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Realizing high-energy physics in topological semimetals Chen, Anffany 2019

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REALIZING HIGH-ENERGY PHYSICS IN TOPOLOGICAL SEMIMETALSbyAnffany ChenA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)July 2019© Anffany Chen, 2019The following individuals certify that they have read, and recommend to the Facultyof Graduate and Postdoctoral Studies for acceptance, the dissertation entitled:Realizing High-Energy Physics in Topological Semimetalssubmitted by Anffany Chen in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin PhysicsExamining Committee:Marcel Franz, Physics and AstronomySupervisorSarah Burke, Physics and AstronomySupervisory Committee MemberIan Keith Aeck, Physics and AstronomyUniversity ExaminerAlireza Nojeh, Electrical & Computer EngineeringUniversity ExaminerAdditional Supervisory Committee Members:Mona Berciu, Physics and AstronomySupervisory Committee MemberJeremy Heyl, Physics and AstronomySupervisory Committee MemberiiAbstractThe discovery of topological phases of matter has brought high-energy and condensedmatter communities together by giving us shared interests and challenges. Onefruitful outcome is the broadened range of possibilities to study high-energy physics incost-effective table-top experiments. I have investigated scenarios in which influentialhigh-energy ideas emerge in solid-state systems built from topological semimetals gapless topological phases which have drawn intense research efforts in recent years.My Thesis details three proposals for realizing Majorana fermions, Adler-Bell-Jackiwanomaly, and holographic black holes in superconductor-Weyl-semimetal heterostruc-tures, mechanically strained Weyl semimetal nanowires/films, and graphene flakessubject to strong magnetic fields, respectively. By analyzing the effects of realisticexperimental conditions, I wish to demonstrate that these proposals are experimen-tally tangible with existing technologies.iiiLay SummaryTopological phases of matter are exotic phases that can only be described by an in-tricate mathematical language  topology. This exciting field of research has broughthigh-energy and condensed matter physicists together by giving us shared interestsand challenges. One fruitful outcome is the broadened range of possibilities tostudy high-energy physics in cost-effective table-top experiments. This Thesis detailsthree proposals to manipulate topological semimetals such that elusive fundamentalparticles, quantum anomalies, or black hole holograms can emerge inside a mesoscopicpiece of material. By analyzing the effects of realistic experimental conditions, Iwish to demonstrate that these proposals are experimentally tangible with existingtechnologies.ivPrefaceThe majority of this Thesis has been published in peer-reviewed journals.Chapter 3 is based on [A. Chen and M. Franz, Phys. Rev. B 93, 201105 (2016)]and [A. Chen, D. I. Pikulin, and M. Franz, Phys. Rev. B 95, 174505 (2017)]. Inthe former, I conducted the numerical analysis while my supervisor M. Franz pennedthe manuscript; in the latter, my collaborator D. I. Pikulin contributed the RandomMatrix Theory analysis (Sec. 3.3.5) while the rest was done by me.Chapter 4 is based on [D. I. Pikulin, A. Chen, and M. Franz, Phys. Rev. X6, 041021 (2016)], in which I contributed the numerical analysis in Sec. 4.5.2. M.Franz wrote most of the paper and worked out the basic theory in Sec. 4.4, 4.5 and4.6. D. I. Pikulin contributed to developing of the ideas behind the basic theory andperformed calculations leading to Figs. A.1, A.2, and C.1.Chapter 5 is based on [A. Chen, R. Ilan, F. de Juan, D. I. Pikulin, and M. Franz,Phys. Rev. Lett. 121, 036403 (2018)]. The idea was conceived jointly by R. Ilan, F.de Juan, D. I. Pikulin, and M. Franz in a conversation held at the 2017 APS MarchMeeting. The paper was written mostly by M. Franz with contributions from R. Ilan,F. de Juan, and D. I. Pikulin. M. Franz performed most of the analytical calculationswith help from D. I. Pikulin. I conducted the numerical simulations which underlieall of the figures in the paper (except for the schematic illustration in Fig. 5.1).vContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Brief Review of Topological Semimetals . . . . . . . . . . . . . . . . 52.1 Weyl Semimetals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Majorana Fermions at Weyl-Semimetal-Superconductor Interface 133.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Superconducting Proximity Effect and Majorana Flat Bands . . . . . 163.2.1 2D Continuum Model of Surface States . . . . . . . . . . . . . 17vi3.2.2 3D Tight-Binding Model . . . . . . . . . . . . . . . . . . . . . 203.2.3 Stability Against Non-Magnetic Disorder . . . . . . . . . . . . 233.2.4 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . 243.3 Experimental Detection by Josephson Current Measurement . . . . . 253.3.1 Heuristic Argument . . . . . . . . . . . . . . . . . . . . . . . . 263.3.2 Experimental Set-Up with Dirac Semimetal Cd3As2 . . . . . . 273.3.3 Majorana Flat Bands at Cd3As2-SC Interface . . . . . . . . . 293.3.4 Current-Phase Relation at Finite Temperatures . . . . . . . . 333.3.5 Random Matrix Theory Analysis of Non-Magnetic Disorder . 363.3.6 Application to Other T -Invariant Weyl Semimetals . . . . . . 394 Chiral Anomalies in Strained Weyl Semimetals . . . . . . . . . . . . 414.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Gauge-Fields Induced by Mechanical Strains . . . . . . . . . . . . . . 444.3 Intuitive Picture of Chiral Anomalies in Weyl Semimetals . . . . . . . 464.4 Analytical Calculations with12-Cd3As2 Model . . . . . . . . . . . . . 484.5 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 554.5.1 Pseudomagnetic Field b from Torsion . . . . . . . . . . . . . . 554.5.2 Pseudoelectric Field e from Unidirectional Strain . . . . . . . 584.6 Experimental Manifestations . . . . . . . . . . . . . . . . . . . . . . . 654.6.1 Topological Coaxial Cable . . . . . . . . . . . . . . . . . . . . 654.6.2 Chiral Torsional Effect . . . . . . . . . . . . . . . . . . . . . . 674.6.3 Ultrasonic Attenuation and EM Field Emission . . . . . . . . 704.6.4 Chiral Anomaly in the Absence of EM Fields . . . . . . . . . 764.7 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 765 Holographic Black Hole on a Graphene Flake . . . . . . . . . . . . . 815.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81vii5.2 Definition of Sachdev-Ye Model . . . . . . . . . . . . . . . . . . . . . 845.3 Graphene Flake in a Magnetic Field . . . . . . . . . . . . . . . . . . . 845.3.1 Tight-Binding Model . . . . . . . . . . . . . . . . . . . . . . . 845.3.2 Energy Spectrum and Zero-Mode Wavefunctions . . . . . . . . 855.3.3 Interaction Matrix Between Zero Modes . . . . . . . . . . . . 875.4 Exact Diagonalization of Many-Body Hamiltonian . . . . . . . . . . . 885.4.1 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.4.2 Level Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 895.5 Further Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 915.5.1 Electron Spins . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.5.2 Chiral-Symmetry-Breaking Disorder . . . . . . . . . . . . . . . 925.6 Prospect of Experimental Realization . . . . . . . . . . . . . . . . . . 926 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Appendix A: Tight-Binding Model, Dispersion Relations, and Parametersfor Cd3As2 and Na3Bi . . . . . . . . . . . . . . . . . . . . . . . . . . 111Appendix B: Nonequilibrium Distribution in a Stretched Weyl Semimetal . 115Appendix C: Hydrodynamic Flow in a Twisted Weyl Nanowire . . . . . . . 118Appendix D: Exchange Splitting and Interaction Matrix Between Zero Modes122Appendix E: Symmetry Breaking Perturbations in Real Graphene . . . . . 127viiiList of Tables5.1 Gaussian ensembles for the SY model. The relevant probabilitydistributions are given by Eq. (5.6) with Z = 827, 4pi81√3, 4pi729√3and β =1, 2, 4 for GOE, GUE, GSE, respectively. . . . . . . . . . . . . . . . . 90A.1 Material parameters taken from Refs. [121] and [139]. The last rowrepresents the effective lattice constant used for our numerical simula-tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112ixList of Figures2.1 This figure is from Ref. [10]. Panel (a) illustrates schematically a Weylsemimetal system (in k-space) with two nodes. At every internodalmomentum, the embedded Chern insulator (red plane) exhibits a chiraledge state. If the system is given an open boundary parallel to thedisplacement of the nodes, as shown by the top surface, all the chiraledge states collectively give rise to the Fermi-arc surface states. Panel(b) shows the band structure (at some fixed kz). At the Fermi energy,the chiral-propagating surface states appear as a Fermi arc ending atthe Weyl nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 This figure is from Ref. [24]. Panel (a) shows the graphene lattice withsublattices A and B. a1 and a2 are the lattice unit vectors and δi's arethe nearest-neighbour vectors. Panel (b) is the corresponding Brillouinzone. The Dirac nodes are located at valleys K and K ′. . . . . . . . 113.1 Surface Fermi arcs in a minimal model of a Weyl semimetal with (a)and without (b) time reversal symmetry T . Weyl nodes are representedby circles with positive (negative) chirality; green arrows indicate thepossible direction of electron spin along the arcs. Panel (c) shows thepi junction setup on the surface of a Weyl semimetal. . . . . . . . . . 17x3.2 Tight binding simulations of the SC/Weyl proximity effect. Panels (a)and (e) show the surface spectral function A(k, ω) for the tight bindingHamiltonian hlatt + δhlatt and ω = 0.15. In all panels we use m = 0.5,λ = 1, Lx = 50 and Lz = 40; while (µ, ) = (0, 0) and (0.1, 1.0) fortop and bottom row respectively. Panel (b) shows the normal statespectrum with flat bands representing the surface states. The effectof the SC proximity effect with ∆0 = 0.5 is indicated in panel (c)while panel (d) shows the effect of two parallel equidistant pi junctions(protected zero modes indicated in red). The bottom row displays ourresults for rotated arcs; panel (f) the uniform SC surface state, panels(g) and (h) two parallel pi junctions along y and x axes, respectively.The small gap at Weyl points in panels (c) and (f) is a finite size effect the gap closes as Lz →∞. . . . . . . . . . . . . . . . . . . . . . . 213.3 By placing a superconducting ring with a narrow opening on top ofa Cd3As2 slab, one obtains a Josephson junction mediated by thesurface states of Cd3As2. We label the length of the junction by d andthe width by W . An external flux going through the ring generatesa phase difference ϕ = 2pi(Φ/Φ0) across the junction, which leadsto supercurrent measurable by a capacitively-coupled SQUID sensor.To study the lowest ABS, we assume d  ξ, the proximity-inducedcoherence length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28xi3.4 Spectrum of the Lxa×Lya×Lza = 40a× 52a× 60a (a = 20Å) slab ofCd3As2 with periodic boundary condition in the y- and z-directions.The material parameters are listed in Table A.1. Energy states arecolor-coded according to the expectation value of distance from thesurface. Panel (a) shows the normal-state energy spectrum with theDirac cones separated along kz and the surface Fermi arcs spannedbetween them. In Panel (b) a uniform pairing potential of magnitude∆0 = 10meV is introduced on the surfaces; the Fermi arcs are gappedout. In Panel (c), two junctions of pi phase difference are implementedon the top surface and the bottom surface remains uniformly supercon-ducting; the resulting zero-energy surface flat bands linking the Diracnodes are four-fold degenerate and decoupled. In Panel (d), the phasedifference deviates from pi to 1.1pi; the flat bands get split from zeroenergy and obtain small dispersion, while the rest of the bands staythe same. In Panels (b)-(d), the small gaps near the Dirac nodes aredue to the finite-size effect. . . . . . . . . . . . . . . . . . . . . . . . . 30xii3.5 The Josephson current (per junction) computed from Eq. (3.18) isshown at various ϕ. At kBT = 0 and 0.1meV, there is a clear currentjump of magnitude 60meVe/~ or 15µA. At kBT = 1meV, the CPRretains little skewness. For comparison, the solid blue curve showsthe sinusoidal CPR of a conventional junction with one ABS. Themagnitude is small (bounded by∆02e~) compared to our numericalresults, mainly because our junction hosts a large number of ABSs.Note that ∆0 = 10meV in the numerics, which is unrealistically largeso that the superconducting gap is larger than the finite-size gaps thatare always present in finite systems. While kBT = 0.1 and 1meVappear to be fairly high temperatures, it is the ratio kBT/∆0 (1% and10% respectively) that dictates the thermalization of quasiparticles.In a more realistic system with ∆0 = 0.1meV, the predicted blue andgreen curves correspond to T ∼10mK amd 100mK respectively. . . . 343.6 The first few quasiparticle excitation energies Ei at kz = 0 are shownat various ϕ. The subgap ABS (in red) has a kink at pi while the otherstates show little phase dependence. Same behaviour holds for anykz ∈ (−Q,Q); outside of this range there is no ABS. . . . . . . . . . 353.7 Panel (a) plots ν(E) at pi and shows that non-magnetic disorder broad-ens the Majorana flat bands by c/∆0. Panel (b) shows ν(E) at c/∆0 =0.1 and different ϕ's. The broadening diminishes as ϕ deviates from pi. 383.8 Panel (a) shows the disordered CPR near pi at kBT/∆0 = 0.01 andvarious c/∆0. The discontinuity is rounded off but the jump size isunaffected. Panel (b) shows the CPR slope at pi as c/∆0 increases(again at kBT/∆0 = 0.01). For comparison, the slope at pi of aconventional sinusoidal CPR is 1. . . . . . . . . . . . . . . . . . . . . 39xiii4.1 With external magnetic field threaded through the Weyl semimetalnanowire, counter-propagating zeroth Landau levels are formed at theopposite Weyl nodes, shown in Panel a. Twisting the nanowire inducesa pseudo-magnetic field b, whose chiral nature is seen in the parallelzeroth Landau levels of the bulk, as illustrated in Panel b. Whena parallel electric field is applied, the electron states begin to evolvesemiclassically towards higher momenta. . . . . . . . . . . . . . . . . 444.2 The effect of strain on the hopping amplitudes in the tight bindingmodel. a) Unidirectional strain along the z axis simply changes thedistance between the neighboring orbitals leading to the modificationof the hopping amplitude t1 that is linear in u33 to leading order insmall displacement. b) Torsional strain changes the relative orien-tation of the orbitals and brings about hopping amplitudes that aredisallowed by symmetry in the unstrained crystal, such as tsp. Thecorresponding mathematical expression encodes the expectation thattsp would become equal to Λ if the p orbital were displaced all the wayto the horizontal position. In the real material one of course expectsEq. (4.12) to be valid only for displacements small compared to thelattice parameter a. . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3 The displacement field u in the presence of torsion. Consecutive layersof the crystal are rotated by relative angle ϕ0 = Ω(L/a). . . . . . . . 54xiv4.4 Tight-binding model simulations of a Weyl semimetal wire under tor-sional strain and applied magnetic field B = zˆB. Top row of figuresshows the band structure of the lattice Hamiltonian defined by Eqs.(4.8) and (4.13) computed for12-Cd3As2 model parameters, for a wirewith a rectangular cross section of 30× 30 sites and a lattice constanta = 40Å. (We use larger lattice constant here and in subsequent simu-lations than in real Cd3As2 in order to be able to model nanowires andfilms of realistic cross sections with available computational resources.Note that this does not affect the physics at low energies becausethe lattice Hamiltonian is designed to reproduce the relevant k · ptheory independent of a.) Open boundary conditions are imposedalong x and y, periodic along z. Parameters appropriate for Cd3As2are used. Middle and bottom rows show spectral functions Abulk(k, ω)and Asurf(k, ω). The former is obtained by averaging the full spectralfunction Aj(k, ω) over sites j in the central 10 × 10 portion of thewire while the latter averages over the sites located at the perimeter ofthe wire. The torsion applied in columns c and d corresponds to themaximum displacement at the perimeter of 0.5a, or ϕ0 ' 2o betweenconsecutive layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56xv4.5 Tight-binding model simulations of a Weyl semimetal under appliedmagnetic field B = zˆB and unidirectional strain. Parameters forCd3As2 listed in Appendix A are used in all panels. Only spin up sectorof the model is considered with B = 10T. a) Band structure of thesystem with periodic boundary conditions in all directions (no surfaces)projected onto the z axis (k denotes the crystal momentum along thez direction). Solid (dashed) lines show occupied (empty) states. Occu-pation of the strained system is determined by adiabatically evolvingthe single-electron states of the unstrained system. b) Band structureof a slab with thickness d = 1000Å (50 lattice sites). Only posi-tive values of k are displayed but the band structure is symmetricabout k = 0. Red (black) lines show occupied (empty) states. Thecentral panel indicates the nonequilibrium occupancy of the strainedsystem obtained by adiabatically evolving the single-electron states ofthe unstrained system. The right panel shows the occupancy of thestrained system once the electrons relaxed back to equilibrium. Allthree panels correspond to the same total number of electrons N . c)Change in the electron density in response to the applied strain as afunction coordinate y perpendicular to the slab surfaces. δρ refers tothe nonequilibrium distribution while δρeq refers to the relaxed state.Note that density oscillations near the edges apparent in δρ average tozero: there is no net charge transfer between the bulk and the surfacein the nonequilibrium state, as can also be deduced from the vanishingδρ in the bulk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59xvi4.6 Numerically calculated change in the bulk charge density δρbulk inresponse to unidirectional strain α = 0.03 as a function of the appliedfield B. Parameters for Cd3As2 are used with µ = 0 and d = 1000Å(50 lattice sites). Solid black symbols give result for the p-h symmetricversion of the12-Cd3As2 model obtained by setting all Cj parametersto zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.7 Band structure of the spin down sector of Cd3As2 in magnetic fieldB = 10T. Two chiral branches are visible at low energy but they arenow strongly distorted by p-h symmetry breaking terms and they nolonger traverse the gap between the valence and the conduction band. 654.8 Equilibrium current density in the Weyl semimetal wire under torsion.a) Schematic depiction of the bulk/surface current flow. b) Groundstate current density computed from the lattice model Eqs. (4.8) and(4.13) at chemical potential µ = 5meV. Warm (cold) colors representpositive (negative) current density j. The ring-shaped inhomogeneityin j apparent in the bulk of the wire reflects Friedel-like oscillations inelectron wavefunctions caused by the presence of the surface. . . . . . 664.9 Proposed geometry for the EM field emission measurement in the limitwhen all the dimensions of the crystal are much larger than the soundwavelength λs. a) A slab of thickness d = 2d′is subjected to magneticfield B and a longitudinal acoustic sound wave propagating along thez direction. b) A snapshot of the electric field distribution near thesurface calculated from Eq. (4.43). As a function of time the entirepattern moves in the z direction at the speed of sound cs. . . . . . . 74xvii5.1 An irregular shaped graphene flake in an applied magnetic field givesrise to the (0+1)-dimensional SY model, holographically dual to a blackhole in (1+1)-dimensional anti-de Sitter space. Inset: lattice structureof graphene with A and B sublattices marked and nearest neighborvectors denoted by δj. . . . . . . . . . . . . . . . . . . . . . . . . . . 835.2 Electronic properties of an irregular graphene flake in the absence ofinteractions. a) Single-particle energy levels j of the Hamiltonian H0as a function of the magnetic flux Φ = SB through the flake. The flakeused for this calculation, depicted in the inset, consists of 1952 carbonatoms with equal number of A and B sites. The energy spectrum,calculated here in the Landau gaugeA = Bxyˆ and with open boundaryconditions, shows the same generic features irrespective of the detailedflake geometry. b) Typical wavefunction amplitudes of the eigenstatesΦj(r) belonging to LL0 at Φ = 40Φ0 and the edge modes. The numeralsabove each panel denote the energy j of the state in eV, scale bar showsthe magnetic length lB =√~c/eB. . . . . . . . . . . . . . . . . . . . 865.3 Statistical properties of the coupling constants and the thermal en-tropy. a) Histogram of |Jij;kl| as calculated from Eq. (5.4) with V1 = 1for the graphene flake depicted in Fig. 5.2 and N = 16, compared to theGaussian distribution (orange line) with the same variance 0.000805V1.Inset shows the histogram of real and imaginary components of Jij;kl.The mirror symmetry about the horizontal follows from the hermiticityproperty Jij;kl = J∗kl;ij. b) Entropy S(T ) of the SY Hamiltonian (5.1)calculated with Js shown in panel (a). . . . . . . . . . . . . . . . . . 88xviii5.4 Many-body level statistics for the interacting electrons in LL0 of thegraphene flake. Blue bars show the calculated distributions for thegraphene flake. Orange, green and red curves indicate the expecteddistributions given by Eq. (5.6) for GOE, GUE and GSE, respectively.To obtain smooth distributions, results for N = 14, 15, (16) have beenaveraged over 8 (4) distinct flake geometry realizations while N =17, 18 reflect a single realization. . . . . . . . . . . . . . . . . . . . . . 91A.1 Dispersion relations for the spin-up sector of the lattice Hamiltonian(4.7) describing Cd3As2 (top row) and Na3Bi (bottom row). Theparameters used in the simulations include the particle-hole symmetrybreaking terms and are summarized in Table A.1. We used a latticewith 40 × 40 sites and the magnetic fields shown in the green boxesfor each of the material. Notice the different magnitude of effectivemagnetic fields for different compounds  this is due to the differentlattice constants, and different sign of the physical magnetic field be-tween the two rows. Different sign of magnetic fields shows that thephysical magnetic field compensates the torsional one in opposite Weylpoints for opposite directions of magnetic field in accordance with theinterpretation in the main text. . . . . . . . . . . . . . . . . . . . . . 113A.2 Persistent currents in a Cd3As2 nanowire under torsion and magneticfield. a) Band structure detail for spin up (blue) and spin down (green)sectors in a 30 × 30 lattice with B = b = 3.2T and other parametersas in Fig. A.1. b) Calculated current density jz for µ = 0 includingcontributions from both spin sectors. . . . . . . . . . . . . . . . . . . 114xixC.1 Ratio of conductance of a disordered W ×W × 20 system to the con-ductance of the clean system averaged over 100 disorder realizations.Green line is the best fit to the data  parabolic, grey curves show thefailure of the linear (with non-negative G(0)) and cubic fits. . . . . . 121E.1 Effects of the second neighbor hopping t′. a) Single-particleenergy spectrum of a flake (the same geometry as Fig. 5.2) with secondneighbor hopping t′ = 0.037t. b) Average shift δ = Kij and standarddeviation K of 40 energy levels that comprise LL0 as a function of t′. 129E.2 Effects of random on-site potential. a) Low-energy part of thenumerically calculated energy spectrum for the flake with nI = 1% ofdefected sites as a function of the disorder potential strength w andN = 40. b) Average shift δ = Kij and standard deviation K of 40energy levels that evolve from the zero modes which comprise LL0 inthe pure sample. These levels are marked in red in panel (a). . . . . . 131xxAcknowledgmentsFirst and foremost, I would like to thank my supervisor Prof. Marcel Franz. Forthe past few years, he provided amazing support, encouragement, and confidence inme. His guidance and instruction were gracious; his excitement and insight in physicswere inspiring. I am truly grateful for the priviledge of being one of his students.I would also like to thank Prof. Mona Berciu, Prof. Sarah Burke, and Prof.Jeremy Heyl for cheerfully accepting the enormous task of being on my committee.Thank you for overseeing the final stage of my PhD.I have benefited tremendously from my thought-provoking colleagues, especiallyDima Pikulin, Tianyu Liu, Chengshu Li, Etienne Lantagne-Hurtubise, Oguzhan Can,Stephan Plugge, and Sharmistha Sahoo. I thoroughly enjoyed every discussion withyou.On a less personal note, I would like to acknowledge support by NSERC, CI-fAR, UBC Quantum Matter Institute through QuEST, and the 2016 Boulder Sum-mer School for Condensed Matter and Materials Physics through NSF grant DMR-13001648. Some numerical simulations in this Thesis were performed using the Kwantcode [1].Lastly, I could not have persevered without my family: Ron Chen, Jennifer Terng,Skylar Zerr, Marcus Zerr, Christal Zerr, Hedy Funk, and Alice Chik. Thank you foryour unfailing support.xxiFor my parentsxxiiChapter 1IntroductionTo see a world in a grain of sand,And a heaven in a wild flower,Hold infinity in the palm of your hand,And eternity in an hour.