions to theseequations, shown in fig.3.2, indicate that the effective nearest-neighbor hop-pings vary monotonically with g. As a consequence, we see that no symmetrybreaking second order transition arises.1.0 0.5 0.0 0.5 1.0g0.51.01.5aaccbb1.0 0.5 0.0 0.5 1.0g0.00.10.20.32A= | a+ a|2B= | a a|2C= | c+ c|2D= | c c|2E= | b+ b|2F= | b b|Figure 3.2: The self-consistent values of {τj} and {∆j}. As expected, {τj} =t = 1 at g = 0.This result can be understood as follows. In the square lattice Majorana-Hubbard model, where there is only one kind of plaquette, pairing MZM’sis conducive because the energy of one-half of the plaquettes is minimized,143.4. Brute-force minimizationthereby making it energetically favorable [17]. In our case, since there arethree kinds of plaquettes, if the MZM were to dimerize along a particulardirection, the energy of one-half of the plaquettes of one kind is minimized.These, however, amount to only one-sixth of the total number of plaquetteinteraction terms and, hence, is not energetically favorable.3.4 Brute-force minimizationInstead of solving for the order parameters self-consistently, one could tryand directly minimize 〈H〉 as a function of the six effective hoppings {τj}.While this works for g > 0, the problem does not seem to have a well definedlower bound when g < 0. In fig. 3.3, we see that for g > 0, the values of theorder parameters exactly match those in fig. 3.2.153.4. Brute-force minimization0.00 0.25 0.50 0.75 1.00g3.653.703.753.80aaccbb0.00 0.25 0.50 0.75 1.00g0.00.10.20.32A= | a+ a|2B= | a a|2C= | c+ c|2D= | c c|2E= | b+ b|2F= | b b|Figure 3.3: Values of {τj} that minimize 〈H〉 for different values of thecoupling g. The corresponding {∆j} have been evaluated using |ΨMF〉.16Chapter 42-leg ladderTo go beyond the mean-field theory, we look at ladder geometries that areamenable to more exact treatments. The simplest ladder has two legs andcorresponds to a having Ny = 1. Building on the convention from chapter2 we denote αr = αj and βr = βj . The system without interactions isdescribed byH0 =∑jit[αjβj + βjαj+1 + αjαj+1 − βjβj+1] (4.1)and the four fermion interactions areHI = g∑j[αjαj+1βj+1βj − αjβj−1βj−2αj−1]= g∑j[αjβjαj+1βj+1 + βjαj+1βj+1αj+2]. (4.2)Note that this is essentially a chain of MZMs in 1D with next-nearest-neighbor hoppings that alternate in sign5. Also, observe that the plaquettesof the third kind, which appear in the 2D case, are not present here. The 2-leg ladder, therefore, misses the crucial competition between the three kindsof plaquettes. We shall, however, study this Hamiltonian a bit further outof mere curiosity.4.1 SymmetriesFor now, we are interested in the discrete symmetries of the effective 1Dlattice model. It is easy to verify that translation T by one site (αj →βj and βj → αj+1) and time reversal Θ (i → −i) are not symmetries ofthe Hamiltonian. The combination, however, leaves H0 invariant. Further,5So far we have denoted a to be lattice spacing in the triangular lattice. Upon mappingthe 2-leg ladder to a chain the effective inter-vortex distance becomes a/2. However, inorder to keep notations clear and play safe with the perilous factors of 2, in this chapterwe shall use a to refer to the spacing between the sites in the chain.174.2. Non-interacting limit<latexit sha1_base64="7HU+8wD+AdEhx68juXXejBW iYUs=">AAACBHicbVDLSsNAFJ3UV62vqks3g0VwVRIVdFnQhcsK9gFtKDfTSTN2MhlmJkII3foBbvUT3Ilb/8Mv8De ctllo64ELh3Pu5d57AsmZNq775ZRWVtfWN8qbla3tnd296v5BWyepIrRFEp6obgCaciZoyzDDaVcqCnHAaScYX0/9 ziNVmiXi3mSS+jGMBAsZAWOlbh+4jGDwMKjW3Lo7A14mXkFqqEBzUP3uDxOSxlQYwkHrnudK4+egDCOcTir9VFMJZA wj2rNUQEy1n8/uneATqwxxmChbwuCZ+nsih1jrLA5sZwwm0oveVPzXk1GmGdEL60145edMyNRQQebbw5Rjk+BpInj IFCWGZ5YAUcw+gEkECoixuVVsMt5iDsukfVb3zuvu3UWtcVNkVEZH6BidIg9doga6RU3UQgRx9Ixe0Kvz5Lw5787Hv LXkFDOH6A+czx9HJJjx</latexit><latexit sha1_base64="Z2NthJQb6+5gqYZ8VqopYE/ doD4=">AAACA3icbVDLSgNBEOyNrxhfUY9eBoPgKeyqoMeAHjxGMA9IljA76U3GzM4uM7PCEnL0A7zqJ3gTr36IX+B vOEn2oIkFDUVVN91dQSK4Nq775RRWVtfWN4qbpa3tnd298v5BU8epYthgsYhVO6AaBZfYMNwIbCcKaRQIbAWj66nf ekSleSzvTZagH9GB5CFn1Fip1Q3Q0N5Dr1xxq+4MZJl4OalAjnqv/N3txyyNUBomqNYdz02MP6bKcCZwUuqmGhPKRn SAHUsljVD749m5E3JilT4JY2VLGjJTf0+MaaR1FgW2M6JmqBe9qfivlwwzzZleWG/CK3/MZZIalGy+PUwFMTGZBkL 6XCEzIrOEMsXtA4QNqaLM2NhKNhlvMYdl0jyreudV9+6iUrvJMyrCERzDKXhwCTW4hTo0gMEInuEFXp0n5815dz7mr QUnnzmEP3A+fwB12ph9</latexit>Figure 4.1: 2-leg ladder maps on to a chain with next-nearest-neighborhopping terms.spatial parity P, inversion about the cross in fig. 4.1, is also a symmetry.These are summarized as follows• T Θ : αj → βj , βj → −αj+1 and i→ −i• P : αj → β−j and βj → −α−j4.2 Non-interacting limitIn the scenario without interactions, after a Fourier transform of the oper-ators and restricting the states to half of the BZ, we haveH0 = it∑k>0[α†kβk(1− e−i2ka) + h.c.] + (ei2ka − e−i2ka)(α†kαk − β†kβk).