Topological Band Theory I. Introduction - PowerPoint Presentation

1. Topological Band TheoryI. Introduction - Insulating State, Topology and Band Theory II. Band Topology in One Dimension - Berry phase and electric polarization - Su Schrieffer Heeger model : domain wall states and Jackiw Rebbi problem - Thouless Charge PumpIII. Band Topology in Two Dimensions - Integer quantum Hall effect - TKNN invariant - Edge States, chiral Dirac fermionsIV. Symmetry Protected topological band insulators. - Z2 Topological Insulators - Topological Superconductors - Ten fold way

2. Organizing Principles for Understanding MatterSymmetryTopologyInterplay between symmetry and topology has led to a new understanding of electronic phases of matter.Conceptual simplificationConservation lawsDistinguish phases of matter by pattern of broken symmetriesProperties insensitive to smooth deformationQuantized topological numbersDistinguish topological phases of matter genus = 0genus = 1symmetry group p31msymmetry group p4

3. Topology and Quantum PhasesTopological Equivalence : Principle of Adiabatic ContinuityQuantum phases with an energy gap are topologically equivalent if they can be smoothly deformed into one another without closing the gap.Topologically distinct phases are separated by quantum phase transition.Topological Band TheoryDescribe states that are adiabatically connected to non interacting fermionsClassify single particle Bloch band structuresEg ~ 1 eVBand Theory of Solidse.g. SiliconEadiabatic deformationexcited statestopological quantumcritical pointGapEGGround state E0EkMany body energy spectrumSingle particle energy spectrum

4. Topological Electronic PhasesMany examples of topological band phenomenaStates adiabatically connected to independent electrons: - Quantum Hall (Chern) insulators - Topological insulators - Weak topological insulators - Topological crystalline insulators - Topological (Fermi, Weyl and Dirac) semimetals …..Many real materialsand experimentsBeyond Band Theory: Strongly correlated statesState with intrinsic topological order - fractional quantum numbers - topological ground state degeneracy - quantum information - Symmetry protected topological states - Surface topological order ……Much recent conceptual progress, but theory isstill far from the real electronsTopological SuperconductivityProximity induced topological superconductivityMajorana bound states, quantum informationTantalizing recent experimental progress

5. Band Theory of SolidsBloch Theorem : Band Structure : Ekxp/a-p/aEgap=kxkyp/ap/a-p/a-p/aBZLattice translation symmetryBloch HamiltonianA mapping

6. Berry PhasePhase ambiguity of quantum mechanical wave functionBerry connection : like a vector potentialBerry phase : change in phase on a closed loop CBerry curvature : Famous example : eigenstates of 2 level HamiltonianCS

7. RTopology in one dimension : Berry phase and electric polarizationClassical electric polarization :+Qend-Qend1D insulatorQuantum polarization : a Berry phaseBZ = 1D Brillouin Zone = S1Bloch states are defined for periodic boundary conditionsDefine localized Wannier States :Wannier states associated with R are localized, but gauge dependent.k-p/ap/a0

8. Gauge invariance and intrinsic ambiguity of P The end charge is not completely determined by the bulk polarization P because integer charges can be added or removed from the ends : The Berry phase is gauge invariant under continuous gauge transformations, but is not gauge invariant under “large” gauge transformations. Changes in P, due to adiabatic variation are well defined and gauge invariantwhenwithk-p/ap/al10CSgauge invariant Berry curvature

9. Su Schrieffer Heeger Modelmodel for polyacetylenesimplest “two band” modelad(k)d(k)dxdydxdyE(k)kp/a-p/aProvided symmetry requires dz(k)=0, the states with dt>0 and dt<0 are distinguished byan integer winding number. Without extra symmetry, all 1D band structures are topologically equivalent.A,iB,idt>0 : Berry phase 0P = 0dt<0 : Berry phase pP = e/2Gap 4|dt|Peierls’ instability → dtA,i+1

10. “Chiral” Symmetry :Reflection Symmetry :Symmetries of the SSH model Artificial symmetry of polyacetylene. Consequence of bipartite lattice with only A-B hopping: Requires dz(k)=0 : integer winding number Leads to particle-hole symmetric spectrum: Real symmetry of polyacetylene. Allows dz(k)≠0, but constrains dx(-k)= dx(k), dy,z(-k)= -dy,z(k) No p-h symmetry, but polarization is quantized: Z2 invariant P = 0 or e/2 mod e

11. Topological Boundary Modes0EMany body ground state:m(x)y0(x)ABBoundary betweentopologically distinct insulatorsTopological boundarymodesSu Schrieffer Heeger Model (1979)Polyacetalene (CH)nJackiw Rebbi Model (1976)Single Particle Spectrum:Topological zero energy bound stateCharge fractionalization:Splitting the indivisibleAABe/2e/2chiral symmetry:

12. Thouless Charge Pumpt=0t=TP=0P=ek-p/ap/a t=Tt=0=The integral of the Berry curvature defines the first Chern number, n, an integer topological invariant characterizing the occupied Bloch states,In the 2 band model, the Chern number is related to the solid angle swept out bywhich must wrap around the sphere an integer n times.The integer charge pumped across a 1D insulator in one period of an adiabatic cycle is a topological invariant that characterizes the cycle.