- from Auguries of Innocence by William BlakeThroughout the history of science and philosophy, the most prominent view ofthe Universe is that it is made up of indivisible fundamental particles. Nevertheless,modern physicists consider the possibility that all things and the interactions amongthem are merely collective excitations from an underlying quantum vacuum. Thisemergent picture is familiar from solid-state physics, where quasiparticles seeminglymoving in free space emerge at long distances from complex many-body systems. Itremains an open question whether we can construct, at least theoretically, a crystallattice from which everything the high-energy physicists know  fermions, bosons,gauge fields, chiral interactions, and so on  emerge. If we dare to include gravity,then we would be searching for the Theory of Everything  inside a rock. The greatpoet probably never thought that one day physicists would take his imagery literally perhaps the whole Universe can be held in the palm of your hand.1It seems that the research community is naturally moving towards that end, asthere has been productive interaction between high-energy and condensed matterphysicists. These two seemingly disparate fields  high-energy physics mainly con-cerns with single-particle processes at extremely high temperatures while condensedmatter physics deals with many-body problems near absolute zero  both foundtremendous benefits in using quantum field theoretical approaches and had manyback-and-forth exchanges of ideas. For example, the Green's function approachand Feynman diagrams were first used in high-energy physics and later adapted inmany-body problems, and spontaneous symmetry-breaking during phase transitionsin condensed matter systems is implemented by high-energy physicists to understandthe very early Universe.These interactions escalated in recent years partly due to the discovery of topo-logical phases of matter. As their names suggest, these phases can only be distin-guished by their underlying topological properties, beyond the traditional Landauclassification by local symmetries. Several epitomes of this growing class of phasesare described in the low-energy regime by the Chern-Simons theory, a topologicalfield theory, originally considered by high-energy physicists [2, 3]. These phases alsoexhibit exotic, topologically-protected boundary states, some of which are physicalrealizations of high-energy theories. For instance, the chiral edge states of FractionalQuantum Hall states are described by chiral conformal gauge theories previouslystudied by string theorists [4], and the surface states of 3D topological insulators aremassless relativistic fermions [5]. Entanglement, the mechanism which gives rise totopological orders [6], is another shared interest with the high-energy community. Itseems to be closely related to the geometry of space-time, as suggested by quantum-gravity research [7]Having high-energy physics hidden inside these materials is an incredible experi-mental advantage. Typically experimentalists must go to great lengths to observe2high-energy physics directly because of the immense energy-scale involved. It istherefore exciting to find table-top testing grounds in topological materials, whichhave become increasingly accessible in laboratories due to the intense interest andcompetition among experimental groups worldwide. The technologies for growing,manipulating, and measuring these materials are mature and innovative. So farimportant topological signatures such as quantized conductance, exotic surface states,and anomalous response functions have been confirmed by transport experiments,angle-resolved photoemission spectroscopy (ARPES), and scanning tunneling mi-croscopy (STM) [8, 9].Exploring this context, my collaborators and I devised ways to realize high-energyphysics in topological semimetals [10]. These topological phases can be studied usingband theory and by definition exhibit band-touching points or lines at the Fermienergy. Gapless topological phases have recently gathered great interests after it wasrealized that topological invariants are well-defined in the gapped subspace of theBrillouin zone. Notable examples include graphene and Weyl semimetals, which arethe focus of this work. Both systems have gapless points around which the energybands disperse linearly. Graphene, an atomic layer of graphite, is well-understood intheory and accessible experimentally thanks to the Nobel-prize-winning "peeling"technique [11]. Only until recently do people recognize its topological attribute.Weyl semimetals are 3D analogies of graphene. Although they entered the scenefairly recently, intense research efforts have made them available in labs and led toobservations of their topological signatures [1216].Our research developed into three distinct projects:ˆ Majorana Fermions at Weyl-Semimetal-Superconductor Interface (Chapter 3)ˆ Chiral Anomalies in Strained Weyl Semimetals (Chapter 4)ˆ Holographic Black Hole on a Graphene Flake (Chapter 5)3Each was completed by extensive analysis done with experimental realization in mind.This Thesis is the culmination of our investigation and results. I will begin by givinga brief review of the topological semimetals in Chapter 2.4Chapter 2Brief Review of TopologicalSemimetalsUp until recently, physicists upheld the traditional Landau theory of phase transition,which states that distinct phases of matter are characterized by the symmetriesdescribing the organizations of the microscopic components. For instance, water turnsinto ice when water molecules lose their freedom to rotate and move continuously;we say that the translational and rotational symmetries of the system are broken.Without breaking symmetries, the topological phases of matter undergo phase tran-sitions when the so-called topological order changes [6]. Therefore its discovery wasrevolutionary and justifiably recognized by the 2016 Nobel prize. We now knowthat there is a realm of quantum (i.e. zero-temperature) materials living beyondsymmetry classification. A topological order is robust against local perturbations ofthe Hamiltonian (caused by impurities inside the material, for example), in muchthe same way that, in mathematics, the topology of a space is invariant undersmooth deformation. It is manifested through exotic properties such as non-abelianexcitations [17] and quantized edge conductance [5].5There are two types of topological phases, depending on the range of quantumentanglement among the microscopic constituents [6]. Most topological phases withlong-range entanglement  such as quantum spin liquids [18] and fractional quan-tum hall effect [4]  occur in strongly correlated systems, which calls for advancedtheoretical techniques. On the other hand, the topological phases with short-rangeentanglement, also known as symmetry-protected topological phases (SPT) [19], canbe described effectively by topological band theory, stemming from conventional bandtheory of non-interacting quasiparticles. The SPT phases are less robust in the sensethat certain symmetries must be present to maintain the topological order.The pioneering examples of the SPT phases are topological insulators (TI) andtopological superconductors (TSC). They have been fully categorized by a ten-fold-way table according to symmetry protection [20] and realized experimentally in a va-riety of materials [5, 21]. Their topological orders are defined by topological invariantscomputed using their ground state wavefunctions [3, 22]. Topological invariants areusually integer-valued and robust against any perturbations that respect the requiredsymmetries. They dictate the quantization of certain physical observables such as thequantized Hall conductance in integer quantum Hall systems.By construction it seems that topological order exists only in a fully gappedsystem, in which the ground state is well-separated from excitations. More recently,physicists pondered the question of whether there exists gapless topological states. Itturns out that one can compute topological invariants in semimetals. A semimetal is asolid-state system in which the conduction and valence bands touch at isolated pointsor along a continuous line, so it is possible to construct a compact subspace of theBrillouin zone that avoids the gapless region and compute topological invariants onit. If a topological invariant is found to be nontrivial, the system is a topologicalsemimetal. Weyl semimetals are the leading examples of topological semimetals[10, 23]. They have topologically protected band-touching points. Graphene can6be seen as their 2D analogue [24]. Nodal-loop semimetals are another example witha continuous line of band-touching points [23, 25, 26]. Nodal superconductors aregeneralizations of topological semimetals [27, 28]. In the following, we focus onintroducing Weyl semimetals and graphene as they are the main materials of interestin this Thesis.2.1 Weyl SemimetalsA Weyl semimetal is a 3D solid-state system in which the valence and conductionbands are nondegenerate and touch at points in the Brillouin zone. The nondegenerateband-touching points are called the Weyl nodes. Ideally, the Fermi level lies right atthe Weyl nodes and there are no other Fermi surfaces. The electronic band structurenear a Weyl node can be effectively described by the Weyl fermion HamiltonianH =∑qc†qvq · σcq, (2.1)where momentum q is measured from the node, σ = (σx, σy, σz) are the Paulimatrices in the pseudospin space, v is the velocity of the massless Weyl fermion.The corresponding energy spectrum is a linearly dispersing Weyl cone. The factthat the low-energy theory describes a Weyl fermion is important for our discussionon chiral anomaly in Chapter 4. Note that all three Pauli matrices are used up bythe Hamiltonian, so any additional mass term only shifts the position or energy ofthe node. In other words, the Weyl node is stable against any perturbations to theHamiltonian.To describe the nontrivial topology of a Weyl semimetal, consider a 2D subspaceof the Brillouin zone defined by an arbitrary closed surface S enclosing a Weyl node.The electronic band structure is fully gapped in this subspace, on which it is thereforepossible to compute topological invariants. In particular, one can compute the Chern7number, which is a measure of the total Berry flux through the surface. One beginsby computing the Berry vector potentialA(k) = −i∑n occupied〈un,k|∇k|un,k〉, (2.2)where un,k is the Bloch wavefunction and n runs through the filled bands. The Berrycurvature is the curl of the Berry vector potentialB(k) =∇k ×A, (2.3)analogous to the fact that magnetic field is the curl of magnetic vector potential.Intuitively, the Berry curvature can be seen as magnetic field in the momentum space.Chern number is the integral of Berry curvature through the surfaceC = 12pi∫SB · ds = n ∈ Z. (2.4)It must be an integer, the proof of which can be found in Sec. 3.6 of Ref. [3]. Asystem with nonzero Chern number is called a Chern insulator. To gain physicalintuition of the Chern number, one can invoke the Stokes' theorem and rewrite Eq.(2.4) as the line integral of the Berry vector potential A over the boundary of S.Since S is a closed surface without a boundary, the integral vanishes as long as A iswell-defined over the momentum space enclosed by S. Nevertheless, a nonzero Chernnumber can result from any nontrivial structure ofA, which obstructs the applicationof Stokes' theorem. Weyl nodes are singularities of the Berry vector potential A, sothe Chern number for the surface enclosing each node is nontrivial. In most Weylsemimetals, each Weyl node is a source or a sink of 2pi Berry flux, corresponding toChern number of 1 or -1  the chirality of the node. As magnetic monopoles in themomentum space, two Weyl nodes of opposite chiralities can annihilate and result in8a band gap; whereas a single Weyl node cannot simply vanish as explained below Eq.(2.1).The nonzero Chern numbers embedded in the band structure of Weyl semimetalsgive rise to oddly-shaped surface states  the Fermi arcs. To illustrate this, we considerthe simplest Weyl semimetal with two nodes of opposite chiralities1. Due to Gauss'law, any closed surface enclosing a single Weyl node experiences 2pi Berry flux, sowe can choose a closed surface such that it touches the boundary of the Brillouinzone, as shown by a combination of the red and blue planes in Fig. 2.1a. Since itcontains a Weyl node, it has Chern number 1. Now let us consider the red and blueplanes separately. Each of them is a closed surface on its own due to the periodicityof the Brillouin zone, so we can compute their Chern numbers. We know that oneof them must be 0 and the other 1 because together they give one. Without lossof generality, let us suppose that the red plane has Chern number 1. The red planecan be anywhere between the two nodes and still has the same amount of Berry fluxthrough it, i.e. the same Chern number. In other words, there is an embedded Cherninsulator in the band structure at every crystal momentum between the nodes. Cherninsulators are known to exhibit chiral edge states [3]. If the Weyl semimetal systemhas a surface parallel to the displacement between the Weyl nodes (so the momentumin this direction remains a good quantum number), then the chiral edge states  oneat each internodal momentum  accumulatively form the surface states of the Weylsemimetal. At the Fermi energy, they form a Fermi arc ending at the Weyl nodes,which is strikingly different from the Fermi surface in a conventional 2D metal. This isone of the most pronounced signatures of the nontrivial topology of Weyl semimetals.Fig. 2.1b shows the band structure of a semi-infinite Weyl semimetal with one openboundary. Between the bulk Weyl nodes span the chiral-propagating surface stateswhich appear as a Fermi arc at the Fermi energy.1Weyl nodes must occur in positive-negative pairs so that the total chirality in the Brillouin zonevanishes according to the fermion-doubling theorem [29].9Figure 2.1: This figure is from Ref. [10]. Panel (a) illustrates schematically a Weylsemimetal system (in k-space) with two nodes. At every internodal momentum, theembedded Chern insulator (red plane) exhibits a chiral edge state. If the system isgiven an open boundary parallel to the displacement of the nodes, as shown by thetop surface, all the chiral edge states collectively give rise to the Fermi-arc surfacestates. Panel (b) shows the band structure (at some fixed kz). At the Fermi energy,the chiral-propagating surface states appear as a Fermi arc ending at the Weyl nodes.The above argument for the existence of Fermi-arc surface states assumes transla-tional symmetry, which is broken in any realistic materials with impurities. Neverthe-less, Fermi arcs have been experimentally confirmed by angle-resolved photoemissionspectroscopy (ARPES) measurements in the first Weyl semimetal candidates  afamily of compounds consisting of TaAs, TaP, NbAs and NbP [15, 16, 30, 31]. Theirtopological response  negative magnetoresistance due to chiral anomaly (more inChapter 4)  has also been observed [13, 14]. Dirac semimetal Cd2As3 [3237]has the topology of a Weyl semimetal, but its Weyl cones coincide pairwise due tocrystal symmetry (hence realizing Dirac fermions in (3+1)-dimension). It is great fortheoretical studies because it has only four Weyl nodes. Its Fermi arcs and negativemagnetoresistance have been observed [12, 3840].10Figure 2.2: This figure is from Ref. [24]. Panel (a) shows the graphene latticewith sublattices A and B. a1 and a2 are the lattice unit vectors and δi's are thenearest-neighbour vectors. Panel (b) is the corresponding Brillouin zone. The Diracnodes are located at valleys K and K ′.2.2 GrapheneGraphene is a 2D honeycomb lattice of carbon atoms, as shown in Fig. 2.2a. It isa semimetal with two band-touching points at K and K ′ in the Brillouin zone (seeFig. 2.2b). Near each node, the electronic band structure can be effectively describedby the Hamiltonian of a massless Dirac fermion in 2D:h(K + q) = qxσx + qyσy (2.5)andh(K ′ + q) = −qxσx + qyσy. (2.6)It can be shown that the gaplessness of the Dirac cones is protected by inversionand time-reversal symmetries. Any perturbation that breaks either of them makesgraphene a gapped insulator. In fact, it is possible to construct a mass term thatrenders graphene a Chern insulator2.The Dirac nodes in graphene, like the Weyl nodes in Weyl semimetals, are topo-logically nontrivial. They are vortices in the Berry vector potential A and canbe detected by a 1D topological invariant. One computes the Berry phase for a2Since our work in Chapter 5 employs the semimetallic nature of graphene, I will leave theinterested readers to learn more about the insulating graphene in Ref. [3].11constant-energy loop C surrounding the nodeγ =∫Cdk ·A(k) (2.7)which gives pi or −pi, the vorticity of the node. Two Dirac nodes of opposite vorticitieswould annihilate when brought close.Similar to the argument for the surface states of Weyl semimetals, the edge statesin graphene can be understood in terms of the embedded 1D topological insulators.At each fixed ky between K and K′is a polyacetylene-type 1D topological insulator[41], which is known to exhibit isolated zero modes at the boundary. As a result, theband structure of graphene with boundary (e.g. a ribbon) displays zero-energy flatbands of edge states spanned between the bulk Dirac nodes. Detailed derivation ofthe edge states can be found in Ref. [3].12Chapter 3Majorana Fermions atWeyl-Semimetal-SuperconductorInterface3.1 OverviewAll fermions in the Standard Model are described by the Dirac equation. Mathemat-ically, one can tweak the equation and find self-conjugating solutions representingthe Majorana fermions. These hypothetical particles, indistinguishable from theiranti-particles, have not been observed in nature, but they play important roles inneutrino physics and theories beyond the Standard Model [42]. Following the dis-covery of neutrino mass in the neutrino-oscillation experiments [4345], Majoranafermions replaced massless Weyl fermions as the most suitable candidate for neutrinos.Notably this would allow the process of neutrinoless double beta decay, which violatesthe conservation of lepton number and thus provides a primordial mechanism thatcould originate the matter, anti-matter asymmetry. Furthermore, Majorana fermions13fill the role of several promising dark-matter candidates and superpartners requiredby Supersymmetry.It is therefore exciting that Majorana fermions have been found as emergent quasi-particles in solid-state systems. In fact, they appear ubiquitously in superconductorsas Bogoliubov quasiparticles, which are superpositions of electrons and holes in theBogoliubov-de-Gennes theory of superconductivity [46]. A more interesting case ariseswhen Majorana quasiparticles sit at zero energy, separated from the other excitedstates by the superconducting gap. These so-called Majorana zero modes (MZMs)exhibit non-Abelian exchange statistics, instead of the usual fermionic statistics, dueto phase contribution from the superconducting condensate. In principle one can storequantum information in a group of these localized modes by spatially braiding them.This method for storing quantum information is more fault-tolerant than many otherproposals and thus has drawn intense theoretical and experimental research interests[47].The canonical example for realizing MZMs is the Kitaev chain, a 1D super-conductor with a topological invariant defined in the bulk and MZMs localized atthe end points [48]. More proposals have ensued and demonstrated that generallyMZMs can be found at the boundary or the defects of topological superconductors[22, 4966]. One particular example is a 2D topological insulator (TI) coupled toa conventional s-wave superconductor [49, 50]. The metallic edge of the 2D TIbecomes superconducting due to proximity effect and resembles the Kitaev chain.One can artificially create an end point at the edge of the 2D TI by implementing aJosephson junction in the superconductor. Two decoupled MZMs show up localizedat the junction when the phase difference across the junction is pi, which restores thetime-reversal-invariant symmetry needed by TI.In Sec. 3.2, we show that Weyl semimetals with time-reversal symmetry T canalso be suitable platforms for hosting MZMs, and the underlying mechanism is similar14to the aforementioned case of 2D TI. The nontrivial topology of a T -invariant Weylsemimetal can be understood from the hidden 2D TIs embedded in the band structure.The Fermi arcs can be seen as stacks of helical edge states of the embedded 2DTIs. Proximity-coupling the surface of the Weyl semimetal by a superconductorthus amounts to proximitizing the edges of the 2D TIs. It follows from the previousparagraph that a junction mediated by the Weyl semimetal surface states would alsolocalize MZMs at pi phase difference. An important difference is that the systemhere is one dimension higher, so the geometric width of the junction and the extramomentum degree of freedom allow the localization of a large population of MZMs.In particular, while the junction mediated by the edge states of a 2D TI localizes apair of MZMs, one mediated by the surface states of a T -invariant Weyl semimetalcontains N pairs of MZMs where N scales linearly as the momentum space distancebetween the Weyl nodes and the junction width.While this result may not be applicable to quantum computation as the zeromodes are not spatially separated, the conglomeration of MZMs is ideal for studyinginteracting Majorana fermions. Moreover, the set-up is ready to be realized withexisting experimental technologies. To determine the presence of MZMs, one canmeasure Josephson current across the junction as phase difference varies. In Sec.3.3, we predict a characteristic step discontinuity in the current-phase relation, whichis a type of fractional Josephson effect previously studied in other unconventionaljunctions [50, 6771]. We also demonstrate that this experimental signature is robustagainst finite temperature and weak non-magnetic disorder.153.2 Superconducting Proximity Effect andMajoranaFlat BandsInterfacing topological materials with conventionally ordered states of matter, suchas magnets and superconductors, has led to important conceptual advances over thepast decade. Notable examples of this approach include the Fu-Kane superconductor[49] that occurs in the interface of a 3D strong topological insulator (STI) and aconventional s-wave SC, the fractional quantum Hall effect that arises when STI isinterfaced with a magnetic insulator [7274], as well as many interesting phenomenathat occur when both SC and magnetic domains are present [75, 76]. Rich physics,including Majorana zero modes, also results when the edge of a 2D topologicalinsulator is interfaced with magnets and superconductors [50, 77, 78]. More recentlyvarious exotic phases of quantum matter have been predicted to occur based on thesesame ingredients in strongly interacting systems [7989].In this section, we explore the physics of the interface between a Weyl semimetaland an s-wave superconductor. Surface states of a Weyl semimetal exhibit character-istic Fermi arcs that terminate at surface projections of the bulk Weyl nodes [90, 91].Such open Fermi surfaces are fundamentally impossible in a purely 2D system andwe thus expect the resulting SC state to also be anomalous. In this respect thesituation is similar to the Fu-Kane superconductor [49] whose existence hinges on theodd number of Dirac fermions occurring on the surface of an STI. As we shall seethere are several notable differences between STI/SC and Weyl/SC interfaces whichmake the latter a distinct and potentially more versatile platform for explorations ofnew phenomena.163.2.1 2D Continuum Model of Surface StatesNondegenerate Weyl points can occur in crystals with broken time reversal symme-try T or broken bulk inversion symmetry P . Recent experimental work reportedconvincing evidence for Weyl nodes and surface Fermi arcs in T -preserving noncen-trosymmetric crystals in the TaAs family of semimetals [15, 16, 9295]. We thereforefocus our discussion on the SC proximity effect in this class of materials. When acrystal respects T the minimum number of Weyl points n is 4. This is because underT a Weyl point at crystal momentum Q maps onto a Weyl point at −Q with thesame chirality. Since the total chiral charge in the Brillouin zone must vanish therehas to be another pair of T -conjugate Weyl nodes with an opposite chirality. Webegin by discussing the SC proximity effect in this minimal case with n = 4. Wenote that although the currently known Weyl semimetals exhibit larger number ofnodes (n = 24 in the TaAs family) recent theoretical work identified materials thatcould realize instances with smaller n [96, 97], including the minimal case with n = 4predicted in MoTe2 [98]. The generic surface state of such a minimal T -preservingWeyl semimetal is depicted in Fig. 3.1a.kxkykxkya) b) c)SC SC' = 0 ' = ⇡(x)Figure 3.1: Surface Fermi arcs in a minimal model of a Weyl semimetal with (a) andwithout (b) time reversal symmetry T . Weyl nodes are represented by circles withpositive (negative) chirality; green arrows indicate the possible direction of electronspin along the arcs. Panel (c) shows the pi junction setup on the surface of a Weylsemimetal.Interfacing such a surface with a spin singlet s-wave SC, one expects formationof a paired state from time-reversed Bloch electrons at crystal momenta k and −k17along the arcs. A minimal model describing this situation is defined by the secondquantized Bogoliubov-de Gennes (BdG) Hamiltonian H = ∑k Ψˆ†khBdG(k)Ψˆk whereΨˆk = (ck↑, ck↓, c†−k↓,−c†−k↑)T is the Nambu spinor andhBdG =h0(k) ∆∆† −syh∗0(−k)sy . (3.1)The normal state surface Hamiltonian for two parallel Fermi arcs with electron spinlocked perpendicular to its momentum as in Fig. 3.1a can be written ash0(k) = vszkx − µ, for |ky| < K. (3.2)Here sα are Pauli matrices acting in electron spin space and K controls the extent ofthe arc in the y direction. It is possible to make this model more generic by allowingthe velocity v and chemical potential µ to depend on ky and the cutoff K on kx (whichwould curve and shift the arcs similar to Fig. 3.1a) but the minimal model definedabove already captures the essential physics we wish to describe. We note that themodel based on Eq. (3.1) is valid only away from the surface projections of the Weylpoints, specifically when v||ky| −K| > ∆, as it does not capture the bulk bands thatbecome gapless near the Weyl points.If we define another set of Pauli matrices τα in the Nambu space we can write Eq.