(4.3)This may be written asH0 = 2t∑k>0(α†k β†k)( −2 sin(2ka) i(1− e−i2ka)−i(1− ei2ka) 2 sin(2ka))(αkβk), (4.4)with a dispersion relation that readsE±k = ±4t√sin2 (ka) + sin2 (2ka) (4.5)with k restricted to 0 ≤ k ≤ pi/2a, as depicted in fig. (4.2). Note that wehave only one node in the spectrum, k = 0, with a linear dispersion.184.3. Low-energy theoryFigure 4.2: The gapless spectrum in with the energy denoted in units of thehopping amplitude t.4.3 Low-energy theoryIn order to study possible symmetry breaking and understand the relevanceof interactions, we focus on the low energy excitations close to the gaplesspoint where the dispersion is linear. If the Majorana fields excite Fouriermodes close to the nodal point k = 0,α(x) ≈ 12√2N∑|k|<Λeikxαk and β(x) ≈ 12√2N∑|k|<Λeikxβk. (4.6)Here, x = 2aj for any given site index j, N is the number of unit cells and Λis a momentum cutoff. As before, the k < 0 modes correspond to excitationsin the positive energy branch.In order to derive the continuum theory, we assume that the fields α(x)and β(x) vary slowly at the scale of vortex spacing. To leading order in thegradient expansion, and after suppressing the position dependence, eq. (4.1)readsH0 ' it∫dx [αβ + β(α+ 2a ∂α) + α(α+ 2a ∂α)− β(β + 2a ∂β)] (4.7)where ∂ represents the derivative along the spatial direction and we havepassed into the continuum limit by∑j →∫dx. Using the fermionic anti-commutation relations to simplify the first two terms and integrating thethird term by parts (to obtain a more symmetric form) givesH0 ' ita∫dx [β∂α+ α∂β + 2(α∂α− β∂β)]. (4.8)194.3. Low-energy theoryThis can be recast into a diagonal form (see Appendix. C.1 for more details)that is reminiscent of a relativistic theory of free Majorana fermionsH0 ' ita√5∫dx [γR∂γR − γL∂γL]. (4.9)The normalization chosen in (4.6) ensures that the new fields obey the stan-dard relativistic anti-commutation relations{γn(x), γn′(y)} = 12δn,n′δ(x− y) (4.10)as shown in Appendix. C.2. Here n, n′ = L or R.In order to incorporate the interactions into the effective low energytheory, we start from eq. (4.2), follow the above prescription. Owing to thefact that we have only two fields and the requirement of locality, the simplestinteraction term must necessarily have two derivativesHI ≈ −8ga2∫dx γL(∂γL)γR(∂γR) + . . . (4.11)Since the action S =∫d2xL, in (1+1)D, is a dimensionless quantity, theLagrangian density must have the mass dimension [L] = −2. And becausethe free theory is quadratic in the Majorana fields, we have [γn] = 1/2.The term in eq. (4.11) then has a dimension 4; a derivative has dimension1. Equivalently the coupling has a negative mass dimension [g] = −2. Thismeans that at a some given energy scale E, a perturbative expansion ismeaningful only in the dimensionless parameter gE2, which is small at lowenergies. The perturbation is, therefore, irrelevant in the renormalizationgroup sense. In other words, the gapless phase should survive in the vicinityof the g = 0. On the other hand, once the symmetries are broken, nothingprecludes the generation of a mass term under renormalization and a phasetransition into a massive phase at some critical value of g/t is, therefore,possible.At this stage, we note that the standard Lorentz invariant mass term ofthe formδHm = im∫dx γLγR (4.12)does not appear in this Hamiltonian (4.9). This is consistent with the sym-metries of the lattice model we started out with. δHm breaks the T Θsymmetry because in the field theory notation the symmetry would require204.4. Strong coupling limitm → −m. It is easiest to see this by writing this term in the language ofthe original fields α and βδHm = im∫dx [c1 αβ + c2 βα] (4.13)where c1 =2−√5√20and c2 =2+√5√20. Now, T Θ would correspond to α → β,β → −α and i → −i. Hence, one must necessarily have m = 0 so long asthe symmetry is unbroken.6In line with [15], this reasoning can also be viewed from the perspectiveof an order parameter B that is defined via 〈iαjβj〉 = A+B and 〈iβjαj+1〉 =A−B. Under the transformation T Θ,〈iαjβj〉 ↔ 〈iβjαj+1〉 . (4.14)That is, when the symmetry is respected, we have B = 0. In other words,at a second order transition both B and m become non-zero.4.4 Strong coupling limitIn order to understand the strong coupling regime of the model, it is in-structive to recast the Hamiltonian in the language of spins. To do so, wefirst rewrite the Majorana operators in the usual complex fermion notationand then perform a Jordan Wigner transformation. This has been outlinedin Appendix C.3. The resulting Hamiltonian is given byH =∑jt [σzj + σxj σxj+1 + σxj σyj+1 + σyj σxj+1]− g [σzjσzj+1 + σxj σxj+2]. (4.15)Let us now look at the limit t = 0. For the moment, suppose that thetwo kinds of interaction terms in the original Majorana model have unequalinteraction strengths g1 and g2HI =∑j−g1[σzjσzj+1]− g2[σxj σxj+2]. (4.16)This can be mapped into a transverse field Ising model with multi-spininteractions using the domain wall variable τ zj = σzjσzj+1. That is,HI = −g1∑τ zj − g2∑τxj−1τxj τxj+1τxj+2 (4.17)6The symmetry P, however, does not mandate a zero mass because it takes α(x) →β(−x) and β(x)→ −α(x), which implies m→ m.214.4. Strong coupling limitIt is known that when g1/g2 = 0, the model has 24−1 groundstates and isrelated to the 8-state Potts model [26, 27]. On the other hand, for g1/g2 →∞, all the spins are aligned along the Z axis because the Hamiltonian reducesto that of a chain of free spins in a magnetic field of strength g1. And atthe point g1/g2 = 1, the system is expected to show a first order transition[28]. It has been argued in [15] that on either side of the line g1 = g2 = g,the system has two groundstates and at the first order transition, all fourstates co-exist.Based on this reasoning, in the infinite coupling limit (g/t → ∞), theHamiltonian (4.15) has four groundstates. One may then identify and ap-proximate these states with the four mean-field states discussed in [15].Namely, we work with the notion that neighboring MZM’s hybridize to formDirac fermion levels. The four states come about because the states thusformed can be filled or empty and, moreover, they can alternate along thechain depending on the sign of the coupling.With this picture in mind, one may treat the effect of turning on a small tas a perturbation. The nearest-neighbor hopping terms split the degeneracyand result in a doublet of groundstates and a low lying doublet of excitedstates. The next-nearest-neighbor terms, on the other hand, vanish in thefirst order perturbative corrections and leading term is of the order O(t2/g).From the above arguments, we expect to see a gapless phase that per-sists upon turning on the interactions between MZMs. But unlike the modelwithout the next neighbor hoppings in [15], where breaking of the translationsymmetry corresponded to a phase transition of tri-critical Ising universal-ity class, translation by one site is no longer a symmetry in our case. Inprinciple, this can be probed numerically, but since our primary interest isin the 2D model, we leave this question to a future study and move on tostudy the 4-leg ladder.22Chapter 54-leg ladderIn the previous chapter, we have seen that only two-thirds of the plaquetteinteractions that are usually present on the full 2D lattice arise in a 2-legladder. The next simplest model is the 4-leg ladder. It can be checked that,as opposed to the 2-leg case, the system now has a gapped spectrum in theabsence of interactions.In order to understand the role of interactions, we start by studyingsmall systems, with up to 32 MZMs, using exact diagonalization. We seethat the gap to the first excited drops sharply upon tuning to a finite valueof attractive interactions. See fig. 5.1 for typical plots.While this transition could represent a symmetry breaking phase tran-sition, it seems unlikely for two reasons. First, the degeneracy is exactdepending on the system size, but it is known that states that break asymmetry are usually separated by a gap that goes down exponentially insystem size. Secondly, the groundstate degeneracy changes upon increasingthe number of legs. This, again, does not corroborate if one expects thesymmetry breaking to be independent of system size.To study how these gaps scale for larger systems and understand thenature of this phase, we turn to the density matrix renormalization group(DMRG) algorithm [29, 30]. DMRG, by design, is best suited to studyquantum systems in one spatial dimension. Crudely, this has to do with thefact that the entanglement across a cut in 1D does not scale with systemsize. For the numerics, we recast the Hamiltonian in the spinless complexfermion basis. In this representation, we have a 2-leg ladder and the numberof complex fermions is given by N = NxNy. Alternatively, one can chooseto work in the spin basis following a Jordan-Wigner transformation.5.1 Gapless phaseTo ascertain that the system transitions into a gapless phase at some criticalcoupling, we begin by measuring the gaps using DMRG. The presence ofedge states, however, complicates the task of determining the bulk gapsand calls for periodic boundaries. In particular, we turn to anti-periodic235.1. Gapless phase−3 −2 −1 0 1g0.00.51.01.52.0Ei−E0Nx = 8, Ny = 2E0E1E2E3E4E5E6−3 −2 −1 0 1g0.00.51.01.52.0Ei−E0Nx = 4, Ny = 4E0E1E2E3E4E5E6Figure 5.1: Behavior of energy gaps as a function of the coupling for differentsystem geometries under periodic boundary conditions. We have set t = 1.boundary condition (APBC) along x because it appears to be better thanPBC for converging on the excited states. The extrapolation of the gap tolarge systems is shown in fig. 5.2.For positive values of the coupling, the gap extrapolates to a finite value.Intriguingly, the gaps lie on a line of zero slope when Nx is odd. When Nxis even, on the other hand, the gap decreases linearly. The constant energygap (for odd Nx) may be interpreted as the excitation being localized inspace.For negative g, the trend is markedly different and the gaps scale tozero. In this case the role of even and odd Nx is switched as compared to245.2. Central charge0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.071/Nx0.000.050.100.150.200.25E1E0g = 2g = 2⇥50<latexit sha1_base64="ws+kJtBE2hfLsSS/EgPI1j+SxdQ=">AAACBXicbVDLSgMxFL3js9ZX1aWbYBFclRkf6EoKblxWsA9ph5 JJ0zY0kxmSO8IwdO0HuNVPcCdu/Q6/wN8wbWehrQcCh3Pu5dycIJbCoOt+OUvLK6tr64WN4ubW9s5uaW+/YaJEM15nkYx0K6CGS6F4HQVK3oo1p2EgeTMY3Uz85iPXRkTqHtOY+yEdKNEXjKKVHjooQm7Ihdstld2KOwVZJF5OypCj1i19d3oRS0KukElqTNtzY/QzqlEw ycfFTmJ4TNmIDnjbUkVtjp9NDx6TY6v0SD/S9ikkU/X3RkZDY9IwsJMhxaGZ9ybiv148TI1gZi4e+1d+JlScIFdslt5PJMGITCohPaE5Q5laQpkW9gOEDammDG1xRduMN9/DImmcVryzint3Xq5e5x0V4BCO4AQ8uIQq3EIN6sAghGd4gVfnyXlz3p2P2eiSk+8cwB84nz8 te5jN</latexit>Figure 5.2: Extrapolation of the energy gap for values of g in the two phases.Separate straight lines have been fit for even and odd Nx owing to theirdistinct behaviors (see text). The systems considered vary from N = 30through N = 50, while keeping a maximum bond dimension of 103 in theDMRG sweeps. The truncation errors are less than 10−8. Note that thegaps for g = 2 have been rescaled by a factor of 50.the gapped phase: gaps vanish independent of the number of sites so longas Nx is even.A conventional way to determine the value of the critical interactionstrength would be to look at the gaps or correlation functions close to thetransition. In this particular problem, we observe that another quantitychanges across the two phases: the fermionic parity of the first excited state.While the groundstate is always in the even parity sector, the parity of thefirst excited state switches from being even in the gapless phase to odd inthe gapped phase. The typical behavior is shown in fig. 5.3. The criticalvalue gc ∼ −0.56 is not altered significantly by system size.5.2 Central chargeIn the critical regime, the physics can be described by a conformal field the-ory (CFT). One of the most relevant numbers in the context of CFTs is thecentral charge, which is a measure of the gapless degrees of freedom. Thisquantity can be determined from the entanglement entropy of the ground-state wavefunction. If a periodic system of size N is described by a CFT255.2. Central charge−0.70 −0.65 −0.60 −0.55 −0.50g−101parityE0E1Figure 5.3: The fermionic parity of the two lowest energy states for the caseof N = 50. The first excited state switches from even to odd across thetransition point; a behavior that is observed for all system lengths.with central charge c, then the entanglement entropy of a subregion of sizex scales as [31]:SN (x) =c3log[Npisin(xpiN)](5.1)Because the system is effectively 2-leg ladder in the language of com-plex fermions, we have a two-site unit cell when it is viewed as a chain.This results in an oscillatory behavior SN (x). Averaging the entropy acrossneighboring bonds (x and x+ 1 and assigning it to x+ 1/2) eliminates theoscillatory sub-leading terms and helps one to extract c. Following this pre-scription, we see that the central charge is 1 to a very good accuracy asshown in fig. 5.4. Interestingly, we find that it continues to remain unity asone further decreases the value g; see fig. 5.5.This suggests that the low-energy sector of the gapless regime, whichappears to persist for all g < gc, is universally described in the continuumlimit by the Tomonaga-Luttinger liquid (TLL) theory [32–34]. This is aclass of CFTs whose excitations are bosonic in nature [35, 36].While the value of central charge in the two extended phases is unam-biguous, its behavior in the vicinity of the transition cannot be ascertainedbecause of the large error bars. The situation is not aided by increasing thebond dimension and more sophisticated techniques from CFT are requiredto understand it further.265.2. Central charge15 20 25 30 35x2.953.003.053.103.153.20SN(x)c = 1.020.84 0.86 0.88 0.90 0.9213 log[Npi sin(xpiN)]3.023.043.063.083.103.12SN(x)1.0158 x + 2.1871Figure 5.4: (top) Entanglement entropy SN (x) for a length x subregion ofa periodic system with length N = 50 at g = −4. (bottom) A linear fitthrough the same data with conformal distance plotted on the horizontalaxis.275.2. Central charge−3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0g01234cN = 38N = 50Figure 5.5: Central charge as a function of the coupling, while keeping upto 103 states and under PBC. We have verified that it continues to remainunity for values up to g = −20 in the gapless phase.28Chapter 6ConclusionIn this work, we have studied a Hubbard like tight-binding model aimed atdescribing the low energy physics of Majorana zero modes that arise in atriangular vortex lattice. The non-interacting Hamiltonian corresponds toa gapped and topologically non-trivial spectrum. A self-consistent mean-field analysis suggests that, unlike the previously studies of the model onother lattices [15, 17], turning on interactions does not lead to a conven-tional symmetry breaking phase transition. Numerical simulation of 4-legladders suggests that at strong enough attractive interactions the systemtransitions into a gapless Luttinger liquid. The aforementioned topologicalphase, however, persists for all values of the coupling that are greater thanthe critical value. While the exact nature of the phase transition remainsto be understood, an analysis of interactions in ladders with more legs, andultimately, extrapolation to the 2D limit is an interesting avenue for futurestudies.There has been a growing body of evidence that supports the obser-vations of MZMs in experiments [37–39]. Recent scanning tunneling mi-croscopy studies of Fe(Se,Te) have shown that at low temperatures, thereindeed are zero-energy states at the cores of vortex defects [40]. In lightof these developments, we hope that our findings of the interaction drivenphase transition will be relevant for experiments.29Bibliography[1] Steven R Elliott and Marcel Franz. Colloquium: Majorana fermions innuclear, particle, and solid-state physics. Reviews of Modern Physics,87(1):137, 2015.[2] Sankar Das Sarma, Michael Freedman, and Chetan Nayak. Majoranazero modes and topological quantum computation. npj Quantum In-formation, 1:15001, 2015.[3] Jason Alicea. Majorana fermions in a tunable semiconductor device.Physical Review B, 81(12):125318, 2010.[4] Chetan Nayak, Steven H Simon, Ady Stern, Michael Freedman, andSankar Das Sarma. Non-abelian anyons and topological quantum com-putation. Reviews of Modern Physics, 80(3):1083, 2008.[5] Gregory Moore and Nicholas Read. Nonabelions in the fractional quan-tum hall effect. Nuclear Physics B, 360(2-3):362–396, 1991.