(3.1) for each |ky| < K ashBdG = (vszkx − µ)τ z + ∆1τx −∆2τ y, (3.3)where ∆ = ∆1 + i∆2. For spatially uniform ∆ the spectrum of hBdG is fully gapped.Its topological character can be exposed by noting that for each allowed value of kyEq. (3.3) coincides with the Hamiltonian describing the edge of a 2D TI in contact18with a SC. It is well known that unpaired MZMs exist in such an edge at domain wallsbetween SC and magnetic regions [50]. This is because T -breaking also opens up agap in 2D TI edge and MZMs occur at boundaries between two differently gappedregions. In our present case of the Weyl semimetal breaking T generically does notopen a gap. However, we can achieve a similar result by creating a domain wall inthe complex order parameter ∆(x). Specifically, we show below that a pi junction(i.e. a boundary between two regions whose SC phase ϕ differs by pi, see Fig. 3.1c)hosts a pair of protected MZMs (one for each arc and for each allowed value of ky),separated by a gap from the rest of the spectrum. As a function of ky, then, the bandstructure of such a pi junction exhibits a flat Majorana band pinned to zero energy.We demonstrate below that such flat Majorana bands are experimentally observableand represent a generic robust property of any T -preserving Weyl/SC interface.To exemplify this property within the effective surface theory (3.3) we replacekx → −i∂x and assume ∆(x) to be purely real with a soliton profile such that ∆(x)→±∆0 as x → ±∞. We next note that Hamiltonian (3.3) can be brought to a blockdiagonal form hBdG = diag(h+, h−) by a unitary transformation that exchanges itssecond and third rows and columns. Here h± are 2× 2 matriceshs =−ivs∂x − µ ∆(x)∆(x) ivs∂x + µ , (3.4)and s = ±. It is easy to show that for each allowed value of ky and each s theabove Hamiltonian (3.4) supports an exact Jackiw-Rossi zero mode [99] with a wavefunctionψs(x) = Ais exp{−1v∫ x0dx′[∆(x′)− isµ]}(3.5)exponentially localized near the junction.19We shall demonstrate below by an exact numerical diagonalization of a 3D latticemodel that the Majorana flat bands remain robust beyond the minimal low energysurface theory. Their stability however can be deduced from the surface theory aloneand relies on the time reversal symmetry T and the bulk-boundary correspondencepresent in the Weyl semimetal. The most general surface Hamiltonian consistentwith these requirements is of the form h(k) = (vky · s)kx − µky for |ky| < K whereboth the velocity vector and the chemical potential are even functions of ky. Timereversal symmetry further permits a term (uky · s)ky but this is not consistent withh(k) describing the surface state of a Weyl semimetal 1. For each ky one can nowperform an SU(2) rotation in spin space to bring h(k) to the form indicated in Eq.(3.2). Because the pairing term in the BdG Hamiltonian (3.3) is not affected bythis transformation the calculation demonstrating the zero modes proceeds as before,with the result (3.5) modified in two minor ways: v and µ may now depend on ky andthe spinor structure reflects the SU(2) rotation. The exact zero modes persist for allallowed values of ky in this more general case.It is easy to see that breaking T lifts the zero modes: for instance interpolatingbetween −∆0 and +∆0 through complex values of ∆(x) moves MZMs to finiteenergies. Flat Majorana bands are thus protected by T and by translation symmetrythat is necessary to establish the underlying momentum space Weyl structure.3.2.2 3D Tight-Binding ModelTo ascertain the validity and robustness of the above analytic results, we now studythe problem using a lattice model for the Weyl semimetal. We consider electronsin a simple cubic lattice with two orbitals per site and the following minimal Bloch1For u nonzero the surface state exhibits a massless Dirac dispersion which would imply that theFermi arc can shrink to a point when µ coincides with the Dirac point.20Hamiltonian,hlatt(k) = λ∑α=x,y,zsα sin kα − µ+ σysyMk. (3.6)Here Mk = (m + 2− cos kx − cos kz) and σα are Pauli matrices acting in the orbitalspace. The Hamiltonian (3.6) is inspired by Ref. [100] and adapted to respect T(generated here by isyK with K the complex conjugation). It has a simple phasediagram. For m > λ (taking λ positive) it describes a trivial insulator. At m = λ,two Dirac points appear at k = (0,±pi/2, 0) which then split into two pairs of Weylnodes positioned along the ky axis. These persist as long as |m| < λ. In this phase,surfaces parallel to the y crystal axis exhibit two Fermi arcs terminating at the surfaceprojections of the bulk Weyl nodes illustrated in Fig. 3.2.−π 0 π−101kyE−π 0 π−101kyEOut[9]=kxkxkykya)−π 0 π−101kyE−π 0 π−101kyE−π 0 π−101kyEb) c) d)e) f) g) h)ky ky kyky ky kxEkEkEkEkEkEk−π 0 π−101kxEnormal SC SC+junctionSC SC+junction SC+junctionFigure 3.2: Tight binding simulations of the SC/Weyl proximity effect. Panels (a)and (e) show the surface spectral function A(k, ω) for the tight binding Hamiltonianhlatt + δhlatt and ω = 0.15. In all panels we use m = 0.5, λ = 1, Lx = 50 and Lz = 40;while (µ, ) = (0, 0) and (0.1, 1.0) for top and bottom row respectively. Panel (b)shows the normal state spectrum with flat bands representing the surface states. Theeffect of the SC proximity effect with ∆0 = 0.5 is indicated in panel (c) while panel(d) shows the effect of two parallel equidistant pi junctions (protected zero modesindicated in red). The bottom row displays our results for rotated arcs; panel (f) theuniform SC surface state, panels (g) and (h) two parallel pi junctions along y and xaxes, respectively. The small gap at Weyl points in panels (c) and (f) is a finite sizeeffect  the gap closes as Lz →∞.21We have performed extensive numerical computations based on hlatt(k) definedabove focusing on the slab geometry with surfaces parallel to the x-y plane andthickness of Lz sites. The proximity effect is studied using the BdG Hamiltonian(3.1) with h0 replaced by hlatt and ∆ taken to be non-zero in the surface layers only2. Top row in Fig. 3.2 summarizes our results. The normal state shows gapless Fermiarcs (Fig. 3.2a,b) while uniform SC order is seen to gap out the Fermi arc states exceptin the vicinity of the Weyl points where they merge into the gapless bulk continuum(Fig. 3.2c). Presence of the pi junctions (which we define parallel to the y crystal axis)generates perfectly flat bands at zero energy between the projected Weyl nodes, Fig.3.2d. We note that due to the periodic boundary conditions adopted in both x and ydirections, our numerics by necessity implement two parallel pi junctions which resultsin twice the number of flat bands compared to a single junction. We also find that,remarkably, the bands remain completely flat even when the system size Lx is smallalong the direction perpendicular to the junctions (in which case one would naivelyexpect a large overlap between the bound state wavefunctions resulting in significantenergy splitting). This property can be understood by noting that wavefunctions (3.5)remain exact zero modes of the Hamiltonian (3.1) for periodic boundary conditionsalong x as long as ∆(x) averages to zero over all x and µ = (2piv/Lx)p with p integer.These conditions are satisfied for two equally spaced junctions and µ = 0 used in Fig.3.2. We checked that violating these conditions indeed leads to zero mode splittingthat depends exponentially on the junction distance. Importantly, the fact that thisdetailed property of the simple model (3.2) is borne out in a more realistic latticemodel gives us confidence that the low energy theory provides a correct descriptionof the physical surface state of a Weyl semimetal.The lattice Hamiltonian (3.6) has high symmetry with Weyl nodes confined to lieon the ky axis. It is important to verify that the phenomena discussed above are not2Other order parameter configurations (e.g. ∆ decaying exponentially into the bulk) were alsoconsidered with substantially similar results.22dependent on such a fine tuned lattice symmetry. To this end we perturb hlatt byadding to itδhlatt(k) =  σysx(1− cos ky − cos kz), (3.7)which respects T but breaks the C4 rotation symmetry around the y axis, thusallowing Weyl nodes to detach from ky. For  6= 0 the arcs rotate and curve asillustrated in Fig. 3.2e. Majorana flat bands however remain robustly present in thislow symmetry case (Fig. 3.2g). They now also appear for a junction parallel to the xcrystal direction (Fig. 3.2h) because the Weyl points project onto distinct kx momentain the boundary BZ.3.2.3 Stability Against Non-Magnetic DisorderWe now address the stability of Majorana bands against nonmagnetic disorder thatwill inevitably be present in any real material. Scaling arguments [23, 101103] andnumerical simulations [104106] show that disorder is a strongly irrelevant perturba-tion in a Weyl semimetal: electron density of states (DOS) at low energies D(ω) ∼ ω2remains unchanged up to a critical disorder strength Uc at which point a transitionoccurs into a diffusive regime with finite DOS at ω =0 . The Fermi arcs likewiseremain stable in the weak disorder regime. These theoretical results are confirmedby experimental studies which show clear evidence for Weyl points and Fermi arcsin real materials using momentum resolved probes such as ARPES [15, 16, 9295]and FT-STS [107, 108], in agreement with the predictions of momentum space bandtheory. In T -preserving Weyl semimetals one furthermore expects the surface su-perconducting order to be stable against non-magnetic impurities due to Anderson'stheorem [109]. Finally, stability of Majorana flat bands can be argued as follows. Inthe clean system our calculations show MZMs localized in the vicinity of the junctionat each momentum k between the projected Weyl points. Except in the vicinity of23the latter, these are separated by a gap ∼ ∆ from the excited states in the system.Turning on weak random potential of strength U will cause mixing between MZMsat different momenta k as well as with the bulk modes. We expect the former to bea more important effect (except for MZMs in the close vicinity of the Weyl points)because of the low bulk DOS and the assumption of predominantly small momentumscattering. Importantly, the disorder averaged spectral function of the system mustevolve continuously with the disorder strength U . For weak disorder, therefore, theMajorana band cannot abruptly disappear; instead disorder will produce a lifetimebroadening, giving the δ-function peak present in the clean spectral function a finitewidth proportional to U . This behavior is indeed observed in numerical simulationsof disorder in Majorana flat bands at the edges of 2D topological superconductors[110]. We expect the broadened Majorana band to remain observable until its widthbecomes comparable to the gap ∆ implying a significant range of stability in a weaklydisordered sample.3.2.4 Summary and OutlookSC proximity effect in the surface of a Weyl semimetal produces an interestinginterfacial superconductor with unique properties. In T -preserving models (relevantto the recently identified Weyl semimetals [15, 16, 9295]) the observable signatureconsists of protected flat Majorana bands that occur at zero energy in a linearJosephson pi-junction defined on the surface. The flat bands span projections ofthe bulk Weyl points with opposite chirality onto the one dimensional edge BZ alongthe junction. We have demonstrated the existence and robustness of such flat bandsin minimal models with n = 4 Weyl nodes but expect the signature to survive inmaterials with larger n. In this case each Kramers pair of Fermi arcs will producea Majorana band at zero energy. In the presence of weak disorder there may besome additional broadening due to interband scattering, but given that Fermi arcs24are quite easy to resolve experimentally [15, 16, 9295] even when n = 24, we expectno significant difficulties to arise when n > 4.Majorana flat bands provide a unique opportunity to study interaction effects inMajorana systems because even weak interactions can have profound consequencesin a flat-band setting [111, 112]. They have been theoretically predicted to occur invarious 2D and 3D topological and nodal superconductors [5559, 61, 66]. With theexception of high-Tc cuprates, where dispersionless edge states are known to exist (andwere reinterpreted recently as Majorana flat bands [111]), these have not been ob-served experimentally because of the general paucity of topological superconductors.Majorana flat bands discussed here only require ingredients that are currently knownto exist  a T -preserving Weyl semimetal interfaced with an ordinary superconductor and are thus promising candidates for experimental detection.3.3 Experimental Detection by Josephson CurrentMeasurementThe Majorana flat bands discussed above should be directly observable in tunnelingspectroscopy of the junction region by a technique developed in the context of othermaterials [113, 114]. At phase difference ϕ = pi, flat bands will manifest througha distinctive zero-bias peak in the tunneling conductance (with the spectral weightproportional to the length of the flat portion of the band). The peak will splitsymmetrically about zero energy as ϕ is tuned away from pi and will merge into thecontinuum when ϕ approaches zero. We note that bulk DOS in a Weyl semimetalD(ω) is vanishingly small near the ω = 0 neutrality point, even in the presence ofweak disorder. The prominent zero-bias peak should thus be well visible in a tunnelingexperiment.25In this section, we consider a more conventional probe for the detection of theMajorana flat bands. We predict that, as the local parity of the junction switchesdue to the zero-mode degeneracy, the current-phase relation I(ϕ) of the junctionwould exhibit a step discontinuity at ϕ = pi. We substantiate this prediction bynumerically simulating a Josephson junction mediated by the surface states of Cd3As2,a T -invariant Dirac semimetal. Our analysis further shows that the distinctive currentjump can withstand the effect of finite temperatures and non-magnetic disorder.3.3.1 Heuristic ArgumentA well-known set-up for hosting MZMs is a Josephson junction mediated by theedge of a 2D topological insulator (TI) [49, 50]. When the phase difference acrossthe junction is pi, the lowest Andreev bound state (ABS) at the junction crosseszero energy and can be represented by two unpaired MZMs. For arbitrary phasedifference the energy of the ABS is proportional to cos(ϕ/2). Since the ABS canhave positive or negative energy depending on ϕ, naively thinking, the ABS wouldalternate between being occupied and unoccupied as ϕ varies in order to minimizethe energy. Nonetheless, because the ABS is nondegenerate, whether it is occupied ornot is determined by the fermion number parity (an odd parity means it is occupiedand vice versa), as long as kBT  ∆0 where ∆0 is the superconducting gap. Themany-body ground state of the system is thus 4pi-periodic because it has contributionsfrom this special 4pi-periodic ABS, on top of the usual 2pi-periodic ABSs at higherenergies and ϕ-independent quasiparticles above the gap ∆0. The 4pi-periodicity iseven more obvious in the limit of short junction, where only the lowest ABS remainswith energy given byE(ϕ) = ∆0cos(ϕ/2) (3.8)26where ∆0 is the superconducting gap. As a consequence, the Josephson currentexhibits a characteristic 4pi-periodicity, in contrast to the 2pi-periodicity for conven-tional Josephson junctions. This so-called fractional Josephson effect would be asubstantial evidence for MZMs [50, 6771]. So far the experiments have shown firstsigns of possible 4pi-periodicity [115119].In Sec. 3.2, we have demonstrated that a Josephson junction with pi phasedifference on the surface of a T -invariant Weyl semimetal would localize many pairs ofdecoupled MZMs labeled by different momenta k. In fact, each pair can be attributedto a 2D TI embedded in the Weyl semimetal band structure [10]. Naturally it seemsthat the fractional Josephson effect should also carry over, but in this case it doesnot violate the fermion parity to change the occupancies of ABSs. Indeed, a Cooperpair can split and occupy ABSs at different k's. The 4pi-periodicity is no longerprotected by parity; instead we have a distinct Josephson current jump at odd-piphase differences where many ABSs simultaneously change their occupancies in orderto minimize the ground state energy.3.3.2 Experimental Set-Up with Dirac Semimetal Cd3As2The above heuristic argument applies to any T -invariant Weyl or Dirac semimetals3. Among all candidates, the Dirac semimetal Cd3As2 has the advantage of beingexperimentally accessible and having the minimal number of four Weyl points [120,121]. For concreteness and experimental implication, we focus our analysis on thismaterial.Consider a linear Josephson junction obtained by proximitizing the top surface ofa Cd3As2 slab with an open superconducting ring, as shown in Fig. 3.3. To study thelowest ABS, we work in the short-junction limit where the junction length d ξ, theproximity-induced coherence length. An applied flux Φ through the ring creates phase3For our purposes, a Dirac semimetal can be considered as a special case of Weyl semimetal withcoinciding Weyl nodes due to point group symmetry.27difference ϕ = 2pi(Φ/Φ0) across the junction, where Φ0 = h/2e is the superconductingflux quantum. The phase difference drives a supercurrent around the ring, which canbe detected by a SQUID sensor as in [122, 123], or by including the junction intoa SQUID with a stronger junction in parallel and measuring the modulation of thetotal supercurrent, as done for example in [124].WFigure 3.3: By placing a superconducting ring with a narrow opening on top of aCd3As2 slab, one obtains a Josephson junction mediated by the surface states ofCd3As2. We label the length of the junction by d and the width by W . An externalflux going through the ring generates a phase difference ϕ = 2pi(Φ/Φ0) across thejunction, which leads to supercurrent measurable by a capacitively-coupled SQUIDsensor. To study the lowest ABS, we assume d ξ, the proximity-induced coherencelength.We predict that the current-phase relation (CPR) exhibits a discontinuous currentjump at ϕ = pi. The jump size is proportional to the number of MZMs, whichis maximized when the junction width is oriented along the direction in which theDirac nodes are separated, and decreases as cos(θ) when the junction is rotated byan angle θ. Low non-zero temperature and weak non-magnetic disorder smoothenthe discontinuity to some extent, but the CPR profile remains highly skewed and thejump size is unaffected. We estimate that a current jump on the order of 1µA canbe observed in typical experiments. Our prediction is substantiated by the followinganalysis.283.3.3 Majorana Flat Bands at Cd3As2-SC InterfaceThe low-energy effective Hamiltonian of tetragonal Cd3As2 has been extracted fromsymmetry considerations [120, 121]. In the basis {|PJ= 32, Jz =32〉, |S 12, 12〉, |S 12,−12〉,|P 32,−32〉}, where S and P refer to spin-orbit-coupled Cd-5s and As-4p states, the4× 4 Hamiltonian takes the formH0(k) = 0(k) +M(k) Ak− 0 0Ak+ −M(k) 0 00 0 −M(k) −Ak−0 0 −Ak+ M(k)(3.9)up to second order in k near the Γ point. 0(k) = C0 + C1k2z + C2(k2x + k2y), M(k) =M0 + M1k2z + M2(k2x + k2y), k± = kx ± iky. The parameters in the Hamiltonianare best-fit values to the ab initio calculation [121], and are listed in Table A.1 ofAppendix A.Each diagonal block describes a Weyl semimetal. Diagonalizing this Hamiltoniangives the energy spectrumE(k) = 0(k)±√M(k)2 + A2(k2x + k2y) (3.10)The Weyl nodes are located at (0, 0,±Q) where Q = √−M0/M1 such that thesquare-root term vanishes. Each location has two Weyl nodes of opposite chiralities,one from each diagonal block in Eq. (3.9), and the point group symmetry preventsthem from mixing. In our analysis, the chemical potential µ is always set to theenergy of Weyl nodes, E0 = C0 −C1M0/M1 to minimize the influence of bulk states.To model a lattice, we turn H0(k) into a tight-binding Hamiltonian Htb(k) via thesubstitutions k2i ∼ 2a2 (1− cos(kia)) and ki ∼ 1asin(kia), where a is the lattice constantfor the low-energy effective model. Q now satisfies cos(aQ) = 1 + (a2/2)(M0/M1).29aaaFigure 3.4: Spectrum of the Lxa × Lya × Lza = 40a × 52a × 60a (a = 20Å) slab ofCd3As2 with periodic boundary condition in the y- and z-directions. The materialparameters are listed in Table A.1. Energy states are color-coded according to theexpectation value of distance from the surface. Panel (a) shows the normal-stateenergy spectrum with the Dirac cones separated along kz and the surface Fermiarcs spanned between them. In Panel (b) a uniform pairing potential of magnitude∆0 = 10meV is introduced on the surfaces; the Fermi arcs are gapped out. In Panel(c), two junctions of pi phase difference are implemented on the top surface and thebottom surface remains uniformly superconducting; the resulting zero-energy surfaceflat bands linking the Dirac nodes are four-fold degenerate and decoupled. In Panel(d), the phase difference deviates from pi to 1.1pi; the flat bands get split from zeroenergy and obtain small dispersion, while the rest of the bands stay the same. InPanels (b)-(d), the small gaps near the Dirac nodes are due to the finite-size effect.We let a = 20Å even though ax = ay = 3Å and az = 5Å are the best-fit values to theab initio calculation [120]. The larger lattice constants make the relevant low-energyband structure more numerical resolvable. This approach is not an issue because ournumerics mainly serve illustrative purposes; later on the quantitative predictions willbe made with realistic parameters. We consider a slab of Cd3As2 with Lx × Ly × Lzsites. It is periodic in the y- and z-directions and finite in the x-direction. WeFourier transform Htb(k) along the x-direction in order to define individual layers.Then numerical diagonalization gives the normal-state energy spectrum, showing twoDirac cones in the bulk and two Fermi arcs on each surface. Fig. 3.4(a) projects thespectrum onto kz-axis, so the Fermi arcs overlap and appear as curved bands (eachcorresponding to a different ky) connecting the Dirac cones.Next we verify that the surface states can be gapped out by s-wave pairing of time-reversed states. We introduce a pairing potential to both top and bottom surfaces.30The matrix of Bogoliubov-de Gennes (BdG) Hamiltonian readsHBdG= Htb − µ ∆∆† −THtbT−1 + µ (3.11)where every entry is a 4Lx × 4Lx matrix whose basis are tensor products of spin-coupled orbitals and x-layers. Htbis as discussed above, and its time-reversal conju-gate THtbT−1 = Htbbecause it is time-reversal invariant. The pairing potential ∆ isa diagonal matrix whose nonzero elements are ∆0 when the x-layer index is 1 or Lx.Diagonalizing HBdGgives a particle-hole symmetric band structure with gapped-outsurface Fermi arcs and intact bulk Dirac cones, as shown by Fig. 3.4(b).Lastly we Fourier transform Htbagain along y; now each block in HBdGis a matrixwhose basis are tensor products of spin-coupled orbitals, x-layers, and y-sites. Thenwe let the nontrivial diagonal elements of ∆ be ∆0 when the x-layer index is Lx and∆0eiϕfor Ly/4 < y/a ≤ 3Ly/4∆0 for y/a ≤ Ly/4 or y/a > 3Ly/4(3.12)when the x-layer index is 1. This defines two evenly spaced short junctions eachof phase difference ϕ on the top surface. Two evenly spaced junctions are neededto satisfy the periodic boundary condition along y. We keep the bottom surfaceuniformly superconducting so that the bottom surface states do not obstruct thezero-energy flat bands in the figures. Note that the junctions are periodic widthwise,but a realistic junction would have a finite width. In principle, sides of the samplein Fig. 3.3 could also carry supercurrent, which is not accounted for in our model.However, in the geometry where the junction width is much larger than the thicknessof Cd3As2 slab, the contribution of the side modes is negligible. This is additionallysubstantiated by Ref. [123], a S/TI/S junction experiment set up like Fig. 3.3: theauthors did not consider this geometric effect, yet found good agreement between their31theoretical model and experimental data. The energy spectrum of HBdG(ϕ = pi) inFig. 