[6] Sankar Das Sarma, Michael Freedman, and Chetan Nayak. Topologi-cally protected qubits from a possible non-abelian fractional quantumhall state. Physical review letters, 94(16):166802, 2005.[7] A Yu Kitaev. Unpaired majorana fermions in quantum wires. Physics-Uspekhi, 44(10S):131, 2001.[8] Roman M Lutchyn, Jay D Sau, and S Das Sarma. Majorana fermionsand a topological phase transition in semiconductor-superconductorheterostructures. Physical review letters, 105(7):077001, 2010.[9] Yuval Oreg, Gil Refael, and Felix von Oppen. Helical liquids andmajorana bound states in quantum wires. Physical review letters,105(17):177002, 2010.[10] Dmitri A Ivanov. Non-abelian statistics of half-quantum vortices inp-wave superconductors. Physical review letters, 86(2):268, 2001.30Bibliography[11] Liang Fu and Charles L Kane. Superconducting proximity effect andmajorana fermions at the surface of a topological insulator. Physicalreview letters, 100(9):096407, 2008.[12] Ching-Kai Chiu, DI Pikulin, and M Franz. Strongly interacting majo-rana fermions. Physical Review B, 91(16):165402, 2015.[13] Meng Cheng, Roman M Lutchyn, Victor Galitski, and S Das Sarma.Tunneling of anyonic majorana excitations in topological superconduc-tors. Physical Review B, 82(9):094504, 2010.[14] Armin Rahmani, Xiaoyu Zhu, Marcel Franz, and Ian Affleck. Emer-gent supersymmetry from strongly interacting majorana zero modes.Physical review letters, 115(16):166401, 2015.[15] Armin Rahmani, Xiaoyu Zhu, Marcel Franz, and Ian Affleck. Phasediagram of the interacting majorana chain model. Physical Review B,92(23):235123, 2015.[16] Edward OBrien and Paul Fendley. Lattice supersymmetry and order-disorder coexistence in the tricritical ising model. Physical review let-ters, 120(20):206403, 2018.[17] Ian Affleck, Armin Rahmani, and Dmitry Pikulin. Majorana-hubbardmodel on the square lattice. Physical Review B, 96(12):125121, 2017.[18] Kyle Wamer and Ian Affleck. Renormalization group analysis of phasetransitions in the two-dimensional majorana-hubbard model. PhysicalReview B, 98(24):245120, 2018.[19] Chengshu Li and Marcel Franz. Majorana-hubbard model on the hon-eycomb lattice. Physical Review B, 98(11):115123, 2018.[20] Yaacov E Kraus and Ady Stern. Majorana fermions on a disorderedtriangular lattice. New Journal of Physics, 13(10):105006, 2011.[21] Eytan Grosfeld and Ady Stern. Electronic transport in an array ofquasiparticles in the ν= 5/ 2 non-abelian quantum hall state. PhysicalReview B, 73(20):201303, 2006.[22] Tianyu Liu and M Franz. Electronic structure of topological super-conductors in the presence of a vortex lattice. Physical Review B,92(13):134519, 2015.31Bibliography[23] F Duncan M Haldane. Model for a quantum hall effect without landaulevels: Condensed-matter realization of the “parity anomaly”. PhysicalReview Letters, 61(18):2015, 1988.[24] Jeffrey CY Teo and Charles L Kane. Topological defects and gap-less modes in insulators and superconductors. Physical Review B,82(11):115120, 2010.[25] Timothy H Hsieh, Ga´bor B Hala´sz, and Tarun Grover. All majoranamodels with translation symmetry are supersymmetric. Physical reviewletters, 117(16):166802, 2016.[26] Lo¨ıc Turban. Self-dual ising chain in a transverse field with multispininteractions. Journal of Physics C: Solid State Physics, 15(4):L65, 1982.[27] KA Penson, R Jullien, and P Pfeuty. Phase transitions in systems withmultispin interactions. Physical Review B, 26(11):6334, 1982.[28] Francisco C Alcaraz. Order of phase transition for systems withmultispin interactions: Monte carlo simulations. Physical Review B,34(7):4885, 1986.[29] Steven R White. Density matrix formulation for quantum renormaliza-tion groups. Physical review letters, 69(19):2863, 1992.[30] Steven R White. Density-matrix algorithms for quantum renormaliza-tion groups. Physical Review B, 48(14):10345, 1993.[31] Pasquale Calabrese and John Cardy. Entanglement entropy and confor-mal field theory. Journal of Physics A: Mathematical and Theoretical,42(50):504005, 2009.[32] Sin-itiro Tomonaga. Remarks on bloch’s method of sound waves ap-plied to many-fermion problems. In Bosonization, pages 63–88. WorldScientific, 1994.[33] JM Luttinger. An exactly soluble model of a many-fermion system. InLuttinger Model: The First 50 Years and Some New Directions, pages3–11. World Scientific, 2014.