3.4(c) clearly shows flat bands of zero-energy surface states, which are Majoranaor self-conjugate for reasons explained in [46, 125]. There are in total four Majoranaflat bands. Each junction locally hosts two such bands, which hybridize and gain anon-zero energy when ϕ moves away from pi, as shown by Fig. 3.4(d). As long as thetime-reversal symmetry T is respected, ϕ must be either 0 or pi, so we can say that theMajorana flat bands are protected by T . Moreover, translational symmetry prohibitsMZMs in the same band from mixing because kz is a good quantum number. Lastly,the flat bands at different junctions are decoupled due to the artificial symmetry ofevenly spaced junctions.The number of MZM pairs, N , in each junction is determined by the junctionwidth, W = aLz, and the distance between the Dirac nodes, 2Q:N = WQ/pi (3.13)If the junction is rotated by an angle θ,N = WQcos(θ)/pi (3.14)because the distance between the Dirac nodes, as projected onto the 1D Brillouinzone parallel to the junction width, becomes 2Q cos θ. In our numerical model, aQ ∼0.21pi and Lz = 60, so N ∼ 13 at each junction. In a realistic model of Cd3As2,Q =√−M0/M1 = 0.033Å−1. Assuming a typical junction width of W = 1µm, onefinds that N ∼ 100. Hence a significant MZM population can be easily obtained inexperiments, especially with a wide junction.323.3.4 Current-Phase Relation at Finite TemperaturesOne can detect the Majorana flat bands by measuring the current-phase relation(CPR). The MZMs cause a current jump at pi phase difference. The jump size isproportional to N . To demonstrate this in Cd3As2, we compute the Josephson currentgiven byJ(ϕ) =2e~dε(ϕ)dϕ(3.15)where ε(ϕ) is the many-body ground state energy of the system [126]. We diagonalizeHBdG(ϕ) for every fixed ϕ. The positive energies, labelled by Ei(ϕ), are the excitationenergies of Bogoliubov quasiparticles. Due to particle-hole symmetry, the negativeenergies of the filled states below EF = 0 are simply −Ei(ϕ), soε(ϕ) = −12∑iEi(ϕ) (3.16)where the factor of12compensates the particle-hole doubling in the BdG construction.As shown in Fig. 3.5, the resulting J(ϕ) displays a 2pi-periodic CPR with a currentjump of magnitude ∼ 60meVe/~ or 15µA.At non-zero temperature, quasiparticles below EF = 0 can be thermally excitedto occupy positive energy states. With the free energy F (ϕ) in place of ε(ϕ), theJosephson current becomes [127, 128]F = −kBT lnZ = −kBT∑iln(2 cosh(Ei(ϕ)2kBT))(3.17)J(ϕ) =2e~dF (ϕ)dϕ= − e~∑idEi(ϕ)dϕtanh(Ei(ϕ)2kBT)(3.18)33Figure 3.5: The Josephson current (per junction) computed from Eq. (3.18) isshown at various ϕ. At kBT = 0 and 0.1meV, there is a clear current jump ofmagnitude 60meVe/~ or 15µA. At kBT = 1meV, the CPR retains little skewness.For comparison, the solid blue curve shows the sinusoidal CPR of a conventionaljunction with one ABS. The magnitude is small (bounded by∆02e~) compared toour numerical results, mainly because our junction hosts a large number of ABSs.Note that ∆0 = 10meV in the numerics, which is unrealistically large so that thesuperconducting gap is larger than the finite-size gaps that are always present infinite systems. While kBT = 0.1 and 1meV appear to be fairly high temperatures,it is the ratio kBT/∆0 (1% and 10% respectively) that dictates the thermalization ofquasiparticles. In a more realistic system with ∆0 = 0.1meV, the predicted blue andgreen curves correspond to T ∼10mK amd 100mK respectively.which agrees with Eq. (3.15) when T → 0 4. Fig. 3.5 shows that the currentjump is slightly rounded at kBT/∆0 = 0.01 but significantly loses skewness whenkBT/∆0 = 0.1. In experiments, one should aim at lowering kBT/∆0 to 1%. For atypical induced gap of 0.1meV, ideally the temperature should be 10mK or lower.Note that electron-phonon interaction would also knock the system off the groundstate, which further smoothens the current discontinuity. The extent of this effect isleft for future studies.How do we understand this current jump? A slice of spin-up sector of the Cd3As2Hamiltonian at any fixed kz ∈ (−Q,Q) combined with a slice of spin-down sector at−kz can be regarded as a 2D TI. Although a short junction on the edge of 2D TI hosts4Our expression for Josephson current differs from that in Refs. 127, 128 by a factor of 2, whichwas there to account for spin degeneracy. In our case, any degeneracy is already included in thesummation because we assign a unique label to every eigenstate.34a 4pi-periodic ABS with energy given by Eq. (3.8), the 2D TI embedded in Cd3As2does not locally conserve the fermion parity, so its ground state energy evolves asε˜(ϕ) = −12∆0∣∣∣cos(ϕ2)∣∣∣ (3.19)following the lower of the two particle-hole-conjugate ABS bands in order to minimizethe energy. Above-gap states are not included in ε˜(ϕ) because they are phase-independent and do not contribute to the supercurrent. ε˜(ϕ) has a kink at pi, resultingin a characteristic supercurrent jump of magnitudeδJ =e~∆0 (3.20)The current jump in Fig. 3.5 can thus be understood as the cumulative effect ofmany Josephson junctions mediated by the edge states of 2D TI. In Fig. 3.6, weconfirm that a slice of our model at kz = 0 indeed shows the same phase dependenceas described.Figure 3.6: The first few quasiparticle excitation energies Ei at kz = 0 are shown atvarious ϕ. The subgap ABS (in red) has a kink at pi while the other states show littlephase dependence. Same behaviour holds for any kz ∈ (−Q,Q); outside of this rangethere is no ABS.35With N ABSs, the total current jump isδJtotal= NδJ (3.21)For our numerics, N ∼ 13 and ∆0 = 10meV, so δJtotal ∼ 31µA, on the same orderof magnitude as what Fig. 3.5 suggests. It is an overestimation because the actualsuperconducting gap as seen in Fig. 3.4(b) is smaller than 10meV and shrinking as kzapproaches the nodes. This happens because the opening of surface superconductinggap is interfered by the Dirac cones in the bulk, which is inevitable when we usean unrealistically large ∆0 (in order to overcome the finite-size effect). In a realisticsetting where ∆0 is much smaller than the normal-state bulk gap, we expect thesuperconducting gap to be uniform except in the vicinity of the nodes. In that caseEq. (3.21) provides an accurate estimate of the jump size. With realistic parameters,N ∼ 100 , so δJtotal∼ 2.4µA assuming a typical induced gap of∆0 = 0.1meV. Currentjump of this magnitude is readily observable in experiments.Due to Eq. (3.14), the jump size varies as cos(θ) where θ is the angle between thejunction width and the direction in which the Dirac nodes are separated (z-axis). Toobserve this, one can prepare multiple samples with different junction orientationsand compare their CPR measurements.3.3.5 Random Matrix Theory Analysis of Non-Magnetic Dis-orderDisorder breaks the translational symmetry, allowing MZMs at different kz's to couple.We show that non-magnetic disorder broadens the Majorana flat bands to cover arange of near-zero energies. Consequently the discontinuity in Josephson current isrounded off but remains a robust signature for weak disorder.36Having in mind an experiment in which the chemical potential is tuned to theneutrality point, we consider only the subgap states, consisting of ABSs at thejunction and bulk states on the Dirac cones. There are many ABSs; whereas thebulk density of states is vanishingly small at low energy: D(ω) ∼ ω2. Moreover, thebulk wavefunctions spread out over the entire bulk, so their probability densities nearthe surface are small. Based on the above arguments, we assume that any scatteringbetween ABSs and bulk states is negligible in comparison to scattering among theABSs, and neglect the presence of bulk states altogether.At each kz ∈ (−Q,Q), there is a pair of particle-hole symmetric Andreev levelsdispersing according to E = ±∆0 cosϕ/2. Non-magnetic disorder randomly couplesN pairs of ABSs such that the effective Hamiltonian reads:H =I∆0 cosϕ/2 MM † −I∆0 cosϕ/2 . (3.22)HereM is a N×N random phase-independent matrix coupling the ABSs at differentkz's, and I is the N ×N identity matrix. Note that we have chosen the shape of Hsuch that the disorder matrix resides in the off-diagonal blocks. One can show thatthe Hamiltonian matrix can be chosen this way by diagonalizing whichever part ofthe disorder matrix is in the diagonal part of the Hamiltonian and noting that thedisorder potential is time-reversal invariant and cannot shift the phase difference atwhich the states cross zero from pi. Let us also use time-reversal and particle-holesymmetries to further constrainM , one of the choices beingM = M∗ andM = −MT .Thus M is real antisymmetric matrix and its eigenvalues are purely imaginary. Thestatistics of energy levels of such a random matrix are well known [129],P (mi) =1pic2Re√c2 −m2i , (3.23)37where c is a constant describing the disorder strength. We can now diagonalize Mand M † to obtain the eigenvalues of the whole matrix Hei =√∆20 cos2(ϕ/2) +m2i . (3.24)By change of variables in the probability distribution (3.23), we obtain disorder-averaged density of states as a function of phase difference and disorder strength:ν(E) ∝ |E| Re√c2E2 −∆20 cos2(ϕ/2)− 1, (3.25)-0.25 0 0.251.1πφ1.01ππν(E)	[4N/Δ 0π]E/Δ00.1c/Δ00.050.01060402080100-0.25 0 0.25 E/Δ0(a) (b)Figure 3.7: Panel (a) plots ν(E) at pi and shows that non-magnetic disorder broadensthe Majorana flat bands by c/∆0. Panel (b) shows ν(E) at c/∆0 = 0.1 and differentϕ's. The broadening diminishes as ϕ deviates from pi.As shown by Fig. 3.7, ν(E) describes the broadening of Andreev levels, whichdiminishes as ϕ deviates from pi, so we expect the CPR to be perturbed mostly nearthe jump. To confirm this, we compute the Josephson current using the free energyintegrated with ν(E).Fig. 3.8(a) shows that the discontinuity is rounded off but the overall jump sizeis not affected. The rounding is akin to that due to nonzero temperature as seen inFig. 3.5, but to a lesser degree because c/∆0 = 0.1 smoothens the discontinuity whilekBT/∆0 = 0.1 almost completely destroys the skewness. We also measure the degreeof rounding by the slope of CPR at pi. It decreases with increasing c/∆0, as shown in380-0.50.5π3π/4 5π/4 c/Δ00.25 0.5 0.75 100510|dJ/dφ|π	[NΔ 0e/ℏ]φJ[NΔ 0e/ℏ](a) (b)0.2c/Δ00.10Figure 3.8: Panel (a) shows the disordered CPR near pi at kBT/∆0 = 0.01 andvarious c/∆0. The discontinuity is rounded off but the jump size is unaffected. Panel(b) shows the CPR slope at pi as c/∆0 increases (again at kBT/∆0 = 0.01). Forcomparison, the slope at pi of a conventional sinusoidal CPR is 1.Fig. 3.8(b), but remains significantly larger than 1, the slope at pi of a conventionalsinusoidal CPR.The above RMT analysis does not consider non-averaged fluctuations, which canin principle obscure the jump. This is however not the case here. Supercurrent inthe form of Eq. (3.18) is essentially a sum of N random variables when the energyeigenvalues are randomized as in Eq. (3.24). By the Central Limit Theorem, the sumgrows as N , but the standard deviation of the sum grows as√N .3.3.6 Application to Other T -Invariant Weyl SemimetalsWe have shown that flat bands of MZMs occur in a linear Josephson junction mediatedby the surface states of Cd3As2. Numerical simulation of CPR shows a significantsupercurrent jump at pi, the size of which is determined by Eqs. (3.14), (3.20) and(3.21). Low temperature (. 10mK) and weak non-magnetic disorder mildly roundoff the jump.This result applies to every T -invariant Weyl/Dirac semimetal, which in generalhas an even number of Weyl nodes. With a junction of pi phase difference on thesurface, each pair of Weyl nodes that used to terminate the same Fermi arc becomethe end points of a Majorana flat band, living in the 1D Brillouin zone parallel tothe junction width. As before the CPR would have a jump, but the dependence of39jump size on junction orientation varies. For example in TaAs, a T -invariant Weylsemimetal with 24 nodes [15, 16], the jump size is the largest when the junction widthis parallel to the [110]-direction onto which every pair of Weyl nodes projects to twodistinct points.40Chapter 4Chiral Anomalies in Strained WeylSemimetals4.1 OverviewThe Dirac equation for spin-12fermions can be decoupled into left- and right-handedWeyl equations if the fermions are massless. In this case the current densities ofleft- and right-handed fermions are separately conserved: ∂tρL/R + ∇ · jL/R = 0 orequivalently ∂tρ5 +∇ · j5 = 0 where (ρ5, j5) = (ρR, jR)− (ρL, jL) is the chiral currentdensity. Further incorporating electric and magnetic fields by coupling the equationsto a gauge potential does not affect the chiral symmetry. However, once the theory isquantized as in massless quantum electrodynamics (QED), the conservation of chiralcurrent is violated in order for the theory to remain gauge invariant. There is anextra anomalous contribution that is nonzero in the presence of parallel electric andmagnetic fields:∂tρ5 +∇ · j5 = e22pi2~2cE ·BThis quantum phenomenon is called the Adler-Bell-Jackiw anomaly or the chiralanomaly [130, 131]. It can be applied to any gauge theories with massless fermions41and has significant implications in the Standard Model where certain light fermionscan be treated as massless. Remarkably, it explains the rapid neutral pion decaypi0 → 2γ and predicts successive generations of quarks and leptons by enforcinganomaly cancellation in the electroweak theory [132].In their seminal paper, Nielsen and Ninomiya demonstrated that the chiral anomalyoccurs in semiconductors where the conduction and valence bands meet at point-like degeneracies [29]. Today we call these materials Weyl semimetals because thelow-energy excitations near the gapless nodes are described by the Weyl equations.They showed that in the presence of parallel electric and magnetic fields, electronssemiclassically evolve along the lowest Landau levels from one Weyl node to anotherwith opposite chirality, resulting in excess of chiral charge. They also argued that themagneto-conduction is strong because any backscattering has to be internodal. Thishas been observed in experiments as a negative magnetoresistance [1214].Recent theoretical work shows that small mechanical strain acts like pseudo-electromagnetic fields e, b in Weyl semimetals [133]. They differ from the ordinaryfields E, B by coupling with an extra minus sign to Weyl cones of different chirality.It is straightforward to derive the chiral anomaly equation with consideration of thechiral gauge fields:∂tρ5 +∇ · j5 = e22pi2~2c(E ·B + e · b) (4.1)∂tρ+∇ · j = e22pi2~2c(E · b+ e ·B) (4.2)where (ρ, j) = (ρR, jR) + (ρL, jL) is the electric current density [134]. It seems oddthat the conservation of electric charge can be violated by parallel ordinary and chiralgauge fields. Indeed this is unphysical in the context of high-energy physics, but in asolid state system the nonconservation only describes what happens inside the bulknear the energy level of Weyl nodes  the electric charge of the entire system, includingboth surface and bulk, is always conserved.42In this work, we investigate the strain-induced chiral anomaly in Weyl semimetals,especially scenarios where the second chiral anomaly equation is nontrivial. By closelyexamining the band structure of a two-node tight-binding model in finite geometries,we shed new light to the anomalous nonconservation of electric charge: there is adynamical charge re-distribution between bulk and surface that produces the chiralanomaly effect near the Weyl nodes while conserving the total charge of the system.First, we show that twisting a nanowire of Weyl semimetal induces a pseudo-magneticfield b, whose chiral nature is seen in the parallel zeroth Landau levels of the bulk,as shown in Fig. 4.1b (normally the zeroth Landau levels at opposite Weyl nodes arecounter-propagating, as in Fig. 4.1a). Numerical simulation of the band structureunveils surface modes travelling in the opposite direction. Hence when a parallelE is turned on, electrons evolve to occupy more bulk modes, in agreement withthe second chiral anomaly equation, and less surface modes. Next, we demonstratethat the repetitive motion of stretching and compressing a film of Weyl semimetalinduces an oscillating pseudo-electric field e. In the presence of a parallel magneticfield B, the oscillating e moves the chemical potential up and down in the zerothLandau levels according to the second chiral anomaly equation. Once again the bandstructure reveals the presence of surface states, which would equilibrate any changein the bulk chemical potential. Therefore the oscillating bulk chemical potential leadsto periodic charge transfers between bulk and surface.43B ΩkεEkεEa bFigure 4.1: With external magnetic field threaded through the Weyl semimetalnanowire, counter-propagating zeroth Landau levels are formed at the opposite Weylnodes, shown in Panel a. Twisting the nanowire induces a pseudo-magnetic fieldb, whose chiral nature is seen in the parallel zeroth Landau levels of the bulk, asillustrated in Panel b. When a parallel electric field is applied, the electron statesbegin to evolve semiclassically towards higher momenta.The interesting interplay between surface and bulk charges has several experimen-tally observable consequences. Since the surface and bulk modes in a twisted Weylsemimetal travel in the opposite directions, any backscattering in such a topologicalcoaxial cable necessarily involves transverse hydrodynamic charge flow. Thus thecurrent driven by longitudinal E has a large relaxation time and gives a conductivityproportional to the square of the torsion strength, which can be measured in transportexperiments. The pseudo-electric field e in practice can be generated by sound waves.In the presence of nontrivial e·B, the periodic bulk-surface charge transfers would leadto anomalous sound attenuation and electromagnetic field emission, both detectableby conventional probes. Lastly, we demonstrate that our results are applicable torealistic Weyl semimetals by conducting additional numerical analysis with ab-initiotight-binding models for Cd3As2 and Na3Bi.4.2 Gauge-Fields Induced by Mechanical StrainsMechanical strain that varies smoothly on the interatomic scale is known to affectthe low-energy Dirac fermions in graphene in a way that is similar to the externally44applied magnetic field. More precisely, strain acts in graphene as a chiral vectorpotential that couples to Dirac fermions oppositely in the two valleysK andK ′ [135].The pseudomagnetic field that arises from this effect in a curved graphene sheet canbe larger than 300T, and has been observed through the spectroscopic measurement ofthe Landau levels in the seminal experiment on graphene nanobubbles [136]. In termsof their low-energy physics Weyl and Dirac semimetals [23, 137, 138] can be thoughtof as three dimensional generalization of graphene. The question thus immediatelyarises whether strain in these materials gives rise to similar effects. Recent theoreticalwork [133] showed that this is indeed the case at least in a simple toy model of a Weylsemimetal with broken time reversal symmetry T . The authors predicted that theelectron-phonon coupling in such a system will lead to non-zero phonon Hall viscosity,an interesting but notoriously difficult quantity to measure. We consider here theeffect of strain in more realistic models relevant to Dirac semimetals Cd3As2 [3237]and Na3Bi [139141] and the related Weyl semimetals [30, 142145].One reason why strain can generate pseudomagnetic fields as large as 300T ingraphene [136] lies in its mechanical flexibility: substantial curvature can be achievedwithout breaking the graphene sheet. This suggests that to probe strain-inducedeffects in Dirac and Weyl semimetals one should focus on films or wires as thesewill be much more flexible than bulk crystals. In this work we thus concentrate onthese geometries and show that strain leads to phenomena that are both striking andexperimentally measurable. We note that high-quality nanowires of Dirac semimetalCd3As2 have been grown and shown to exhibit giant negative magnetoresistance dueto the chiral anomaly [39] as well as Aharonov-Bohm oscillations indicative of theprotected surface states [40]. These wires bend easily and show mechanical flexibilitythat is required to study strain related phenomena. We also discuss consequences oflattice distortions caused by sound waves (phonons). These can be used to study the45above phenomena in crystalline flakes and films which are readily available for nearlyall known Dirac and Weyl materials.4.3 Intuitive Picture of Chiral Anomalies in WeylSemimetalsOur results can be most easily understood by thinking about the simplest Weylsemimetal with a single pair of Weyl points [146] although many aspects translateto more complicated Weyl and Dirac semimetals. The low-energy effective theory isthen defined by the Hamiltonian H =∫d3rΨ†rh(r)Ψr where Ψ†r = (ψ†r,R, ψ†r,L) andh = vχzσ · (p− eA− χzea)− µ. (4.3)Here ψ†r,R/L represent two-component right and left handed Weyl fermion creationoperators, χz = ±1 labels the chirality of the two Weyl nodes, σ is a vector of Paulimatrices in the pseudospin space and p = −i~∇. A and a denote gauge potentialsof the ordinary EM and the chiral field, respectively.To develop some intuition for the chiral anomaly let us consider the Hamiltonian(4.3) in the presence of a static uniform (pseudo)magnetic field. We begin with theordinary magnetic field B = Bzˆ. The solution of the corresponding Schrödingerequation hΦ = Φ is well known and consists of the set of Dirac Landau levels withenergies [29]n(k) = ±~v√k2 + 2ne|B|~c, n = 1, 2, ..., (4.4)for each Weyl fermion. There is also one chiral n = 0 level per valley with 0(k) =χzsgn(B)~vk. If a parallel electric field E = Ezˆ is now applied to the system then theelectron momenta begin to evolve according to the semiclassical equation of motionk(t) = k(0) − eEt/~. Because of the existence of the two chiral branches in the46spectrum this leads to charge pumping between the two Weyl points, as illustratedin Fig. 4.1a, at a rate consistent with the chiral anomaly equation (4.1). The keypoint here is that in a real solid where the Hamiltonian is defined on the latticethe two chiral branches are connected away from the Weyl points and the chiralanomaly equation simply describes the semiclassical evolution of the electron statesthrough the Brillouin zone [29]. In the presence of relaxation processes a steady statenon-equilibrium distribution of electrons with nonzero chiral density ρ5 is obtainedwhich is responsible for the anomalous ∼ B2 contribution to the magnetoresistance.Now consider the effect of the chiral magnetic field b = bzˆ. The solution consistsof the same Dirac Landau levels Eq. (4.4) but the n = 0 levels now disperse in thesame direction for the two Weyl points, 0(k) = sgn(b)~vk, as illustrated In Fig.4.1b. Now if a parallel electric field E = Ezˆ is applied to the system we see that thecharge density seemingly begins to change. Since the total charge is conserved thisextra charge density must come from somewhere. We will demonstrate below that itcomes from the edge of the system. Indeed this is plausible if we note that the energyspectrum sketched in Fig. 4.1b does not represent a legitimate dispersion of a latticesystem which, due to the periodicity of the energy bands in the momentum space,must exhibit the same number of left and right moving modes. Since the Landaulevels are the correct eigenstates in the bulk we conclude that the missing left movingmodes must exist at the boundary. Our numerical simulations of a lattice modelbelow indeed confirm this conclusion. Thus, in the presence of b and E the chiralanomaly can be understood as pumping of charge between the bulk and the edge ofthe system. The effects of nonzero e·B and e·b terms are more subtle, as they involverelaxational dynamics, but can be understood from similar arguments. Indeed, thedifference between the effects lies in the directions of magnetic and electric fields asapplied to the two Weyl cones. These effects and their experimental consequencesconstitute the main result of this work.47Several interesting observation follow from the above discussion. First, we con-clude that electric transport in a twisted Weyl semimetal wire will be highly unusualbecause the right-moving modes occur in the bulk whereas the left-moving modesare localized near the boundary. (More precisely we may say that there is a netimbalance between the number of left and right moving modes in the bulk and atthe boundary.) Since the left and right moving modes are spatially segregated oneexpects backscattering to be suppressed in such wires giving rise to anomalouslylong mean free paths. In addition, transport will sensitively depend on the appliedtorsion, giving rise to the new chiral torsional effect (CTE) that we describe in detailbelow. Second, we will see that charge transfer between the bulk and the boundaryleads to interesting effects when time-dependent e field is generated e.g. by drivinga longitudinal sound wave through the crystal when B field is also present. Such asound wave will experience an anomalous attenuation that can be attributed to thechiral anomaly. It will also produce charge density oscillations in the crystal that canbe observed through electric field measurement outside the sample. Third, the chiralanomaly can be observed even in the complete absence of real EM fields when thecrystal is put simultaneously under torsion and time-periodic uniaxial strain. Thennonzero e · b term is generated and according to Eq. (4.1) the chiral charge fails tobe conserved. We argue that this has observable consequences for sound attenuationin the crystal.4.4 Analytical Calculations with12-Cd3As2 ModelWe now proceed to justify the above claims by detailed model calculations. Forsimplicity and concreteness, we once again adopt the low-energy effective model ofDirac semimetal Cd3As2 introduced in Chapter 2. The model captures the bandinversion of the atomic Cd-5s and As-4p levels near the Γ point. In the basis of the48relevant spin-orbit coupled states |P 32, 32〉, |S 12, 12〉, |S 12,−12〉 and |P 32,−32〉 it is definedby a 4× 4 matrix Hamiltonian [32]H(k) = 0(k) +M(k) Ak− 0 0Ak+ −M(k) 0 00 0 −M(k) −Ak−0 0 −Ak+ M(k). (4.5)Here 0(k) = C0 + C1k2z + C2(k2x + k2y), k± = kx ± iky, and M(k) = M0 + M1k2z +M2(k2x + k2y). Parameters Cj, A and Mj follow from the k · p expansion of the firstprinciples calculation [32] and are summarized in Appendix A. We note that H(k)(with different parameters) also describes Dirac semimetal Na3Bi [139].The low-energy spectrum of the model (4.5) consists of a pair of Dirac pointslocated atKη = (0, 0, ηQ), Q =√−M0/M1, (4.6)where η = ± is the valley index. The model respects time reversal symmetry T =iσyτxK, where K denotes complex conjugation and σ, τ are Pauli matrices in spinand orbital space, respectively. T maps the upper diagonal (spin up) block h(k) ofH(k) onto the lower diagonal (spin down) block −h(k) and vice versa.Since spin up and spin down blocks are effectively decoupled in the model Hamil-tonian (4.5), we can analyze them separately. It is easy to see that each diagonalblock taken in isolation can be regarded as describing a minimal T -breaking Weylsemimetal with one pair of Weyl nodes located at K±. In the following we will oftenfocus our discussion on the spin up block of Hamiltonian (4.5) and refer to it as12-Cd3As2 model. Once we have understood the physics of this12-Cd3As2 model,it will be straightforward to deduce the behavior of the actual Cd3As2 by simplyadding a time-reversal conjugate set of states to the results obtained for12-Cd3As2.49We emphasize that although12-Cd3As2 model taken on its own does not describe anyspecific real material, the results we report for this model are relevant to a broad classof Weyl semimetals with broken T such as the Burkov-Balents layered heterostructure[146] and more recently proposed magnetic Weyl materials [147, 148]. We will explainin detail how these results apply to T -preserving Weyl and Dirac semimetals.For many considerations and for numerical calculations, it will be useful to regu-larize the model defined by Eq. (4.5) on a lattice. Although real Cd3As2 crystal hasa complex structure with 40 atoms per unit cell, Ref. [32] showed that its low-energyphysics can be well described by an effective tight binding model with s and p orbitalson vertices of the tetragonal lattice and lattice constants ax, ay = 3.0Å and az = 5.0Å.Here we simplify the model one step further and assume a simple cubic lattice witha lattice constant a. We checked that this leads to only minor deviations from thetetragonal model of Ref. [32]. We construct the tight-binding model for Cd3As2, asfurther explained in Appendix A, such that in the vicinity of the Γ point it matchesthe k · p Hamiltonian (4.5) to the leading order in the expansion in small ak. Forquantitative estimates we use a = 4Å while in the numerics we use larger valuesof a as this will allow us to simulate systems of sufficient size with the availablecomputational resources. This does not affect the qualitative features of the physicswe wish to describe. The Cd3As2 Hamiltonian regularized on the lattice thus becomesH latt = k + hlatt 00 −hlatt , (4.7)where k is the lattice version of 0(k) given in Appendix A whilehlatt(k) = mkτz + Λ(τx sin akx + τy sin aky). (4.8)50Here mk = t0 + t1 cos akz + t2(cos akx + cos aky) and t0 = M0 + 2(M1 + 2M2)/a2,t1/2 = −2M1/2/a2, Λ = A/a.The Hamiltonian (4.8) exhibits a single pair of Weyl nodes at Kη = (0, 0, ηQ),with Q given by cos(aQ) = −(t0 + 2t2)/t1 which coincides with Eq. (4.6) in the limitaQ 1. In the vicinity of the nodes, we can expand hlatt(K±+q) in q to obtain theWeyl Hamiltonianhη(q) = ~vjητ jqj, (4.9)with the velocity vectorvη = ~−1a(Λ,Λ,−ηt1 sin aQ). (4.10)For Cd2As3 parameters and a physical lattice constant a = 4Å, this gives ~vη =(0.89, 0.89,−1.24η)eVÅ. From Eq. (4.10), we can read off the chiral charge of theWeyl node located at valley η,χη = sgn(vxηvyηvzη) = −η. (4.11)The effect of strain on the lattice Hamiltonian (4.8) is implemented using themethod developed in Refs. [133, 149]. The key observation is that certain tunnelingamplitudes that are prohibited by symmetry in the unstrained crystal become allowedwhen the strain is applied because of the displacement and rotation of the relevantorbitals in the neighboring atoms. For our purposes the most important modificationof the Hamiltonian (4.8) comes from the replacement of the hopping amplitude alongthe zˆ-direction [133, 149],t1τz → t1(1− u33)τ z + iΛ∑j 6=3u3jτj, (4.12)51where uij =12(∂iuj + ∂jui) is the symmetrized strain tensor and u = (u1, u2, u3)represents the displacement vector. The physics of Eq. (4.12) has been discussedat length in Ref. [149] and is easy to understand intuitively by inspecting the twoexamples of strain configurations given in Fig. 4.2. The first term in Eq. (4.12) reflectsthe change in the hopping amplitude t1 between two like orbitals (Fig. 4.2a) whenthe distance d between the neighboring atoms changes due to strain. The amplitudedepends exponentially on d, but for small strain it can be expanded to leading orderin the atomic displacements which leads to a correction proportional to u33. Thesecond term describes generation of hopping processes along the zˆ-direction betweendifferent orbitals (Fig. 4.2b) which are prohibited in the unstrained crystal due totheir s- and p- symmetry. The underlying mechanism is outlined in the caption ofFig. 4.2.sst1u33xzyt1 ! t1(1 u33)ps+ -tsp+ -u31⇤tsp ! i⇤u31a bFigure 4.2: The effect of strain on the hopping amplitudes in the tight binding model.a) Unidirectional strain along the z axis simply changes the distance between theneighboring orbitals leading to the modification of the hopping amplitude t1 thatis linear in u33 to leading order in small displacement. b) Torsional strain changesthe relative orientation of the orbitals and brings about hopping amplitudes that aredisallowed by symmetry in the unstrained crystal, such as tsp. The correspondingmathematical expression encodes the expectation that tsp would become equal to Λ ifthe p orbital were displaced all the way to the horizontal position. In the real materialone of course expects Eq. (4.12) to be valid only for displacements small comparedto the lattice parameter a.As a simple example consider stretching the crystal along the zˆ-direction. Thisis represented by a displacement field u = (0, 0, αz) where α = ∆L/L measures theelongation of the crystal. The only nonzero component of the strain tensor is u33 = α52and Eq. (4.12) thus gives t1 → t1(1 − α). It is easy to deduce that for small α thischanges the value of Q→ Q−αQ/(aQ)2 thus moving the Weyl nodes closer togetheror farther apart depending on the sign of α. We see that stretching the crystal hasthe same effect on the Weyl fermions as the z-component of the chiral gauge field a.More generally, elastic distortion expressed through Eq. (4.12) generates addi-tional terms in the lattice Hamiltonian (4.8) of the formδhlatt(k) = −t1u33τ z cos akz + Λ(u13τx − u23τ y) sin akz. (4.13)Expanding again in the vicinity of K±, we obtain the linearized Hamiltonian of thedistorted crystalhη(q) = vjητj(~qj − ηecaj), (4.14)where the gauge potential is given bya = −~cea(u13 sin aQ, u23 sin aQ, u33 cot aQ). (4.15)For aQ 1, we may approximate sin aQ ' aQ ' a√−M0/M1 and cot aQ ' 1/aQ.We thus conclude that in a Weyl semimetal with nodes located on the kz axis,components uj3 of the strain field act on the low-energy fermions as a gauge potential.a represents a chiral gauge field because it couples with the opposite sign to the Weylfermions with different chirality χ.We saw above that a3 ∼ u33 can be generated by stretching or compressing thecrystal along its zˆ axis. Time-dependent distortion of this type will thus producea pseudoelectric field e = −1c∂ta directed along zˆ. In combination with an appliedmagnetic field B ‖ zˆ, this will generate nonzero e · B term and, as we discussbelow, allow to test the second chiral anomaly equation (4.2). It is also possibleto generate the pseudomagnetic field by applying torsion to the crystal prepared in53a wire geometry. To see this, consider the displacement field u that results fromtwisting a wire-shaped crystal of length L by angle Ω. As illustrated in Fig. 4.3, wehaveu = ΩzL(r × zˆ), (4.16)where r denotes the position relative to the origin located on the axis of the wire.Nonzero components of the strain field are u13 = (Ω/2L)y and u23 = −(Ω/2L)x. ViaEq. (4.15) we then get the pseudomagnetic fieldb = ∇× a = b0zˆ, b0 = Ω ~c2Laesin aQ. (4.17)Figure 4.3: The displacement field u in the presence of torsion. Consecutive layersof the crystal are rotated by relative angle ϕ0 = Ω(L/a).To close this section, we estimate the magnitude of the strain-induced field b thatcan be achieved in a typical Cd3As2 nanowire described in Ref. [39]. We consider acylindrical wire with a diameter d = 100nm, length L = 1µm and lattice parametera = 4Å. Eq. (4.6) gives Q = 0.033Å−1, so the the condition aQ 1 is satisfied and wemay expand the sine in Eq. (4.17). Recalling further that Φ0 = hc/e ' 4.12×105TÅ2,we find b0 ≈ 1.8× 10−3T per angular degree of twist. The maximum attainable fieldstrength in a given wire will depend on how much torsion can the wire sustain beforebreaking. While we were unable to find any data on the mechanical properties ofCd3As2, we note that Ref. [39] characterized the nanowires as greatly flexible. We54take this to imply that they can withstand substantial torsion. Based on this, a twistangle Ω ' 180o would appear sustainable and will produce b0 ≈ 0.3T. For the wireunder consideration, such a twist translates to a maximum displacement at the outerradius of the wire of about 0.3Å between the neighboring atoms, or about 8% of theunit cell. Because the maximum twist angle is limited by the maximum distortion,higher effective fields can be achieved in thinner wires.4.5 Numerical SimulationsTo further confirm the validity of the analytical results presented in the previoussection, we carried out extensive numerical simulations of the lattice Hamiltonian(4.8) in the presence of magnetic field B as well as torsional and unidirectional strainimplemented via Eq. (4.13). Magnetic field was implemented through the usual Peierlssubstitution. Our results below indeed validate the general concepts discussed aboveand illustrate them in a concrete setting of a lattice model relevant to Cd3As2 andNa3Bi.4.5.1 Pseudomagnetic Field b from TorsionWe start by studying a wire grown along the crystallographic z axis in the presence ofmagnetic field B = zˆB and torsion. Representative results are displayed in Fig. 4.4.For simplicity and ease of interpretation, we used here parameters appropriate forCd3As2 (summarized in Appendix A), neglecting terms in k. We have verified thatsubstantially similar results are obtained when k is retained as well as for parametersappropriate for Na3Bi. These results are given in Appendix A.55-π 0 πk-20-1001020E [meV]-π 0 πk -π 0 πk -π 0 πkk-20-1001020E [meV]k k k-π 0 πk-20-1001020E [meV]-π 0 πk-π 0 πk-π 0 πkOut[20]=BULKSURFACEBAND STRUCTUREb=0, B=0  b=0, B=3.2Ta b  b=3.2T, B=0c  b=3.2T, B=3.2TdFigure 4.4: Tight-binding model simulations of a Weyl semimetal wire under torsionalstrain and applied magnetic field B = zˆB. Top row of figures shows the bandstructure of the lattice Hamiltonian defined by Eqs. (4.8) and (4.13) computed for12-Cd3As2 model parameters, for a wire with a rectangular cross section of 30 × 30sites and a lattice constant a = 40Å. (We use larger lattice constant here and insubsequent simulations than in real Cd3As2 in order to be able to model nanowiresand films of realistic cross sections with available computational resources. Note thatthis does not affect the physics at low energies because the lattice Hamiltonian isdesigned to reproduce the relevant k · p theory independent of a.) Open boundaryconditions are imposed along x and y, periodic along z. Parameters appropriate forCd3As2 are used. Middle and bottom rows show spectral functions Abulk(k, ω) andAsurf(k, ω). The former is obtained by averaging the full spectral function Aj(k, ω)over sites j in the central 10 × 10 portion of the wire while the latter averages overthe sites located at the perimeter of the wire. The torsion applied in columns c andd corresponds to the maximum displacement at the perimeter of 0.5a, or ϕ0 ' 2obetween consecutive layers.Column (a) in Fig. 4.4 shows the spectrum of an unstrained wire in zero field.Gapless Weyl points are apparent at k = ±Q and are connected by surface states thatoriginate from the Fermi arcs, expected to occur in the surface of a Weyl semimetal.56Spectral functions computed in the bulk, Abulk(k, ω), and at the surface, Asurf(k, ω),confirm this identification of bulk and surface electron states. Column (b) exhibitsour results for an unstrained wire in magnetic field B = 3.2T along the axis of thewire. As expected on the basis of arguments that led to Fig. 4.1a, we observe at lowenergies a pair of left and right moving chiral modes. These originate from the n = 0Landau level and occur in the bulk of the sample. We also observe that the surfacestates remain largely unaffected by the field.Our main finding is illustrated in column (c). Torsional strain applied to the wireproduces two right-moving chiral modes that are localized in the bulk of the sampleas evidenced by Abulk(k, ω). The bulk spectrum has the structure depicted in Fig.4.1b expected to occur in the presence of the chiral magnetic field b. We are thusled to identify the torsional strain with the chiral vector potential a. Surface statesdiscernible in the corresponding Asurf(k, ω) are seen to compensate for the bulk bandstructure by providing the required left moving chiral modes.Column (d) shows the spectrum for the case when the strength of B is chosen toexactly equal b. As a result, vector potentials A and a add in one Weyl point butcancel in the other. The resulting spectrum exhibits a set of right moving bulk chiralmodes present in only one of the two Weyl points. This establishes the completeequivalence of the real magnetic field B and the strain-induced pseudomagnetic fieldb insofar as their action on the low-energy Weyl fermions is concerned.We note that pseudomagnetic field b ' 3.2T indicated in Fig. 4.4 is larger thanthe maximum achievable field in the realistic Cd3As2 wire estimated in the previoussection. This is because for clarity we employed here larger torsion (resulting inthe maximum displacement of about half the lattice spacing) than can likely besustained in a real wire. For weaker torsion strengths, the effect remains qualitativelyunchanged but becomes less clearly visible in the numerical data for system sizes thatare accessible to our simulations.57Results presented in Fig. 4.4 pertain to a Weyl semimetal described by Hamil-tonian (4.8) but are easily extended to Cd3As2 as long as we continue neglectingthe particle-hole symmetry breaking term k. In this limit, spectra for Cd3As2 areobtained by simply superimposing bands Ek and −Ek shown in Fig. 4.4 or by formingspectral functions A(k, ω)+A(k,−ω). Full spectra, including the p-h breaking terms,are more complicated but show the same qualitative features. Some relevant examplesare given in Appendix A.4.5.2 Pseudoelectric Field e from Unidirectional StrainAccording to our previous discussion, pseudoelectric field e should emerge when theu33 component of the strain tensor becomes time dependent. This can be achievedthrough dynamically stretching and compressing the crystal along its z axis, e.g. bydriving longitudinal sound waves through the crystal. To see how the lattice modelrealizes the chiral anomaly under these conditions, we first consider an infinite bulkcrystal in the presence of a uniform magnetic field B = zˆB and investigate the effectof the static u33 strain. The spectrum of an unstrained crystal in the field B = 10T isdisplayed in Fig. 4.5a (we use once again Cd3As2 parameters and include this time alsok). At low energies the spectrum exhibits the expected chiral branches that resultfrom the n = 0 Dirac Landau level. We assume the system is initially in its groundstate with all energy levels below the chemical potential µ0 occupied and all levelsabove µ0 empty. We now implement unidirectional strain through Eq. (4.13) whichamounts to rescaling the hopping amplitudes t1 → t1(1−α) and c1 → c1(1−α). Herec1 is the hopping amplitude along the z direction in k defined below Eq. (A.1). Weimagine doing this sufficiently slowly so that the ground state evolves adiabaticallyin response to the increasing strain. The new ground state for α = 0.03 is depicted inFig. 4.5a. It exhibits a slightly modified band structure with the chemical potentialshifted to a new value µ′. The shift in µ occurs because under adiabatic evolution an58electron initially in the quantum state with momentum k in the nth band remains inthat state as the band energy En(k) evolves in response to strain.-1 0 1k-30-20-100102030E [meV]3% strainunstrained1D, a=20, 100x80x80, flux 0.01 (B=10T) str 0.95a bμ0μ᾽μ᾽k k kunstrained equilibrium3% strain non-equilibrium3% strain equilibriumμ0μ᾽eqy/a0.0010.00080.00060.00040.00020-0.0002-0.000420100-10-20Change in Charge Density [e per site]cδρδρeqE [meV]bulkstatessurfacestatesFigure 4.5: Tight-binding model simulations of a Weyl semimetal under appliedmagnetic field B = zˆB and unidirectional strain. Parameters for Cd3As2 listedin Appendix A are used in all panels. Only spin up sector of the model is consideredwith B = 10T. a) Band structure of the system with periodic boundary conditions inall directions (no surfaces) projected onto the z axis (k denotes the crystal momentumalong the z direction). Solid (dashed) lines show occupied (empty) states. Occupationof the strained system is determined by adiabatically evolving the single-electronstates of the unstrained system. b) Band structure of a slab with thickness d = 1000Å(50 lattice sites). Only positive values of k are displayed but the band structure issymmetric about k = 0. Red (black) lines show occupied (empty) states. The centralpanel indicates the nonequilibrium occupancy of the strained system obtained byadiabatically evolving the single-electron states of the unstrained system. The rightpanel shows the occupancy of the strained system once the electrons relaxed backto equilibrium. All three panels correspond to the same total number of electronsN . c) Change in the electron density in response to the applied strain as a functioncoordinate y perpendicular to the slab surfaces. δρ refers to the nonequilibriumdistribution while δρeq refers to the relaxed state. Note that density oscillations nearthe edges apparent in δρ average to zero: there is no net charge transfer between thebulk and the surface in the nonequilibrium state, as can also be deduced from thevanishing δρ in the bulk.From the point of view of the low-energy theory, the lateral shift of the chiralbranches is consistent with the effect of the uniform chiral gauge potential az whichaccording to our discussion below Eq. (4.12) moves the Weyl points closer togetherfor α > 0. From Eqs. (4.14) and (4.15) we can estimate the amount of this shiftδQ ' (e/~c)az = −u33 cot aQ/a. This in turn gives an estimate for the required59change in the chemical potential δµ = µ′ − µ0 = −~vδQ, orδµ = −vceaz = α~vacot aQ. (4.18)For Cd3As2 parameters including the particle-hole symmetry breaking terms in k,we have ~v ' 1.94eVÅ, which implies δµ = 3.75meV for α = 0.03. This estimatecompares favorably with the value δµnum = 3.46meV obtained from our lattice modelsimulation presented in Fig. 4.5a.If we continue focusing solely on the low energy degrees of freedom, we wouldconclude that a change δµ in the chemical potential in a linearly dispersing bandwith degeneracy (B/Φ0) brings about a change in the electron densityδρ = 2δµ2pi~v(BΦ0), (4.19)where the factor of 2 accounts for two chiral branches. Using Eq. (4.18) it is easyto verify that Eq. (4.19) coincides exactly with the prediction of the second chiralanomaly equation (4.2) for uniform static magnetic field and a time dependent pseu-doelectric field e = −1c∂ta.If on the other hand we espouse a band theory point of view, then we see thatin reality the charge density remains unchanged. This is because precisely the samenumber of single electron states are filled before and after the deformation. Thechemical potential changes in order to accommodate the fixed number of electronsin the modified band structure. We may thus conclude that in an infinite crystalpseudoelectric field induced by strain does not bring about any change in chargedensity. The chiral anomaly equation (4.2) however correctly predicts the straininduced change in the chemical potential δµ.A change in the chemical potential, even if time dependent (as would be the casewhen strain is induced by a sound wave), is not easily measurable when not accompa-60nied by a density change. So it would seem that this effect does not have observableconsequences. Consider however a finite system with boundaries. The key pointis that topologically protected surface states that are present in a Weyl semimetalwill generally not respond to strain in the same way as the bulk states. To a goodapproximation one may consider the surface state to remain basically unaffected by asmall unidirectional strain. This is verified by our numerical simulations summarizedin Fig. 4.5b. In that case application of strain will bring about a nonequilibriumdistribution of electrons (µ changes in the bulk but remains unchanged at the surface).This is illustrated in Fig. 4.5b where we simulate the effect of a 3% strain in a slab ofthickness d with surfaces perpendicular to the y direction and magnetic field along z.We observe that strain shifts the chemical potential for the bulk states by the sameamount as in the infinite system but leaves it essentially unchanged for the surfacestates.Several interesting consequences follow from the above observation. First, we mayexpect the charge density to remain essentially unchanged in the strained crystal withnonequilibrium distribution of electrons. This is because the bulk density remainsunchanged (as per our discussion above) and since the total charge is conserved therecan be no charge transfer to the surface. Second, in a real material the nonequilibriumelectron distribution brought about by strain will relax towards equilibrium, causingdissipation in the system which is in principle observable. When the strain is inducedby a sound wave this dissipation will provide a new mechanism for sound attenuationrelated to the chiral anomaly. Third, the relaxed charge density ρ′(y) in the strainedcrystal will differ from the the original charge density ρ0(y) of the unstrained crystalbecause relaxation necessarily involves transfer of charge between the bulk and thesurface of the sample. This is illustrated in Fig. 4.5c which shows the numericallycalculated change in the charge density δρ(y) = ρ′(y)− ρ0(y) in both nonequilibriumand equilibrium state following the application of a 3% strain. We note that modulo61some local fluctuations near the edge the charge density indeed behaves as expectedon the basis of the above arguments.We conclude by elaborating on this last effect. If the sound frequency ω issmall compared to the electron relaxation rate τ−1, as it will be the case in thetypical experimental situation, the electron distribution will always remain close toan equilibrium characterized by a global chemical potential µ′eq. The correspondingcharge density should then exhibit significant variations as the chemical potentialoscillates. Such a time dependent variation in the charge density will produce EMfields outside the sample which are measurable and can provide direct experimentalevidence for the strain-induced chiral anomaly. We shall estimate the distributionand the amplitude of these fields in the next section.To this end it will be useful to estimate the chemical potential µ′eq of the equili-brated strained system (see also Fig. 4.5b). A straightforward calculation for a slabof thickness d (summarized in Appendix B) givesµ′eq= µ0 +δµ1 + ξB/d, (4.20)where ξB = 2Q`2B is the characteristic lengthscale and δµ is the chemical potentialchange in the system without surfaces given by Eq. (4.18). The physics of Eq. (4.20)is quite simple: it reflects the fact that a surface can accommodate only a limitedamount of charge from the bulk. For a thick slab d  ξB we recover the bulk resultµ′eq ≈ µ0 + δµ because the effect of the surface becomes negligible.From Eq. (4.20) it is easy to obtain an estimate for the corresponding change inthe bulk charge densityδρbulk = − αpia(BΦ0)cot aQ1 + d/ξB. (4.21)62In the limit of a thin slab, d ξB, this result approaches the charge density change(4.19) derived based on the naive application of the chiral anomaly equation, exceptfor the opposite overall sign. In this limit, physically, almost all the non-equilibriumcharge density produced in the bulk can be absorbed by the surface. The bulk chargedensity thus goes down by the amount close to that predicted by the chiral anomaly.Fig. 4.6 shows the bulk charge density δρbulk in response to the unidirectionalstrain α = 0.03 as a function of the applied field B in a relaxed system, numericallycalculated from the lattice model. A good agreement with Eq. (4.21) is seen bothin the magnitude of the effect and its functional form. The lattice model shows asomewhat stronger response than expected which we attribute to the p-h anisotropythat was not included in the analytical calculation. That this is indeed the case isconfirmed by the same calculation performed for the p-h symmetric version of the12-Cd3As2 model which shows closer agreement with Eq. (4.21), modulo finite sizeeffect induced fluctuations (solid black symbols in Fig. 4.6). We however note that inthis case the contribution from the spin-down sector exactly cancels that from spinup so p-h asymmetry is required to obtain a nonzero result.Change in Bulk Charge Density [e per site]B [Tesla]0-0.00002-0.00004-0.00006-0.0000814121086420Eq. (3.4)p-h symmetric, spin-upp-h asymmetric, spin-upp-h asymmetric, both spinsFigure 4.6: Numerically calculated change in the bulk charge density δρbulk in responseto unidirectional strain α = 0.03 as a function of the applied field B. Parameters forCd3As2 are used with µ = 0 and d = 1000Å (50 lattice sites). Solid black symbolsgive result for the p-h symmetric version of the12-Cd3As2 model obtained by settingall Cj parameters to zero.63We note that Eqs. (4.20) and (4.21) were derived assuming quantum limit, i.echemical potential in the n = 0 Landau level. The corresponding results valid awayfrom the quantum limit are given in Appendix B.We close this subsection by considering