[34] FDM Haldane. Effective harmonic-fluid approach to low-energy prop-erties of one-dimensional quantum fluids. Physical Review Letters,47(25):1840, 1981.32[35] Alexander O Gogolin, Alexander A Nersesyan, and Alexei M Tsvelik.Bosonization and strongly correlated systems. Cambridge universitypress, 2004.[36] Thierry Giamarchi. Quantum physics in one dimension, volume 121.Clarendon press, 2003.[37] Jin-Peng Xu, Mei-Xiao Wang, Zhi Long Liu, Jian-Feng Ge, XiaojunYang, Canhua Liu, Zhu An Xu, Dandan Guan, Chun Lei Gao, DongQian, et al. Experimental detection of a majorana mode in the core ofa magnetic vortex inside a topological insulator-superconductor bi 2 te3/nbse 2 heterostructure. Physical review letters, 114(1):017001, 2015.[38] C Chen, Q Liu, TZ Zhang, D Li, PP Shen, XL Dong, Z-X Zhao,T Zhang, and DL Feng. Quantized conductance of majorana zero modein the vortex of the topological superconductor (li0. 84fe0. 16) ohfese.arXiv preprint arXiv:1904.04623, 2019.[39] Stevan Nadj-Perge, Ilya K Drozdov, Jian Li, Hua Chen, Sangjun Jeon,Jungpil Seo, Allan H MacDonald, B Andrei Bernevig, and Ali Yazdani.Observation of majorana fermions in ferromagnetic atomic chains on asuperconductor. Science, 346(6209):602–607, 2014.[40] T Machida, Y Sun, S Pyon, S Takeda, Y Kohsaka, T Hanaguri,T Sasagawa, and T Tamegai. Zero-energy vortex bound state in the su-perconducting topological surface state of fe (se, te). Nature materials,page 1, 2019.33Appendix ASymmetriesHere we denote the MZM operators of the unit cell located at r = md1+nd2by αm,n and βm,n. PBC along x and y correspond to αNx+m,Ny+n = αm,n.Similar relation holds for the β sub-lattice.A.1 TranslationsIt is easy to see that Ta is a symmetry because it corresponds to translationby a unit cell along the horizontal axis. We note that translation along thediagonal direction c also leaves the Hamiltonian invariant, provided that itis accompanied by a particular gauge transformation:αm,n → (−1)m+nβm,nβm,n → (−1)m+n+1αm+1,n+1 (A.1)PBC also gives us the following relations between the TµsT NxNyc = 1, T Nxa = 1, T 2Nyc = T Nya (A.2)The first equation can be derived from the third by taking the N thx poweron both sides and using the second relation to simplify.A.2 Reflections and time reversalAssuming that the α sites are along the x axis, the transformation for acombined action of reflection about x and time reversal ΘRx is given byαm,n → (−1)mαm,−nβm,n → (−1)m+1βm,−n−1i→ −i (A.3)34A.3. Rotation by pi/3Again, with the convention that the α sites lie on the y axis, ΘRy cor-responds toαm,n → (−1)nα−m,nβm,n → (−1)nβ−m−1,ni→ −i (A.4)A.3 Rotation by pi/3In order to study the six-fold rotation symmetry, which arises because of theunderlying triangular geometry, it is convenient to switch to a notation thatis natural to a triangular lattice. In this labeling of the MZMs, the two-siteunit cell is not explicit. In the new notation, the Hamiltonian readsH0 = it∑m,nγm,n [(−1)nγm+1,n + (−1)nγm,n+1 − γm−1,n+1] , (A.5)where r = mp + nq with here p = (a, 0) and q =(12a,√32 a). A rotation bypi/3 in the clockwise direction corresponds toγm,n → sm,n γm+n,−m (A.6)such thatsm,n = (−1)m(−1)n−12 (n odd)= (−1)n2 (n even) (A.7)To prove this, we first note that the coordinates transform asmp + nq→ mp′ + nq′→ m(p− q) + np = (m+ n)p + (−m)q (A.8)Accordingly, suppose that γm,n → sm,n · γm+n,−m. Then,H → it∑sm,nγm+n,−m[sm+1,n(−1)nγm+n+1,−m−1+sm,n+1(−1)nγm+n+1,−m − sm−1,n+1 · γm+n,−m+1] (A.9)Interchanging m↔ n and calling −n→ n,H′ = it∑s−n,mγm−n,n[s−n+1,m(−1)mγm−n+1,n−1+s−n,m+1(−1)mγm−n+1,n − s−n−1,m+1 · γm−n,n+1]. (A.10)35A.3. Rotation by pi/3Renaming m− n→ m givesH′ = it∑s−n,m+nγm,n[s−n+1,m+n(−1)m+nγm+1,n−1+s−n,m+n+1(−1)m+nγm+1,n − s−n−1,m+n+1 · γm,n+1] (A.11)Now, shifting the sum by m→ m− 1 and n→ n+ 1 in the first term aloneand changing the order of operators, it becomess−n−1,m+n · s−n,m+n(−1)m+n(−1) · γm,nγm−1,n+1 (A.12)Finally, comparing the coefficients with the product of γ operators in (A.5),s−n,m+n · s−n,m+n+1 = (−1)ms−n,m+n · s−n−1,m+n+1 = (−1)n+1s−n,m+n · s−n−1,m+n = (−1)m+n (A.13)It is easy to verify that (A.7) satisfies these relations.36Appendix BMean field theory of thesquare latticeMajorana-Hubbard modelApplying the same arguments as in sec. 3.3 to the square lattice