Dirac semimetals. Naively one could thinkthat the effects discussed above will cancel once we include both spin sectors. Thiswould indeed be the case in a perfectly particle-hole symmetric system. However, theband structures of both Cd3As2 and Na3Bi exhibit strong particle-hole asymmetrywhich prevents such cancellations. To elucidate this we show in Fig. 4.7 the bandstructure of the spin-down sector of Cd3As2 in the field of 10T. Compared to thespin-up sector (Fig. 4.5a) it indicates a spectral gap at low energies. It is clear thatwhen µ lies inside this gap then all the physics comes exclusively from the spin-upsector. Specifically, there is nothing here to cancel or even weaken the effects discussedabove. We find that this remains true more generally. Even when µ is outside thegap the contributions to various effects discussed above generically do not cancel butremain of a similar magnitude as they would be in a Weyl semimetal with a single pairof Weyl points. This is illustrated in Fig. 4.6 where the chemical potential is chosento lie outside the bandgap; the effect is only slightly reduced when contributions fromboth sectors are added up. We thus expect the effects discussed above to genericallyremain present in Dirac semimetals such as Cd3As2 and Na3Bi.64-1 0 1k-30-20-100102030E [meV]unstrained3% strain1D, a=20, 100x80x80, flux 0.01 (B=10T) str 0.95Figure 4.7: Band structure of the spin down sector of Cd3As2 in magnetic field B =10T. Two chiral branches are visible at low energy but they are now strongly distortedby p-h symmetry breaking terms and they no longer traverse the gap between thevalence and the conduction band.4.6 Experimental Manifestations4.6.1 Topological Coaxial CableThe phenomena discussed above have several observable consequences which we nowdiscuss. According to Fig. 4.4c Weyl semimetal wire under torsion exhibits spatialseparation between left and right moving modes at low energies: the former arelocalized near the boundary while the latter occur in the bulk. At a generic chemicalpotential we thus expect persistent equilibrium currents to flow in such a wire asindicated in Fig. 4.8a. This can be argued as follows. Suppose the current densityjz(r) is uniformly zero at some reference chemical potential µ0. If we now changethe chemical potential to µ = µ0 + δµ we are populating additional right movingmodes in the bulk and left moving modes at the surface of the wire. Although thetotal current carried by the wire remains zero, as it must be in any normal metalin equilibrium [150], there is now a non-vanishing positive current density flowing inthe bulk compensated by the negative current density flowing along the surface. We65have verified numerically that this is indeed the case in the lattice model (4.8) and(4.13): for any chemical potential µ 6= 0 a ground-state current density develops asillustrated in Fig. 4.8b.0 10 20 303020100a bxyjFigure 4.8: Equilibrium current density in the Weyl semimetal wire under torsion.a) Schematic depiction of the bulk/surface current flow. b) Ground state currentdensity computed from the lattice model Eqs. (4.8) and (4.13) at chemical potentialµ = 5meV. Warm (cold) colors represent positive (negative) current density j. Thering-shaped inhomogeneity in j apparent in the bulk of the wire reflects Friedel-likeoscillations in electron wavefunctions caused by the presence of the surface.Such a current flow generates magnetic fields outside the wire which are, at leastin principle, measurable e.g by scanning SQUID microscopy. In practice, however,we expect this to be a challenging experiment. The currents occur only in a Weylsemimetal with broken T which is most likely to be realized in a magnetic material.It might be difficult to distinguish the fields produced by torsion-induced persistentcurrents from the sample magnetization. We note that in Dirac semimetals, likeCd3As2 or Na3Bi, the total current density will vanish upon including the contributionfrom the lower diagonal block in the Hamiltonian (4.5). This has to be the casebecause non-zero j would violate the T symmetry of the material, which shouldremain unbroken under strain. The current density can be nonzero, however, whenboth torsion and magnetic field are applied. This is demonstrated in Fig. A.2 ofAppendix A.664.6.2 Chiral Torsional EffectThe physics described above however has a simple manifestation observable in trans-port measurements in both Weyl and Dirac semimetals. Consider a measurement oflongitudinal resistivity in a twisted wire. Once again we start by discussing a Weylsemimetal. When electric field E is applied to the twisted wire it begins to producecharge density δρ = ρ − ρ0 in the bulk at the rate given by the anomaly equation(4.2). In view of our discussion above we interpret δρ as charge density imbalancebetween the bulk and the surface of the wire. Such an imbalance can relax back toequilibrium only through processes that induce backscattering between the bulk rightmoving modes and the surface left moving modes. If we denote the relevant scatteringtime by τ , we get an equationddtδρ =e22pi2~2cE · b− δρτ. (4.22)At long times t τ the steady state solution readsδρ =e2τ2pi2~2cE · b. (4.23)The wire clearly carries non-zero electrical current. The expression for the currentdepends on the relative position of the chemical potential µ and the bottom of the firstLandau level 1(0) = ~v√2eb/~c. In the quantum limit, |µ| < 1(0), only the chiralmodes in the n = 0 Landau level are populated. These all move at the same velocityvsgn(b) and the non-equilibrium charge density δρ thus gives electrical currentJCTE= −evsgn(b)δρ = e3vτ2pi2~2cE · bsgn(b). (4.24)For a constant relaxation time τ we thus have chiral torsional contribution to theconductivity σCTE ∼ |b|, similar to the ordinary chiral magnetic effect σCME ∼ |B|67in the quantum limit [29]. However, if the wire radius R significantly exceeds themagnetic length `b =√~c/eb ' 256Å√1T/b, then we find that the appropriaterelaxation time becomes field-dependent, namely,τ ' τ0R2`2b∼ |b|, (4.25)where τ0 is the microscopic scattering time. This is because the bulk electron wave-functions have spatial extent `b in the direction transverse to the axis of the wire.Deep in the bulk impurities cause scattering between the individual bulk modes butsince these are all right moving such processes cannot relax the current. Only thoseelectrons that have diffused all the way to the boundary through repeated scatteringprocesses can backscatter into left moving surface modes. Electrons thus experiencehydrodynamic flow whereby dissipation occurs only at the boundary. Eq. (4.25) isderived in Appendix C and expresses the fact that an electron that is produced nearthe center of the wire has to travel distance R to the boundary and this takes onaverage (R/`b)2scattering events. We conclude that σCTE ∼ b2 in the quantum limitwhen `b  R.In the semiclassical limit, |µ|  1(0), we must take into account the additionalequilibration that occurs between the individual Landau levels within a given Weylpoint. We assume that the relaxation time for this process is very short and essentiallyinstantaneous compared to τ . In this case, electron density produced through Eq.(4.23) gets distributed among all the bulk states and leads to a shift in the chemicalpotential µ → µ + δµ. In the semiclassical limit we can approximate the density ofstates by the expression valid in the zero field, D() = 2/pi2~3v3, where for simplicitywe also assume isotropic velocities. In the limit of interest, δµ  kBT  µ, it iseasy to find from this the shift in the chemical potential caused by a small change in68density,δµ ' 2pi2~3v3µ2 + 2pi23k2BT2δρ. (4.26)We can now calculate the current by noting that, once again, only the chiral branchescontribute. We thus obtainJCTE= −ev(δµ2pi~v)(bΦ0), (4.27)where the first bracket represents the number of extra modes δn that have beenpopulated on the chiral branch and the second reflects their degeneracy. Combiningthis with Eqs. (4.26) and (4.23) we findJCTE=e4v38pi3~c2τµ2 + 2pi23k2BT2(E · b)b. (4.28)In view of Eq. (4.25) in a Weyl semimetal under torsion (parametrized here byb ∝ Ω) we thus predict a positive contribution to the conductivityσCTE∝b2, µ < ~v√2eb~c quantum limit,|b|3, µ ~v√2eb~c semiclassical limit.(4.29)The predicted field dependence is different from the analogous effect encountered inthe presence of the real magnetic field B (where σCME behaves as ∼ B and ∼ B2in the two limits). This reflects the hydrodynamic nature of the electron flow thatoccurs when R `b. The right and left moving modes are then segregated to the bulkand the boundary respectively, which leads to an extra power of b due to b-dependenttransport scattering rate (4.25). We also note that when R  `b, Eq. (4.25) impliessignificant enhancement of the transport lifetime and thus leads us to expect a strongeffect. In the opposite limit, R . `b, the transport scattering rate becomes field69independent and the more conventional behavior with σCTE ∝ b (b2) in quantum(semiclassical) limit is restored.The effect will persist in a Dirac semimetal such as Cd3As2 and Na3Bi, which canbe thought of as two T -conjugate copies of the Weyl semimetal discussed above. Inthe presence of a twist the spectrum will consist of that indicated in Fig. 4.4c for thespin-up sector plus a time-reversed copy (obtained by reversing k → −k) for the spindown sector. The same analysis we just performed applies unchanged for each spinsector if one can ignore spin-flip scattering events. In this case Eq. (4.29) continuesto hold in a Dirac semimetal. Spin-flip processes, if present, open additional channelfor relaxation by scattering between left and right moving bulk modes. In the limitwhen the spin-flip relaxation rate τ−1sf exceeds τ−1the hydrodynamic flow will ceaseand the behavior will cross over to the regular chiral anomaly with σCTE ∝ b (b2)in the quantum (semiclassical) limit. In clean samples of T -preserving Cd3As2 andNa3Bi we expect the hydrodynamic behavior to prevail. This is because ordinarynon-magnetic impurities cannot cause spin-flip scattering. Time reversal symmetrypermits spin-orbit scattering terms of the form zˆ · (σ × k). These do contribute toτ−1sf but we expect such contributions to be small.4.6.3 Ultrasonic Attenuation and EM Field EmissionWe now consider the experimental manifestations of the e·B term in the second chiralanomaly equation (4.2). For concreteness we again start with a Weyl semimetal andconsider a sample in the shape of a slab with thickness d and surfaces perpendicularto the y axis. Magnetic field B is applied along the z-direction. The requisite e fieldis generated by a longitudinal sound wave with frequency ω that is driven along thez direction. This produces a time dependent displacement fieldu = u0zˆ sin(qz − ωt), (4.30)70where q = ω/cs is the wavenumber and cs the sound velocity. The nonzero componentof the strain tensor is u33 = u0q cos (qz − ωt) which through Eq. (4.18) yields anoscillating component of the bulk chemical potential,δµ(t) = u0q(~vacot aQ) cos(qz − ωt). (4.31)As mentioned in Sec. 4.5.2, electron relaxation dissipates energy which will be man-ifested by the attenuation of the sound wave as it propagates through the medium.Specifically, as explained e.g. in Ref. [151] the energy flux I carried by the soundwave obeys I(z) = I0e−2Γzwhere Γ is the sound attenuation coefficient. We nowproceed to estimate Γ which is given by Γ = Q/2I, where Q denotes the amountof energy dissipated in a unit volume per unit time. To provide a crude estimateof Q we assume for a moment that the electron relaxation rate τ−1 is comparableto the driving frequency, ωτ ≈ 1. In this case relaxational dynamics is maximallyout of phase with the sound wave and we can estimate Q simply by calculating theenergy difference between the nonequilibrium distribution of electrons (see Fig. 4.5b)reached at the crest of the wave (assuming no dissipation has occurred until then)and the corresponding equilibrium distribution with the chemical potential µ′eq. Forthis estimate consider a slice of the system perpendicular to z of length l such thatl  λs. Inside the slice the strain can be considered uniform, implying a uniformchemical potential δµ(t) ∝ cosωt. We may thus estimate the total dissipated electronenergy per cycle asEdis= lwd∫ µ′µ′eqDb()d− lw∫ µ′eqµ0Ds()d, (4.32)where w is the width of the slab along the x direction and Db/s() is the bulk/surfacedensity of states given in Appendix B. It is easy to evaluate the requisite integrals.71After some algebra and with help of Eq. (4.20) one obtains, assuming quantum limit,Edis≈ lwd2pi~v(BΦ0)11 + d/ξBδµ¯2, (4.33)where δµ¯ is the amplitude of δµ(t) given in Eq. (4.31).A more complete treatment of the relaxational dynamics, which we omit here forthe sake of brevity, gives a result for the energy dissipated per cycle valid for anyfrequency,Edis=lwd2pi~v(BΦ0)ωτ(1 + d/ξB)2 + (ωτ)2δµ¯2. (4.34)The energy density of the sound wave, averaged over one cycle, is ρE =12ρc2su20q2,where ρ denotes the mass density of the crystal. Noting that the corresponding energyflux is I = csρE one obtains the sound attenuation coefficientΓ =(ωEdis/lwd)2csρE. (4.35)To estimate its magnitude we assume the limit of a thin slab d  ξB and fastrelaxation ωτ  1 in Eq. (4.34). In this limit Γ becomes independent of d:Γ '(ω2pics)~va2(BΦ0)2 cot2 aQρc2s(ωτ). (4.36)For our estimate we take f = ω/2pi = 200MHz, the mass density of Cd3As2 isρ = 7.0× 103 kg/m3 while the speed of sound is cs = 2.3× 103m/s [152] which givesλs ' 11µm at this frequency. For these parameters we obtainΓ ' 3.6× 103m−1[B1T](ωτ). (4.37)We see that depending on the magnitude of the electron scattering rate thesound attenuation can be substantial. There are of course many conventional sources72of ultrasonic attenuation in metals [151]. Given the specific dependence of Γ onfrequency, magnetic field and the fact that it depends only on the component of Bparallel to q, it should be possible to separate the contribution of the chiral anomalyfrom the more conventional contributions.At B = 1T for material parameters relevant to Cd3As2 we have ξB ' 430nm sothe above estimate applies to thin films or wires. For thicker films one must includethe additional suppression factor (1 + d/ξB)−2from Eq. (4.34) that we neglectedso far. This factor reflects the fact that the relaxation mechanism involves chargetransfer from the bulk to the surface of the sample. For the same reason, however,we expect in this limit to obtain an enhanced relaxation time τ ' τ0(d/`B)2, whereτ0 is the microscopic relaxation time as in Eq. (4.25). This is because to relax thenon-equilibrium distribution brought about by the chiral anomaly bulk electrons mustdiffuse to the surface and this takes on average (d/`B)2scattering events. In the endwe expect only a weak dependence of Γ on the sample thickness d although a detailedtreatment of the combined spatial and temporal distribution of electrons during therelaxation process is an interesting topic for future research.The oscillating charge density that occurs in the system in response to the soundwave will generate EM fields that can be detected outside the sample. We show belowthat in a typical situation the electric field close to the surface can be substantial andthus detectable. The field decays as ∼ e−r/λs away from the surface but since λs istens or hundreds of microns at typical ultrasonic frequencies the detection of suchfields should not be difficult [153].To estimate the amplitude of the electric field we assume once again that electronrelaxation is fast compared to the driving frequency, ωτ  1. This means thatelectrons will locally always be close to equilibrium characterized by the chargedensity ρ¯ + δρbulk(z, t) where ρ¯ is the bulk charge density of the unstrained crystaland δρbulk(z, t) is given by Eq. (4.21) with α now describing the local strain field at73(z, t). In a slab of thickness d = 2d′ illustrated in Fig. 4.9 the oscillating componentof the charge density therefore readsρbulk = ρ0 cos(qz − ωt), (4.38)withρ0 = −u0qpia(BΦ0)cot aQ(1 + d/ξB). (4.39)ba2d’yxzB qy/λsz/λsFigure 4.9: Proposed geometry for the EM field emission measurement in the limitwhen all the dimensions of the crystal are much larger than the sound wavelengthλs. a) A slab of thickness d = 2d′is subjected to magnetic field B and a longitudinalacoustic sound wave propagating along the z direction. b) A snapshot of the electricfield distribution near the surface calculated from Eq. (4.43). As a function of timethe entire pattern moves in the z direction at the speed of sound cs.From our previous discussion, we know that the charge generated in the bulkcomes from the boundary. The total charge density that reflects the overall chargeconservation in each constant-z slice of the sample can thus be written asρ = ρ0 [θ(d′ − |y|)− d′δ(y ± d′)] cos(qz − ωt), (4.40)where ρ0 is given by Eq. (4.39). In the near field (static) region we may neglect themagnetic effects and determine the electric field E = −∇Φ by solving the Poissonequation ∇2Φ = −4piρ. Adopting the ansatz Φ(r, t) = f(y) cos(qz − ωt) we are led74to a 1D equation for f(y) of the form(∂2y − q2)f = −4piρ0 [θ(d′ − |y|)− d′δ(y ± d′)] . (4.41)This has a solutionf(y) =4piρ0q2+B cosh qy |y| < d′Ae−q|y| |y| > d′.(4.42)The function f(y) must be continuous at y = ±d′ and the discontinuity in its firstderivative must match the surface charge in Eq. (4.41), f ′(d′+)−f ′(d′−) = 8piρ0d′.This determines constants A and B. The full solution for the potential outside thesample readsΦ(r, t) = 4piρ0Ae−q|y|q2cos(qz − ωt), (4.43)with A = sinh qd′ − 2qd cosh qd′ ≈ −qd′ eqd′ . The electric field that follows fromthis potential is depicted in Fig. 4.9b. For d = 1mm, u0 = 0.01a and all the otherparameters as before the maximum electric field (that occurs right at the samplesurface) can be estimated as |E| ' 4piρ0ed ' 2.4 × 104V/m. This is a large fieldwhich should be easily detectable.In a realistic semimetal we should include screening effects which can substantiallyreduce the electric field amplitude estimated above. A crude estimate of the screenedfield can be obtained by replacing Φ(q) → Φ(q)/(q) where (q) = 1 + k2TF/q2 is thedielectric function in the Thomas-Fermi approximation and k2TF = 4pie2D(µ). It isphysically more transparent to write (kTF/q)2 = (λs/λTF)2where λTF = 1/kTF isthe Thomas-Fermi screening length. Using the experimentally determined electronvelocity v ' 1.5 × 106m/s [34] to obtain density of states D() = 2/pi2~3v3 wefind that λTF ' 32µm[1meV/µ]. Thus, depending on the chemical potential, thescreening length can be quite long. For instance if µ = 10meV the screening lengthλTF ' 3.2µm is comparable to λs = 11µm and the electric potential will be suppressed75only modestly. Even for the experimentally observed µ ≈ 200meV [34] the suppressionis about a factor of 1.6 × 104 which still leaves a significant field strength of severalV/m at the surface. We conclude that the effect remains measurable even in thepresence of realistic levels of screening that can be expected in a Dirac semimetalwith the chemical potential not too far from the neutrality point.4.6.4 Chiral Anomaly in the Absence of EM FieldsWe finally mention an attractive possibility of testing the chiral anomaly using purelystrain-induced gauge fields and no real EM fields. According to our previous discus-sion, a simultaneous application of torsion and time-dependent unidirectional strain ina wire geometry generates both b and e pointed along the z direction of the crystal.In this situation the right hand side of the first anomaly equation (4.1) becomesnonzero even in the complete absence of E and B and pumping of charge betweenthe Weyl nodes occurs. The nonequilibrium electron distribution thus created willrelax via internodal scattering and produce dissipation of energy. This dissipationis in principle measurable. For instance when the pseudoelectric field e is generatedby a longitudinal sound wave its amplitude will be attenuated by this effect and theattenuation coefficient will be proportional to the amount of torsion on the wire.This can be demonstrated by considerations that are similar to those that lead toEq. (4.36). We shall not repeat this analysis here but simply note that a substantialcontribution to the attenuation can be achieved by this effect.4.7 Summary and OutlookWe studied the chiral anomaly in Weyl semimetals in a new context, when the signof the anomaly is the same in the two Weyl cones. This takes place when a chiralgauge field is present in addition to the ordinary EM gauge field. Specifically, this76type of chiral anomaly occurs when pseudomagnetic field b, produced by torsionin the material, is present together with the real electric field E. Alternatively,pseudoelectric e field produced by unidirectional strain combined with a real magneticfield B gives rise to the anomaly. Contrary to the usually discussed chiral anomaly,density of electron grows in both Weyl cones when the fields are applied, thus makingthe bulk theory of the material truly anomalous. The apparent contradiction withcharge conservation is resolved when one takes into account the surface of the material fermions are taken from the surface into the bulk.In the presence of the b · E term the transfer of charge from the surface to thebulk occurs through the ordinary semiclassical evolution of the electron states in theBrillouin zone. This is facilitated by the phenomenon of spatial segregation of theright and left moving modes between the surface and the bulk of the wire undertorsion as discussed in Sec. 4.5.1. In the presence of the e ·B term the situation isdifferent: here the charge transfer occurs through relaxation of the nonequilibriumstate that is generated by the chiral anomaly. This disparity in the action of the twotypes of terms has a very simple physical reason. A uniform b field can only exist ina finite system with boundaries because it requires an increasing strain field in somespatial directions (just like the uniformB field requires increasing vector potentialA).However, in a realistic crystal strain can only increase to a certain point after whichthe crystal breaks. We thus see that a uniform b necessarily implies the existence ofsurfaces. A consequence of this is that the band structure of the relevant system willhave an equal number of left and right moving modes which are however unbalancedbetween the surface and the bulk. Semiclassical evolution in the presence ofE ‖ b willthus transfer charge from the bulk to the surface (or vice versa). By contrast the efield can be created by a time-dependent unidirectional strain which does not requirespatial boundaries. We have seen that in a system without boundaries the e ·B termsimply changes the chemical potential in accord with the chiral anomaly equation77(4.2). If, however surfaces exist, this creates a nonequilibrium distribution whichcan relax by transferring charge to the surface. Furthermore, if sufficient number ofsurface states are available, then the bulk charge density change can be close to thatpredicted by the chiral anomaly.When both torsion and unidirectional strain are applied a new form of the chiralanomaly can be created via the e·b term in Eq. (4.1). In this case charge is transferredbetween the two Weyl points with opposite chirality but remarkably no physical EMfields are required.Based on these general concepts we make several specific predictions for theexperimentally observable signatures of the anomaly. In the case of the b ·E term wepredict a negative contribution to the resistance that has a square or cubic dependenceon the torsion strength, depending on the regime. In the case of the e ·B term wepredict bulk-boundary charge transfer, resulting in EM field emission and ultrasonicattenuation. Similarly, we predict that the e · b term will contribute to ultrasonicattenuation. These predictions are most clear cut in a semimetal with a singlepair of Weyl points. We showed, however, that substantial observable consequencesoccur also in Dirac semimetals Cd3As2 and Na3Bi whose electronic structure can beviewed as two time-reversed copies of such an elemental Weyl semimetal. On generalgrounds we also expect these phenomena to be manifested in more complex Weylsemimetals such as TaAs, ZrTe5 or WTe2 which host several pairs of Weyl points.Because their electronic structures are complex, detailed quantitative modeling ofstrain effects will require delving into the details of the band structures and we leavethis for future study. We nevertheless anticipate that in these materials each pairof Weyl points will contribute to various chiral anomaly related effects discussed inthis work. The contributions will have different magnitudes and signs depending onthe relative positions and Fermi velocities of the Weyl cones. Partial cancellationscan occur but it appears unlikely that the effect would vanish completely, except78perhaps for very specific strain patterns with high symmetry. Pronounced transportsignatures of the ordinary chiral anomaly have already been detected in severalmaterials [13, 154157] including some with multiple Weyl points. This stronglysuggests that the novel strain-induced effects predicted in this work should also beobservable in these materials.Our work draws upon several previous studies. Some of our results regarding thephysical consequences of the second anomaly equation (4.2) have been foreshadowedin Ref. [158] which considered Weyl fermions in magnetically doped topological insu-lators. Here the chiral gauge field can arise from magnetic fluctuations in the systemand was predicted to produce one-dimensional chiral modes in a ferromagnetic vortexline and a novel plasmon-magnon coupling. As far as we know Weyl fermions havenot yet been observed in magnetically doped topological insulators. Also, it may bechallenging to create and control the magnetic textures envisioned in Ref. [158]. Bycontrast the phenomena predicted in our work only require existing materials, such asCd3As2 or Na3Bi. Also, producing the chiral gauge field from strain is not expected topose an exceptional experimental challenge. Our work also draws upon the results ofRefs. [133, 135, 149] which established the equivalence between strain and chiral gaugefield in various materials ranging from graphene and topological insulators to a simplemodel of a Weyl semimetal. Our study however goes far beyond the scope of Ref.[133] by considering the effect of strain in specific materials and geometries and byproviding concrete quantitative predictions for experimentally measurable quantitiesrelated to the chiral anomaly.When our work was substantially completed we became aware of a preprint [159]which discussed a fictitious magnetic field in a Weyl semimetal created by crystaldislocations. This effect is closely related to our b field but is different in thatunlike externally applied strain, dislocations in a crystal cannot be easily controlled.Therefore, experimental detection of this effect may prove challenging. Very recently79Schuster et al. [160] discussed the concept of a topological coaxial cable in a gappedWeyl semimetal with a vortex in the Higgs field that is responsible for the gap.In this situation the vortex line is predicted to carry protected fermionic modes andcontribute exactly quantized conductance. This effect is very interesting but also verydifferent from our concept of topological coaxial cable which occurs in an ungappedDirac or Weyl semimetal and does not in general produce quantized conductance.Given both the fundamental nature of the new anomaly discussed here and itsobvious potential for future applications, we envision numerous possible extensions ofthis work. On the theory side there are multiple questions that one can ask: Which ofthe EM effects in solids translate to pseudo-EM fields discussed here? What are thebest materials to study the effects? Do the chiral states predicted by our work haveprospects for designing more exotic many-body states in the presence of interactions?We also expect experimental activity to be stimulated since our predictions madefor real materials yield effects that should be both unusual and eminently observableby conventional experimental probes such as charge transport, ultrasonic attenuationand EM field emission.80Chapter 5Holographic Black Hole on aGraphene Flake5.1 OverviewOriginally proposed to settle a debate over the black hole information paradox [161],the holographic principle has evolved into a universal conjecture that the informationcontent of all the physics in a volume of spacetime is bounded by the surface area[162]. Relating spacetime geometry to number of quantum states, it opens up anew direction for the unification of quantum mechanics and gravity. If the conjecturecontinues to withstand the test of time, a revolution in physics may be in sight becausethe holographic principle challenges the principle of locality fundamental to existingtheories.The most successful example of the holographic principle is the AdS/CFT corre-spondence, which is the duality between string theories in (n+1)-dimensional anti-de Sitter (AdS) spacetime and conformal field theories (CFT) defined on the n-dimensional spacetime boundary [163]. It is a concrete starting point from whichthe more general holographic principle is probed. Moreover, it provides a novel81approach for tackling mathematically intractable problems. For example, stronglycoupled quantum field theories can be translated with one-to-one correspondenceto weakly interacting string theories on the other side of the duality. It has alsosparked interaction among the string theory, nuclear physics, and condensed mattercommunities.The Sachdev-Ye-Kitaev (SYK) model, a fermionic system with all-to-all randominteractions, is recently shown to be dual to a planar charged black hole [164167].More specifically, the SYK model in the strong coupling regime displays an emergent(0+1)-dimensional CFT, dual to a charged black hole in the (1+1)-dimensional AdS2spacetime described by the Einstein-Maxwell-dilaton (EMD) theory, a quantum grav-ity theory. Remarkably the strongly interacting SYK model can be solved exactlywhen the number of fermionic modes N approaches infinity. It is shown to be maxi-mally chaotic and exhibit finite entropy at zero temperature, which are characteristicof black holes. Outside of its intriguing connection with black holes, the SYK model isa fascinating quantum system in its own right. It is a non-Fermi liquid and potentiallya microscopic model for the strange metal phase in hole-doped cuprates.The SYK models can be written in terms of either Majorana fermions or ordi-nary complex fermions, with the latter having an additional U(1) charge conservingsymmetry. Since Kitaev's work mostly concerns the Majorana SYK model, we dubthe complex SYK model the Sachdev-Ye (SY) model. In this project, we propose anexperimental realization of the SY model using graphene. Consider a mesoscopicgraphene flake with irregular boundary, as shown in Fig. 5.1. In the presenceof a strong perpendicular magnetic field, Dirac Landau levels are formed in thebulk. In particular, the zeroth Landau level is highly degenerate with the number ofzero-energy modes proportional to the total flux through the flake. The wavefunctionsof the zero modes are randomly distributed over the flake in response to the irregularboundary. Since the highly disordered wavefunctions overlap significantly for a suf-82ficiently small flake, one can imagine that the interaction arising from the Coulombrepulsion (as projected onto the zero modes) would be random and all-to-all. Indeedour numerical simulation shows that the four-fermion coupling constants are complexrandom variables with a distribution similar to the Gaussian distribution used in thetheoretical SY models.BA Bδ1δ2δ3Figure 5.1: An irregular shaped graphene flake in an applied magnetic field givesrise to the (0+1)-dimensional SY model, holographically dual to a black hole in(1+1)-dimensional anti-de Sitter space. Inset: lattice structure of graphene with Aand B sublattices marked and nearest neighbor vectors denoted by δj.Furthermore, our proposal reproduces known signatures of the SY models. Weshow that the entropy dependence on temperature agrees well with the Gaussian-randomized SY model consisting of a finite N number of degenerate modes. Thelimit N → ∞ needed for achieving a nonvanishing zero-temperature entropy cannotbe simulated numerically. However, we argue that with a reasonably strong field,a much larger N can be obtained in experiments. The quantum chaotic nature ofthe SY model is revealed when one compares its many-body spectral statistics withthat of the random matrices generated by the Gaussian orthogonal (GOE), unitary(GUE), and symplectic (GSE) ensembles. We show that the distribution of the ratiosof consecutive level spacings follows the Wigner-Dyson level statistics in a N(mod 4)cycle, as predicted by previous studies [168].Lastly, we investigate various issues present in realistic experimental settings,such as impurities and second-nearest-neighbour hoppings. After examining their83effects through numerical studies, we conclude that the SY physics remains robustgiven a strong field, low temperature, and a graphene flake with irregular boundaryand clean interior. Our proposal is relatively simple when compared to the otherswhich involve superconductivity, advanced fabrication techniques, or ultra-cold gases[169171]. Furthermore, follow-up research has indicated that the non-Fermi liquidproperties can been verified in transport experiments [172].5.2 Definition of Sachdev-Ye ModelThe Sachdev-Ye (SY) model [173176], is defined by the second-quantized Hamilto-nianHSY =∑ij;klJij;klc†ic†jckcl − µ∑jc†jcj, (5.1)where c†j creates a spinless fermion, Jij;kl are zero-mean complex random variablessatisfying Jij;kl = J∗kl;ij and Jij;kl = −Jji;kl = −Jij;lk and µ denotes the chemicalpotential. In what follows, we derive the effective low-energy Hamiltonian for electronsin the zeroth Landau level (LL0) of a graphene flake with an irregular boundary andshow that, under a broad range of conditions, it is given by Eq. (5.1). The system,therefore, realizes the SY model.5.3 Graphene Flake in a Magnetic Field5.3.1 Tight-Binding ModelAt the non-interacting level a flake of graphene is described by a simple tight-bindingHamiltonian [24]H0 = −t∑r,δ(a†rbr+δ + h.c.), (5.2)84where a†r (b†r+δ) denotes the creation operator of the electron on the subblatice A(B) of the honeycomb lattice. These satisfy the canonical anticommutation relations{a†r, ar′} = {b†r, br′} = δrr′ appropriate for fermion operators. r extends over the sitesin sublattice A while δ denotes the 3 nearest neighbor vectors (inset Fig. 5.1). t = 2.7eV is the tunneling amplitude [177]. For simplicity we first ignore electron spin butreintroduce it later. The chiral symmetry χ is generated by setting (ar, br)→(−ar, br)for all r which has the effect of flipping the sign of the Hamiltonian H0 → −H0.Magnetic field B is incorporated in the Hamiltonian (5.2) by means of the standardPeierls substitution which replaces t→ tr,r+δ = t exp [−i(e/~c)∫ r+δrA · dl] where Ais the vector potential B = ∇ × A. In the presence of χ the Aharonov-Casherconstruction [178] implies N = NΦ exact zero modes in the spectrum of H0 whereNΦ = SB/Φ0 denotes the number of magnetic flux quanta Φ0 = hc/e piercing thearea S of the flake. It is clear that a flake with an arbitrary shape described by H0respects χ which underlies the robustness of LL0 invoked above.Hopping t′ between second neighbor sites and random on-site potential are exam-ples of perturbations that break χ and are therefore expected to broaden LL0. Theseeffects can be modeled by adding to HSY defined in Eq. (5.1) a termH2 =∑ijKijc†icj (5.3)which describes a small (undesirable) hybridization between the states in LL0 thatwill generically be present in any realistic experimental realization. We discuss theeffect of these terms in Sec. 5.5.2.5.3.2 Energy Spectrum and Zero-Mode WavefunctionsIn Fig. 5.2a we show the single-particle energy spectrum of H0 for a graphene flakewith a shape depicted in the inset. As a function of increasing magnetic field B we85observe new levels joining the zero-energy manifold LL0 such that the number of zeromodes follows N ' NΦ in accordance with the Aharonov-Casher argument. HigherLandau levels and topologically protected edge modes are also visible. Despite therandomness introduced by the irregular boundary, LL0 remains sharp as expected onthe basis of the arguments presented above. This is the key feature in our constructionof the SY Hamiltonian, which guarantees that the H2 term defined above vanishes aslong as the chiral symmetry is respected. In the presence of e-e repulsion, the leadingterm in the effective description of LL0 will therefore be a four-fermion interactionwhich we discuss in Sec. 5.3.3.Electron wavefunctions Φj(r) belonging to LL0 exhibit random spatial structure(Fig. 5.2b) owing to the irregular confining geometry imposed by the shape of theflake.a bLL0LL1LL2edge modeslBedge modesLL0  bulk modesFigure 5.2: Electronic properties of an irregular graphene flake in the absence ofinteractions. a) Single-particle energy levels j of the Hamiltonian H0 as a functionof the magnetic flux Φ = SB through the flake. The flake used for this calculation,depicted in the inset, consists of 1952 carbon atoms with equal number of A and Bsites. The energy spectrum, calculated here in the Landau gauge A = Bxyˆ and withopen boundary conditions, shows the same generic features irrespective of the detailedflake geometry. b) Typical wavefunction amplitudes of the eigenstates Φj(r) belongingto LL0 at Φ = 40Φ0 and the edge modes. The numerals above each panel denote theenergy j of the state in eV, scale bar shows the magnetic length lB =√~c/eB.865.3.3 Interaction Matrix Between Zero ModesFrom the knowledge of the wavefunctions it is straightforward to evaluate the cor-responding interaction matrix elements1between the zero modes. The leadingmany-body Hamiltonian for electrons in LL0 will thus have the form of Eq. (5.1)withJij;kl =12∑r1,r2[Φi(r1)Φj(r2)]∗V (r1 − r2)[Φk(r1)Φl(r2)], (5.4)where V (r) = (e2/r)e−r/λTF is the screened Coulomb potential with Thomas-Fermilength λTF and dielectric constant . The summation extends over all sites of thehoneycomb lattice. It is to be noted that only the part of Jij;kl antisymmetric in(i, j) and (k, l) contributes to the many-body Hamiltonian (5.1) so in the followingwe assume that Jij;kl has been properly antisymmetrized.We numerically evaluated Jij;kl for various values of λTF . The resulting Js arecomplex valued random variables satisfyingJij;kl = 0, |Jij;kl|2 = 12N3J2, (5.5)where J measures the interaction strength and the bar denotes averaging over random-ness introduced by the irregular confining geometry. Fig. 5.3a shows the statisticaldistribution of Jij;kl calculated for the nearest-neighbor interactions V (r) = V1∑δ δr,δand the single-particle wavefunctions Φj(r) depicted in Fig. 5.2b. The distribution ofJij;kl shows the expected randomness with some deviations from the ideal Gaussian.1See Appendix D for more details.87baGrapheneflakeRandom GaussianT/JFigure 5.3: Statistical properties of the coupling constants and the thermal entropy.a) Histogram of |Jij;kl| as calculated from Eq. (5.4) with V1 = 1 for the graphene flakedepicted in Fig. 5.2 and N = 16, compared to the Gaussian distribution (orange line)with the same variance 0.000805V1. Inset shows the histogram of real and imaginarycomponents of Jij;kl. The mirror symmetry about the horizontal follows from thehermiticity property Jij;kl = J∗kl;ij. b) Entropy S(T ) of the SY Hamiltonian (5.1)calculated with Js shown in panel (a).5.4 Exact Diagonalization of Many-Body Hamilto-nianTo show that the low-energy fermions in the graphene flake are described by the SYmodel, we next perform numerical diagonalization of the many-body Hamiltonian(5.1) with coupling constants Jij;kl obtained as described above. We then calculate88physical observables and compare them to the results obtained with random indepen-dent Jij;kl.5.4.1 EntropyThe SY model is known to exhibit non-zero ground state entropy per particle in thethermodynamic limit N → ∞, so we are motivated to compute the entropy of theflake. Given the many-body energy levels Ei, we compute the partition function Z =∑ie−Ei/T , the free energy F = −kBT lnZ, and the total energy 〈E〉 =∑iEie−Ei/T/Z.Then entropy can be found using the thermodynamic law F = 〈E〉 − TS. Fig. 5.3bshows the thermal entropy per particle S(T )/N of the flake. At high temperature,it goes to ln2 as expected from the fact that each zero mode is equally likely tobe occupied or unoccupied. At low temperature, our numerical simulation of 16zero modes cannot demonstrate a nonzero entropy because S(T → 0) vanishes forany finite N [179]. Nevertheless, the entropy calculated with random Gaussian Jij;klindicates no significant difference, which is a good sign that our flake may indeedexhibit SY physics.5.4.2 Level StatisticsMany-body energy level statistics provide another useful tool to validate our hypoth-esis that LL0 electrons in the graphene flake behave according to the SY model. Wethus arrange the energy eigenvalues En of the many-body Hamiltonian (5.1) in in-creasing order and form ratios of the subsequent levels rn = (En+1−En)/(En−En−1).According to the random matrix theory applied to the SY model [168] probabilitydistributions P ({rn}) are given by different Gaussian ensembles, depending on N(mod 4) and the eigenvalue q of the total charge operator Q =∑j(c†jcj − 1/2) assummarized in Table 5.1. Here GOE, GUE and GSE stand for Gaussian orthogonal,89unitary and symplectic ensembles, respectively andP (r) =1Z(r + r2)β(1 + r + r2)1+3β/2, (5.6)with constants Z and β listed in Table 5.1. Since HSY commutes with Q it can beblock diagonalized in sectors with definite charge eigenvalue q. As emphasized inRef. [168], the level statistics analysis must be performed separately for each q-sector.Note that q has integer (half-integer) values for N even (odd) and this is why theneutrality condition q = 0 can be met only for even values of N . Also note that q = 0corresponds to N/2 particles.N( mod 4) 0 1 2 3q = 0 GOE GSEq 6= 0 GUE GUE GUE GUETable 5.1: Gaussian ensembles for the SY model. The relevant probabilitydistributions are given by Eq. (5.6) with Z = 827, 4pi81√3, 4pi729√3and β = 1, 2, 4 for GOE,GUE, GSE, respectively.Fig. 5.4 shows our results for the level statistics performed for a graphene flake withN = 14 through 18 and various values of q. The obtained level spacing distributionsare seen to unambiguously follow the prediction of the random matrix theory forthe SY model summarized in Table 5.1. We are thus led to conclude that interactingelectrons in LL0 of a graphene flake with an irregular boundary indeed exhibit spectralproperties characteristic of the SY model.90N=14 q=0N=15 q=1/2N=16 q=0N=17 q=1/2N=18 q=0GSE GUEGUE GSEGOE N=16 q=1GUEFigure 5.4: Many-body level statistics for the interacting electrons in LL0 of thegraphene flake. Blue bars show the calculated distributions for the graphene flake.Orange, green and red curves indicate the expected distributions given by Eq. (5.6)for GOE, GUE and GSE, respectively. To obtain smooth distributions, results forN = 14, 15, (16) have been averaged over 8 (4) distinct flake geometry realizationswhile N = 17, 18 reflect a single realization.5.5 Further ConsiderationsWe now discuss various aspects of the problem relevant to the laboratory realization.5.5.1 Electron SpinsElectrons in graphene possess spin which we so far ignored. Given the weak spin-orbitcoupling in graphene we may model the non-interacting system by two copies of theHamiltonian Eq. (5.2) plus the Zeeman term, H = H0 +g∗µBB ·Stot where Stot is thetotal spin operator and µB = 5.78× 10−5 eV/T is the Bohr magneton. For graphene91on the SiO2 or hBN substrate we may take g∗ ' 2 which gives the bare Zeemansplitting ∆ES(B) ' 0.12 meV/T, or about 2.4 meV at B = 20 T. We expect thisrelatively small spin splitting to be significantly enhanced by the exchange effect ofthe Coulomb repulsion. The strength of the exchange splitting ∆EC ' 8.8 meV/T isestimated in Appendix D. For such a large spin splitting one may focus on a partiallyfilled LL0 for a single spin projection and disregard the other. The spinless modelconsidered so far should therefore serve as an excellent approximation of the physicalsystem in the strong field.5.5.2 Chiral-Symmetry-Breaking DisorderDisorder that breaks chiral symmetry will inevitably be present in real graphenesamples. Such disorder tends to broaden LL0 and compete with the interaction effectsthat underlie the SY physics. It is known that bilinear terms H2 that arise from suchdisorder constitute a relevant perturbation to HSY and drive the system towards adisordered Fermi liquid (dFL) ground state. In Appendix E we analyze the symmetry-breaking effects and estimate their strength in realistic situations. We conclude thatin carefully prepared samples a significant window should remain open at non-zerotemperatures and frequencies in which the system exhibits behavior characteristic ofthe SY model.5.6 Prospect of Experimental RealizationAn ideal sample to observe the SY physics is a graphene flake with a highly irregularboundary and clean interior. These conditions promote random spatial structure ofthe electron wavefunctions and preserve degeneracy of LL0. Disordered wavefunctionsgive rise to random interaction matrix elements Jij;kl while near-degeneracy of statesin LL0 guarantees that the two-fermion term H2 remains small. To observe signatures92of the emergent black hole the LL0 degeneracyN = SB/Φ0 must be reasonably large numerical simulations indicate that N & 10 is required for the system to start showingthe characteristic spectral features. Aiming at N ' 100, which is well beyond whatone can conceivably simulate on a computer, implies the characteristic sample size L '√S =√NΦ0/B ' 150 nm at B = 20 T. Signatures of the SY physics can be observedspectroscopically, e.g. by the differential tunneling conductance g(V ) = dI/dV whichis predicted [169] to exhibit a characteristic square-root divergence g(V ) ∼ |V |−1/2 inthe SY regime at large N , easily distinguishable from the dFL behavior g(V ) ∼ constat small V . We predict that a tunneling experiment will observe the SY behaviorwhen the chemical potential of the flake is tuned to lie in LL0 and dFL behavior forall LLn with n 6= 0. We also expect the two-terminal conductance across the flake toshow interesting behavior in the SY regime [172].In the limit of a large flake the irregular boundary will eventually become unimpor-tant for the electrons in the bulk interior and the system should undergo a crossoverto a more conventional clean phenomenology characteristic of graphene in appliedmagnetic field. The exact nature of this crossover poses an interesting theoretical aswell as experimental problem which we leave to future study.93Chapter 6ConclusionThis Thesis shows that topological materials can be testing grounds for high-energyphysics ideas, especially those that may be keys to unsolved fundamental problems.This alternative approach for probing high-energy physics is an appealing opportunitybecause direct observation or detection is often difficult. Specifically, we illustratedthree scenarios in which high-energy physics ideas are realized in topological semimet-als. In Chapter 3, we proposed a superconductor-Weyl-semimetal-superconductorjunction that can host Majorana flat bands at pi-phase difference. Majorana flatbands are ideal for studying the interaction effects of Majorana fermions, whichare favorable candidates for solving some crucial mysteries in the Standard Model.We further demonstrated that Josephson current measurement is a promising andexperimentally feasible venue for detecting these flat bands. In Chapter 4, we in-vestigated chiral anomaly present in Weyl semimetals subject to a combination ofelectomagnetic fields and mechanical strains. One key feature is the anomalousnonconservation of electron charges inside the material bulk, balanced out by thesurface charges. Such phenomenon does not happen in the realm of high-energyphysics since the Universe has no boundary. It is fascinating that solid-state systemscan reveal more properties of a well-established high-energy concept. Furthermore,94in quantitative details, we pictured several novel experimental manifestations whichcould have potential technological applications. In Chapter 5, we provided evidencesthat a mesoscopic graphene flake subject to a perpendicular magnetic field gives riseto the SY model, a strongly correlated fermionic quantum dot holographically dual toa (1+1)-dimensional black hole. To our current knowledge, our proposal is the mostexperimentally feasible testing ground for the holographic conjecture.Throughout the Thesis, we leaned heavily on numerical simulations to bring ourtheoretical proposals closer to experimental reality. The effects of realistic laboratoryconditions  such as finite temperatures and impurities  were gauged by numeri-cal analysis. Nevertheless, numerical approach is limited by computational powerand simplifying assumptions in the theories. Real solid-state systems are extremelycomplex, and laboratory environments difficult to control. It is hard to predictwhat actually happens when one attempts the proposed experiments. Therefore torealize our proposals we must work closely with the experimentalists. Thus far ouron-going discussions with experimental groups have already refined our interpretationof results. Once realized, these experiments could either confirm the physics we knowor challenge our current understanding through unexpected findings.Our work also begets more theoretical investigations. How can the interactionsamong Majorana zero modes be explicitly modeled? Would strain-induced chiralanomaly in the presence of interactions give rise to exotic many-body states? FromChapter 5, it seems that a lot more can be done along the line of experimentalholographic principle. Can we find quantum matter systems that are holographicallydual to black holes in higher dimensions, preferably the (3+1)-dimensional spacetimethat we live in? What about other mysterious gravitational objects such as thewormholes [180, 181]? From the reverse perspective, how can we better understandstrongly-correlated systems such as the high-Tc superconductors using the holographic95principle? These open questions may just lead us one step closer to unifying thedisjointed worlds of high-energy and quantum matter physics.96Bibliography[1] C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal, New J. Phys. 16,063065 (2014).[2] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78, 195424 (2008).[3] B. A. Bernevig and T. L. Hughes, Topological Insulators and TopologicalSuperconductors (Princeton University Press, Princeton, 2013).[4] N. Read and D. Green, Phys. Rev. B 61, 10267 (2000).[5] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).[6] X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 82, 155138 (2010).[7] M. Van Raamsdonk, Gen. Rel. Gravit. 42, 2323 (2010).[8] X. Lin, R. Du, and X. Xie, Natl. Sci. Rev. 1, 564 (2014).[9] M. Z. Hasan, S.-Y. Xu, and G. Bian, Phys. Scr. T164, 014001 (2015).[10] A. M. 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Aharonov and A. Casher, Phys. Rev. A 19, 2461 (1979).109[179] W. Fu and S. Sachdev, Phys. Rev. B 94, 035135 (2016).[180] J. Maldacena, D. Stanford, and Z. Yang, Fortschritte der Phys. 65, 1700034(2017).[181] J. Maldacena and X.