Majorana-Hubbard model, we have the following mean-field HamiltonianHMF = i∑rτx(γprγqr − γsrγrr) + τx¯(γqrγpr+x − γrrγsr+x)+τy(γprγsr + γqrγrr) + τy¯(γsrγpr+y + γrrγqr+y) (B.1)where x = (2a, 0), y = (0, 2a) and the sites are labeled as shown in fig. B.1.Figure B.1: A Z2 gauge choice for the square vortex lattice with a four-siteunit cell.37Appendix B. Mean field theory of the square lattice Majorana-Hubbard modelIn this case, the order parameters can be defined as∆x = 〈iγprγqr〉 = −〈iγsrγrr〉∆x¯ =〈iγqrγpr+x〉= − 〈iγrrγsr+x〉∆y = 〈iγprγsr〉 = 〈iγqrγrr〉∆y¯ =〈iγsrγpr+y〉=〈iγrrγqr+y〉, (B.2)and the gap equations are given byτj = t− 2g∆j , (B.3)where j = x, y, x¯, y¯. And, as before, we have∆j =12N∂EMF∂τj. (B.4)Solving these self-consistently, we indeed see a phase transition that ismarked by a non-zero order parameter D. See fig. B.2. The choice of initialτj biases the system to form dimers either along y (non-zero D) or along x(non-zero B). Note that the MZM pairing also breaks pi/2 rotation symme-try [17].38Appendix B. Mean field theory of the square lattice Majorana-Hubbard model0.0 2.5 5.0 7.5 10.0g1234 xxyy0.0 2.5 5.0 7.5 10.0g0.00.10.22A= | x+ x|2B= | x x|2C= | y+ y|2D= | y y|Figure B.2: The mean-field parameters {τj} and {∆j} that have been ob-tained by solving eq. (B.3) and eq. (B.4) self-consistently.39Appendix C2-leg ladder: field theory etc.C.1 Diagonalization and Lorentz invarianceTo get to a Lorentz invariant form of the Hamiltonian, we write (4.9) asH0 ' it∫dx(α β)( ∂x 2∂x2∂x −∂x)(αβ). (C.1)The matrix can be written compactly as (σz + 2σx)∂x and, henceforth, wedenote Γ = (α β). This can be diagonalized using the orthogonal matrixconstructed from its normalized eigenvectors:O = 2−√5√10−4√52+√5√10+4√51√10−4√51√10+4√5 . (C.2)Following this transformation, the kinetic part isH0 '√5 it∫dx(γL γR)(∂x 00 −∂x)(γLγR)' iv∫dx [χTσz∂xχ], (C.3)where χT = ΓTO = (γL γR) and the velocity is v =√5ta.Let us now verify that the theory is indeed Lorentz invariant. Firstly,note that above diagonalization does not influence the time dependence ofthe fields becauseO is independent of time: ΓTOOT∂tOOTΓ = χTOOT∂tχ =χT∂tχ. Although we have used H0 to denote the Hamiltonian thus far, weshall now abuse notation and use it to denote the Hamiltonian density inthe following. The Lagrangian density is given byL0 = iχT∂tχ−H0= i χT∂tχ− iv χTσz∂xχ. (C.4)40C.2. Normalization of the field operatorsNow define the Dirac matrices as γµ ≡ (σy,−iσx), such that {γµ, γµ} =2ηµν1 and ηµν = diag(−1, 1). And finally, with the identification χ¯ = χTγ0,the Lagrangian density takes the formL0 = iv χ¯γµ∂µχ (C.5)This is invariant under any generic Lorentz transformation given byχ→ eiγµaµ/2χ (C.6)where γ0 and γ1 are the generators of rotations and boosts respectively.C.2 Normalization of the field operatorsLet us briefly look at the normalization in eq. (4.6). Consider the followinganti-commutation relation{α(x), α(y)} = 18N∑|k|<Λeik(x−y)→ 12N∫|k|<Λdk( 2piN2a)eik(x−y)≈ 14δ(x− y). (C.7)In the first line, we have plugged in the definition of α(x) and used therelation {αk, αk′} = δ−k,k′ to simplify. We then use the result ∆k = 2piN2a toget the integral representation.With this and the expression (α β) O = (γL γR), it is easy to checkthat{γn(x), γn′(y)} = 12δn,n′δ(x− y), (C.8)as had been claimed in the main text.C.3 Jordan-Wigner transformFor this section, we will work with the convention that αj and βj composea Dirac fermion at site j, that is, cj ≡ (αj + iβj)/2 orαj = c†j + cjβj = i(c†j − cj). (C.9)41C.3. Jordan-Wigner transformIn this basis, the Hamiltonian readsH =∑jt [pˆj + (c†j + cj)(c†j−1 − cj−1) + 2i(c†jc†j+1 + cjcj+1)]−g [pˆj pˆj+1 + (c†j + cj)pˆj−1(c†j−2 − cj−2)]. (C.10)where pˆj = (2c†jcj − 1) measures the parity of the site j. Now, the usual 1DJordan-Wigner transform, in 1D, is defined asc†j =∏i<jσziσ+j and cj =∏i<jσziσ−j . (C.11)One may use these to obtain a transformation that is directly applicable inthe Majorana representation,αj =∏i<jσzi (σ+j + σ−j ) =∏i<jσziσxjβj =∏i<jσzi i(σ+j − σ−j ) = −∏i<jσziσyj . (C.12)We plug these into eq. (4.1) and eq. (4.2) and simplify the expressionsusing an identity of Pauli matrices: σajσbj = δabI + iabcσcj . One then arrivesat the spin Hamiltonian in eq. (4.15).42

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