-L. Qi, ArXiv e-prints (2018), 1804.00491.110AppendicesAppendix A: Tight-Binding Model, Dispersion Rela-tions, and Parameters for Cd3As2 and Na3BiFrom the low energy k · p Hamiltonian (4.5) we construct the requisite lattice modelby replacing Ak± → (A/a)(sin akx ± i sin aky), C1k2z → (2C1/a2)(1 − cos akz), etc.For example 0(k) defined below Eq. (4.5) becomesk = c0 + c1 cos akz + c2(cos akx + cos aky), (A.1)with c0 = C0 + 2(C1 + 2C2)/a2, c1 = −2C1/a2 and c2 = −2C2/a2. The constants cjare chosen such that k matches 0(k) for small ak independent of the chosen valueof the lattice constant a. Treating all other terms in the Hamiltonian (4.5) in thesimilar fashion leads to the lattice Hamiltonian given by Eqs. (4.7) and (4.8).In addition to the results presented in the main text, we performed detailedbandstructure simulations for Dirac semimetals Cd3As2 and Na3Bi. Parameters forthe model Hamiltonian (4.5) are taken from [32, 121] and [139] correspondingly andare summarized in Table A.1. In the main text we only presented results for theparameters of Cd3As2 with the asymmetry parameters Ci set to zero. In the Fig.A.1 we present the dispersion relation computation for the models of12-Cd3As2 and12-Na3Bi with all the asymmetry terms taken into account. The effects discussed in111the main text are present even in this more general case although to see them clearlynow requires more effort due to the more complicated structure of the energy bands.For instance the equivalence of the torsional strain and magnetic field, pointed out inthe main text, here can be only identified by a trained eye. One needs to notice thatthe right Weyl point is at E = 0 in the rightmost graph of the first column of Fig.A.1. Results between the two parameter sets are similar, but notice the larger gapsin Na3Bi, which may make the experimental realization easier.Cd3As2 Na3BiC0 [eV] -0.0145 -0.0638C1 [eVÅ2] 10.59 8.75C2 [eVÅ2] 11.5 -8.4M0 [eV] 0.0205 0.869M1 [eVÅ2] -18.77 -10.64M2 [eVÅ2] -13.5 -10.36A [eVÅ] 0.889 2.46a [Å] 20 7.5Table A.1: Material parameters taken from Refs. [121] and [139]. The last rowrepresents the effective lattice constant used for our numerical simulations.112Figure A.1: Dispersion relations for the spin-up sector of the lattice Hamiltonian (4.7)describing Cd3As2 (top row) and Na3Bi (bottom row). The parameters used in thesimulations include the particle-hole symmetry breaking terms and are summarizedin Table A.1. We used a lattice with 40 × 40 sites and the magnetic fields shown inthe green boxes for each of the material. Notice the different magnitude of effectivemagnetic fields for different compounds  this is due to the different lattice constants,and different sign of the physical magnetic field between the two rows. Different signof magnetic fields shows that the physical magnetic field compensates the torsionalone in opposite Weyl points for opposite directions of magnetic field in accordancewith the interpretation in the main text.To further confirm the validity of our ideas relating the torsional strain to thepseudomagnetic field we computed the equilibrium currents flowing along the wireusing the full Hamiltonian including the p-h symmetry breaking terms. The resultsare as follows: (i) For both Cd3As2 and Na3Bi we find equilibrium currents with thepattern similar to the one displayed in Fig. 4.8 in each spin sector when nonzerotorsion is present. The total current density (summing up contributions from bothspin up and down sectors) vanishes, as it must be in a T -invariant system. (ii) Whenonly magnetic field B is present and no torsion, the current densities are zero in bothsectors separately, in accord with the expectation. (In this case the band structurein each spin sector shows the same number of left and right moving modes in the113bulk of the system). (iii) When both torsion and magnetic field are present then wefind non-zero persistent current density in both spin sectors. In this case T is absentand the currents from the two sectors generically do not cancel. This is illustrated inFig. A.2. We observe an asymmetric band structure that supports different numberof bulk left and right moving modes at various energies, leading to a net imbalancein the current flow between the bulk and the surface. The total current carried bythe wire however still vanishes.-1 0 1k-20-1001020E [meV]a bxy0 10 20 303020100Figure A.2: Persistent currents in a Cd3As2 nanowire under torsion and magneticfield. a) Band structure detail for spin up (blue) and spin down (green) sectors in a30×30 lattice with B = b = 3.2T and other parameters as in Fig. A.1. b) Calculatedcurrent density jz for µ = 0 including contributions from both spin sectors.114Appendix B: Nonequilibrium Distribution in a StretchedWeyl SemimetalIn this appendix, we derive a quantitative estimate for the chemical potential denotedby µ′eq in Fig. 4.5 as well as the corresponding bulk electron density. As discussedin Sec. 4.5.2, upon introducing strain the chemical potential in the bulk of thesystem rises from µ0 to µ′while that at the surfaces remains unchanged. In theprocess, the charge density also remains unchanged (except possibly for perturbationsnear the surface that average to zero and do not affect the bulk). This creates anonequilibrium distribution of electrons illustrated in Fig. 4.5b which then relaxes toa new equilibrium characterized by a global chemical potential µ′eq. The latter canbe calculated by demanding that the total electron number N is conserved.In the unstrained system we have N = Ns + Nb where the subscripts refer tothe surface and the bulk, respectively. In the nonequilibrium strained system Nsand Nb remain the same as per our discussion above. In the new equilibrium theychange to N ′s/b = Ns/b + δNs/b, where δNs/b ' κs/bδµs/b. Here κs/b = dNs/b/dµ isthe compressibility and δµs/b denotes the change in the chemical potential that isresponsible for the change in Ns/b. Number conservation dictates that δNs = −δNbwhich impliesκs(µ′eq− µ0) = −κb(µ′eq− µ′). (B.1)We can solve for µ′eq to obtainµ′eq= µ0 +δµ1 + κs/κb, (B.2)115where δµ = µ′−µ0. The corresponding change in the bulk number is δNb = κb(µ′eq−µ′)which, together with Eq. (B.2) gives the change in the bulk densityδρbulk = − 1Sdκb1 + κb/κsδµ, (B.3)where S is the area of the slab.To complete the calculation we require the surface and bulk electron compress-ibilities. For the surface we assume that we have a single linearly dispersing bandsk = ~v kx on each surface that extends between the surface projections of the twoWeyl points, |kz| < Q. We furthermore assume that the surface state is essentiallyunaffected by the magnetic field, in accord with the results of out lattice simulations.This givesκs = SDs(µ) =SQpi2~v, (B.4)where Ds() = Q/pi2~v is the surface density of states (counting both surfaces).For the bulk we similarly have κb = SdDb(µ). We now must distinguish betweenthe quantum and the semiclassical limits, as defined in Sec. 4.6.2. In the quantumlimit we have a pair of linearly dispersing n = 0 Dirac Landau levels with degeneracy(B/Φ0) whereas in the semiclassical limit many Landau levels are populated so it ispermissible to approximate the density of states by that of a zero-field system. Wethus obtainκb =Sdpi~v(BΦ0)quantum limitSdµ2pi2~3v3 semiclassical limit.(B.5)Substituting these results into Eqs. (B.2) and (B.3) we obtain results for µ′eq andδρbulk quoted in the main text (Eqs. 4.20 and 4.21) for the quantum limit. In thesemiclassical limit, we similarly obtainµ′eq= µ0 +δµ1 + λQ/d(B.6)116andδρbulk = − αpia(BΦ0)cot aQ1 + d/λQ, (B.7)where λQ = Q(~v/µ)2/2 is the characteristic length.117Appendix C: Hydrodynamic Flow in a Twisted WeylNanowireConsider a cylindrical nanowire of radius R made of a Weyl semimetal. Both torsionand electric field E are applied along the axis of the wire (taken here along the zˆdirection), giving a non-zero right hand side ∝ b ·E in the second anomaly equation(4.2). Denoting the right hand side by g(r), we may write∂tρ+∇ · j = g(r), (C.1)whereg(r) = g0[θ(R− r)− 12Rδ(r −R)], (C.2)with g0 = (e2/2pi2~2c)bE. The first term in g(r) describes uniform production ofelectrons in the bulk of the wire at a rate given by the chiral anomaly. The secondterm reflects the fact that those electrons are removed from the surface, in accord withour discussion in Sec. 4.5. The total production in the wire is zero,∫ R+0rdrg(r) = 0,and the charge is conserved.We now assume that the dominant relaxation mechanism for the nonequilibriumelectrons produced in the bulk of the wire is diffusion towards the boundary. Electronsmove ballistically along the zˆ direction and undergo occasional collisions that scatterthem into neighboring Landau level states. Near the boundary bulk electrons canfinally backscatter into the surface modes which are moving in the opposite direction.Under this assumption the diffusion current isj = −D∇ρ, (C.3)118where D = `2b/τ0 is the diffusion constant (`b =√~c/eb is the magnetic length andτ−10 the microscopic scattering rate). The form of the diffusion constant reflects thefact that electron wavefunctions have Landau level character with the spatial extent`b in the direction perpendicular to zˆ and scattering occurs predominantly betweenneighboring Landau level orbitals.Substituting Eq. (C.3) into (C.1) and specializing to long time steady state with∂tρ = 0, we obtain−D∇2ρ = g(r). (C.4)Writing the Laplacian in the polar coordinates and assuming radially symmetricsolution, we findρ(r) =g0D(R2 − r24)θ(R− r). (C.5)The corresponding radial diffusion current density is jr(r) = −D∂rρ = 12g0r. Thetotal non-equilibrium charge in the bulk modes isδQ = −e∫ R0dr2pirρ(r) = −epi8g0DR4. (C.6)Since all these modes move in the same direction with velocity v this gives the totalcurrent along the zˆ direction in the wire,JCTE= 2vδQ = −evpi4(e22pi2~2c)τ`2bR4bE. (C.7)The factor 2 in the first equality reflects the fact that non-equilibrium charge −δQmust exist in the surface left moving modes to maintain overall charge neutrality. Weassume for simplicity that these modes move at the same speed v.As mentioned, the transport current along the wire exhibits all the characteristicsof the hydrodynamic flow: it is largest at the center and vanishes at the boundary.This is because momentum can only be relaxed by electrons that have reached the119boundary and can scatter into surface modes. The amount of current through thewire scales with R4, just like fluid flowing through a pipe.From Eq. (C.7) one can read off the chiral torsional conductivity σCTE which canbe written suggestively in the following way:σCTE=e2v4pihτN , (C.8)where N = piR2/`2b is the number of chiral bulk modes in the wire and τ = τ0R2/`2bis the effective transport scattering time. The form of the latter reflects the fact thatunder diffusion the electron produced near the center of the wire must scatter onaverage (R/`b)2times before it reaches the boundary.To illustrate this point we have performed simulation of conductance in a disor-dered symmetric12-Cd3As2 model. The Hamiltonian parameters are the same as usedin Fig. 4.4. We have performed the conductance simulations for µ = 5meV and for thesystem of W ×W × 20 sites. We have added on-site disorder δµi taken from normaldistribution of width 10meV to simulate the hydrodynamic flow described above. Theratio of conductance of disordered system to the conductance of the clean system isplotted in Fig. C.1. Best fit to the data is in accordance with (C.8), where τ ∝ R2.120Figure C.1: Ratio of conductance of a disordered W × W × 20 system to theconductance of the clean system averaged over 100 disorder realizations. Green lineis the best fit to the data  parabolic, grey curves show the failure of the linear (withnon-negative G(0)) and cubic fits.121Appendix D: Exchange Splitting and Interaction Ma-trix Between Zero ModesHere we discuss the enhancement of the Zeeman splitting due to the exchange inter-action, derive the form of coupling constants Jij;kl quoted in Eq. (5.4) and estimatethe characteristic interaction strength J .General ConsiderationsWe begin by writing the Hamiltonian for the electrons in graphene as H = H0 +HintwhereH0 = −∑〈r,r′〉,σtrr′f†rσfr′σ +g∗µBB2∑r(ρr↑ − ρr↓), (D.1)Here f †rσ creates an electron with spin σ on the site r of the honeycomb lattice,trr′ = t exp [−i(e/~c)∫ r′rA · dl] is the hopping integral in the presence of the magneticfield B = ∇×A and ρrσ = f †rσfrσ. Interactions are described byHint =12∑r,r′ρrV (r − r′)ρr′ , (D.2)where ρr = ρr↑+ ρr↓ represents the total charge on site r and V (r) = (e2/r)e−r/λTFis the screened Coulomb potential.Our strategy is to first solve the non-interacting problem defined by H0 on a flakewith an irregular boundary. This yields a set of single-particle energy levels j and thecorresponding eigenstates Φj(r). As discussed in Sec. 5.3, the energy levels consistof bulk Landau levels and edge modes. The Zeeman term simply offsets the spin-upbands by ∆ES(B) = g∗µBB with respect to spin-down bands.Next we write the interaction term Hint in the basis defined by the eigenstatesΦj(r). If c†jσ creates a particle with spin σ in eigenstate Φj(r) we have ρr =122∑i,j,σ Φ∗i (r)Φj(r)c†iσcjσ. Substituting into Eq. (D.2) and rearranging we findHint =∑i,j,k,l∑σ,σ′Jij;klc†iσc†jσ′ckσ′clσ, (D.3)where Jij;kl is given by Eq. (5.4).Henceforth we focus on the states belonging to LL0, that is, we consider electrondensities such that all Landau levels with negative energies are filled, while LL0 ispartially filled. Given the LL degeneracy N = SB/Φ0 per spin we define the totalnumber of LL0 electronsNF such thatNF = 0 andNF = 2N correspond to completelyempty or filled LL0, respectively. Because higher LLs are separated by an energy gap,for sufficiently weak interactions we can disregard virtual transitions into these bandsand project Hint onto LL0 by simply restricting all indices (i, j, k, l) in Eq. (D.3) tothose labeling eigenstates Φj in LL0.Exchange SplittingWe expect electrons to occupy LL0 in such a way as to maximize the total spin Stotwith Sztot aligned with the field. Such a state will minimize the Zeeman energy as wellas the Coulomb repulsion due to the exchange effect. The latter arises because whenthe spin part of the many-body electron wavefunction is symmetric in spin degrees offreedom the spatial part must necessarily be antisymmetric. This forces Ψ(r1, r2, . . . )to vanish whenever two electron positions coincide, which tends to minimize theshort-range part of the Coulomb repulsion energy. While the Zeeman splitting is easyto determine (see Sec. 5.5), estimation of the exchange splitting magnitude for NFfermions described by Eq. (D.3) is a non-trivial task. This is because couplings Jij;klare all-to-all and essentially random. To get an idea about the expected magnitudeof the exchange splitting we consider below a simple case of N = NF = 2.123For two electrons the position space wavefunction can be either symmetric orantisymmetric under exchange depending on the spin state,Ψ±(r1, r2) =1√2[Φ1(r1)Φ2(r2)± Φ1(r2)Φ2(r1)] . (D.4)The corresponding Coulomb energy isE±C =∑r1,r2|Ψ±(r1, r2)|2V (r1 − r2). (D.5)The exchange splitting, then, becomes simply ∆EC = E+C − E−C and reads∆EC = 2∑r1,r2Re [Φ∗1(r1)Φ2(r1)V (r1 − r2)Φ∗2(r2)Φ1(r2)] (D.6)In order to estimate ∆EC from Eq. (D.6) we make an assumption, motivated byour extensive numerical work, that on lengthscales larger than the magnetic lengthlB wavefunctions Φj(r) behave as random uncorrelated variables. We thus coarsegrain the wavefunctions on a grid with sites denoted by R and spacing lB. Thecoarse-grained wavefunctions Φj(R) are then treated as complex-valued independentrandom variables withΦj(R) = 0, Φ∗i (R)Φj(R′) =1MsδijδRR′ . (D.7)Overbar denotes averaging over independent realizations of the flake geometry. Thesecond equality in Eq. (D.7) follows from the normalization of Φj andMs = S/l2B = 2piN (D.8)denotes the number of grid sites in the flake.124With this preparation we now recast Eq. (D.6) as a sum over the coarse grainedgrid,∑r1,r2→ ∑R1,R2 and Φj(r) → Φj(R). Using Eq. (D.7) we then obtain anestimate for the typical exchange splitting∆EC ' 2M2s∑R1,R2δR1R2V (R1 −R2) =2MsV (0). (D.9)Here V (0) must be interpreted as the average Coulomb potential in a grid patchof the size lB, that is V (0) ' (1/pil2B)∫ lB0V (r)2pirdr = 2e2/lB, where we assumedλTF  lB. Taking the dielectric constant  = 2 and N = 2 we find the typicalexchange splitting ∆EC ' 8.8 meV/T. We expect this result to remain at leastapproximately valid for N > 2. Therefore, when NF < N , electrons will fill thespin-down states of LL0 with empty spin-up states separated in energy by a significantexchange gap. The physics of such partially filled spin-down LL0 can be described bythe Hamiltonian (D.3) with σ = σ′ =↓ which is precisely the SY Hamiltonian.Coupling Strength JTo estimate the typical strength of couplings Jij;kl that enter the SY Hamiltonian itis useful to first recast Eq. (5.4) such that it is explicitly antisymmetric in indices(i, j) and (k, l)Jij;kl =12∑R1,R2Ω∗ij(R1,R2)V (R1 −R2)Ωkl(R1,R2), (D.10)where Ωij(R1,R2) =12[Φi(R1),Φj(R2)]. We also passed to the coarse-grained vari-ables, as described above. With help of Eq. (D.7) it is straightforward to show thatJij;kl = 0 and|Jij;kl|2 = 1M3s∑R 6=0V (R)2. (D.11)125The sum can be approximated by an integral,∫ ∞lB22piRdRl2B(e2e−R/λTFR)2=(e2lB)22piΓ(0,lBλTF), (D.12)where Γ(0, x) =∫∞xdy e−y/y is the incomplete gamma function. Combining with Eq.(5.5) we thus obtain an estimateJ ' 2(e2lB)(NMs)3/2√piΓ(0,lBλTF). (D.13)For  = 2 this amounts toJ ' 6.04 meV√B[T] Γ(0,lBλTF). (D.14)For x = lB/λTF  1, which is the limit of interest, Γ(0, x) ' ln(1/x) so J is onlyvery weakly dependent on the screening length. For B = 20 T and λTF/lB = 4 weobtain J ' 25 meV.It is to be noted that our numerical calculations of Jij;kl described in Sec. 5.3[discussion below Eq. (5.5)] give larger values of J than the above estimate, insome cases by as much as an order of magnitude. The discrepancy is most likelyattributable to the fact that LL0 wavefunctions are in fact disordered on a somewhatlonger lengthscale than lB. This would modify the relation between Ms and N givenby Eq. (D.8) and increase the ratio (N/Ms) that enters the estimate for J in Eq.(D.13). We may therefore regard Eq. (D.13) as a conservative lower bound on theexpected magnitude of J . This is already a large energy scale which should make themanifestations of the SY physics experimentally observable at low temperatures inclean graphene flakes.126Appendix E: Symmetry Breaking Perturbations in RealGrapheneTo ascertain the experimental feasibility of our proposal, we discuss the effect ofvarious chiral symmetry breaking perturbations that exist in real graphene. Suchperturbations tend to broaden LL0 and can be modeled by a bilinear term H2 definedby Eq. (5.3). The matrix elements areKij =∑rΦ∗i (r)H′(r)Φj(r), (E.1)where H ′ denotes the Hamiltonian of the perturbation. The strength of these pertur-bations is measured by parameter K defined asK2 = N(|Kij|2 − |Kij|2). (E.2)It is known that since H2 is a relevant perturbation to HSY (in the renormalizationgroup sense) the ground state of the system becomes a (disordered) Fermi liquid forany nonzero K. Nevertheless, if K  J , a significant crossover region can existat finite frequencies and temperatures in which the system behaves effectively asa maximally chaotic SY liquid. According to the analysis of Ref. [169] the zero-temperature propagator of the system with both K and J nonzero exhibits the SYconformal scaling G(ω) ∼ |ω|−1/2 for frequencies satisfying16√piK2/J < ω  J. (E.3)In the following we consider two specific perturbations that are present in real graphene,the second neighbor hopping t′ and random on-site potential. Both break the chiralsymmetry χ and produce non-zero parameter K. We derive limits on the admissible127strength of these perturbations based on the requirement that Eq. (E.3) yields asignificant window in which SY behavior can be observed.Second-Neighbor HoppingWe first consider second neighbor hopping with the Hamiltonian acting asH ′(r)Φj(r) =t′∑a Φj(r + a). Here a denotes the 3 second neighbor vectors in the honeycomblattice. Since |a|  lB we find, upon coarse graining the sum in Eq. (E.1),Kij ' 3t′∑RΦ∗i (R)Φj(R). (E.4)With help of Eq. (D.7) it is straightforward to show thatKij ' 3t′δij, |Kij|2 ' 9t′2(δij +M−1s). (E.5)From Eq. (E.2) we get K ' 3t′√N/Ms ' 3t′/√2pi independent of the field.Experimentally reported values of t′ range between[24] 1-3% of t which wouldproduce a rather large broadening of LL0 in real graphene, K ' 30− 90 meV. On theother hand existing experiments [177] indicate much smaller broadening of Landaulevels in graphene of at most several meV which also includes broadening due toimpurities and other defects. We therefore conclude that the above method mustseverely overestimate the contribution of second neighbor hopping to parameter K.This conclusion is supported by our numerical results presented below.The numerically computed energy spectrum of the graphene flake with secondneighbor hopping t′ = 0.037t is displayed in Fig. E.1a. We observe that while LL0is now significantly shifted away from zero energy it remains sharp and well defined.The overall upward shift of LL0 by about 0.25 eV is consistent with the estimate givenin Eq. (E.5) which implies Kij ' 0.30 eV. The broadening induced by t′ is quantifiedin Fig. E.1b and is well approximated by a linear dependence K ' 0.022t′. This is128about a factor of 50 smaller than the estimate implied by Eq. (E.5). For t′ = 0.02t weobtain K ' 1.2 meV, a result that is much more in line with the experimental data.abLL0LL1LL-1*Figure E.1: Effects of the second neighbor hopping t′. a) Single-particle energyspectrum of a flake (the same geometry as Fig. 5.2) with second neighbor hoppingt′ = 0.037t. b) Average shift δ = Kij and standard deviation K of 40 energy levelsthat comprise LL0 as a function of t′.The discrepancy between the analytical estimate and the numerical result canbe understood as follows. In a large, disorder-free sample of graphene, inclusion ofthe second neighbor hopping produces changes in the band structure (and thus theposition and spacing of LLs) but does not give rise to any LL broadening as longas t′ remains spatially uniform. The sharpness of LLs is protected by translationalinvariance, not the chiral symmetry. In our mesoscopic flake we see that the inclusion129of a spatially uniform t′ primarily shifts the position of LLs, as expected from theargument given above. Because randomness is present in the system due to itsirregular geometry some broadening occurs. This broadening is, however, muchweaker than what is predicted by the naive estimate.Random On-Site PotentialRandom on-site potential is implemented by taking H ′(r) = w∑r′∈I δrr′ , where Idenotes a set of randomly chosen sites with number density nI in the graphene latticeand w controls the disorder potential strength. Substituting H ′ into Eq. (E.1) leadsto the same result as indicated in Eq. (E.5) with 3t′ replaced by wnI . We thereforeexpect an overall energy shift of LL0 by wnI accompanied by a broadeningK ' wnI√NMs. (E.6)Fig. E.2a shows the numerically computed energy eigenvalues as a function ofw for a flake with N = 40 flux quanta and nI = 1%. We observe that LL0is shifted upward as well as broadened with increasing disorder strength. Thisshift δ and broadening K are quantified in Fig. E.2b. At small w these satisfyδ ' 0.8nIw and K ' 1.3nIw while at larger values the dependence is no longerlinear, presumably because the system enters a non-perturbative regime when wbecomes comparable to the bandwidth. We see that the numerically obtained shiftin LL0 is well aligned with the analytical estimate. The broadening K also agreesif we take√N/Ms ' 1.3 (instead of 1/√2pi ' 0.4 implied by Eq. (D.8)). Thisresult reinforces the conclusion, reached in Appendix D by comparing the interactionstrength estimate to the numerical calculation, that the zero mode wavefunctiondisorder scale is somewhat longer than lB.130abFigure E.2: Effects of random on-site potential. a) Low-energy part of thenumerically calculated energy spectrum for the flake with nI = 1% of defected sitesas a function of the disorder potential strength w and N = 40. b) Average shiftδ = Kij and standard deviation K of 40 energy levels that evolve from the zeromodes which comprise LL0 in the pure sample. These levels are marked in red inpanel (a).We finally remark that in the above example 40 flux quanta through a flake with1952 carbon atoms correspond to an unrealistically high magnetic field of ∼ 3200T. Such high fields are needed for us to be able to numerically simulate meaningfulnumber of zero modes N with available computational resources. To make a closercontact with experiment we may however reinterpret these results by viewing thehoneycomb lattice not as the atomic carbon lattice but as a convenient regularizationof the low energy theory of Dirac electrons in graphene. In such low energy theorythe only important parameter is the Dirac velocity vF =32ta ' 1.1 × 106 m/s.The velocity is clearly unchanged if we rescale the lattice constant a → λa and the131tunneling amplitude t → t/λ with λ an arbitrary positive parameter. Under therescaling B → B/λ2 and all energy parameters defined through t are changed asE → E/λ. Thus, if we take λ = 10 in the above example we get a more reasonablefield B = 32 T. According to Eq. (D.14) this corresponds to J ' 34 meV. Eqs. (E.6)and (E.3) then stipulate an upper bound on the disorder strength nIw  9 meV.Clearly, like fractional quantum Hall effect and other exotic phases driven byinteractions, observing the SY physics will require high fields, low temperatures andcarefully prepared graphene flake with an irregular boundary and clean interior.132

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