13. TKNN InvariantThouless, Kohmoto, Nightingale and den Nijs 82Physical meaning: Quantized Hall conductivitykxkyC1C2p/ap/a-p/a-p/aFor 2D band structure, defineLaughlin argument: Thread flux DF = h/eAlternative calculation: compute sxy via Kubo formulaEIne-neDF=BZThouless pump: Cylinder with circumference 1 lattice constant (a)(Faraday’s law)F plays role of ky

14. Realizing a non trivial Chern numberInteger quantum Hall effect: Landau levelsChern insulator: e.g. Haldane model Band Inversion ParadigmEEgkChern BandC=1kEkEconductionbandvalenceband

15. Lattice model for Chern insulator|DEsp| > 4t : Uninverted Trivial Insulatord(k)dzdxdydzdxdyd(k)Chern number 0Chern number 1Square lattice model with inversion of bands with s and px+ipy symmetry near G|DEsp| < 4t : Inverted Chern InsulatorRegularized continuum model for Chern insulatorInverted near k=0 for m<0. Uninverted for  m = 4t0-DEspa = t0 av = 2tsp akEconductionband tz = +1valenceband tz = -1  p+ips

16. Edge StatesGapless states at the interface between topologically distinct phasesIQHE staten=1EgapDomain wallbound state y0Vacuumn=0Edge states ~ skipping orbitsLead to quantized transportBand inversion transition : Dirac EquationkyE0xyChiral Dirac Fermionsm<0m>0 n=1m = -m0 n=0m = +m0

17. Chiral Dirac FermionSingle particle picture:Bulk – Boundary correspondence # chiral edge modes = D (Chern Number)“One way” propagation perfect transmission, responsible for quantized conductance insensitive to disorder, impossible to localizeDoubling Theorem Chiral Dirac fermions can not exist in purely 1D systemEkx0conduction bandvalence bandp/a-p/aEFInsulatorQuantum HallState

18. Quantizd Response: Quantized electrical conductance Quantized thermal conductance: chiral centralchargeMany-body edge spectrum : “chiral Fermi liquid”Free Dirac fermion conformal field theory Neutral excitations: particle-hole pairs ~ chiral phonons Charged excitations: electron charge e kEqEqvqChiral Anomaly :In the presence of electric field, edge charge is not conserved

19. 1D electrical conductorLow energy excitations: Right/Left moving “chiral fermions” Localized by commensurate periodic potential or disorder. Split the 1D chiral modes in a 2D insulator: “wire construction”Trivial InsulatorTrivial InsulatorTopological “Chern” insulatoraka quantum Hall stateseparatedchiralboundary modesRL

20. Symmetry Protected Topological Band InsulatorsThe presence of symmetry can rule out some topological states, but canalso introduce new onesTime Reversal Symmetry Chern number n=0 Z2 topological insulator in d=2,3Superconductivity: particle-hole symmetry of BdG Hamiltonian Z2 topological superconductor d=1 Z topological superconductor d=2Crystal symmetry translation : weak topological insulator reflection : topological crystalline insulator rotation non symmorphic glides, screws

21. Time Reversal SymmetryKramers’ Theorem: for spin ½ all eigenstates are at least 2 fold degenerateAnti Unitary time reversal operator :Spin ½ : Proof : for a non degenerate eigenstate EFconduction bandvalence band↓↑0p/a-p/aEQuantum spin Hall insulator : Split 1D conductor preserving T symmetryprotected by time reversalRequires spin – orbit interactionSimplest model: two copies of chern insulatorR ↑,↓L ↑,↓Helical Edge States :2D QSHI↑↓↑↓1Dsplit

22. Z2 Topological InsulatorThere are two classes of time reversal invariant insulatorsDistinguished by Z2 topological invariant n = 0, 1Two patterns of edge states:n=0 : Conventional Insulatorn=1 : Topological InsulatorKramers degenerate attime reversal invariant momenta k* = -k* + Gk*=0k*=p/ak*=0k*=p/aEven number of bandscrossing Fermi energyOdd number of bandscrossing Fermi energy

23. Physical Meaning of 2 InvariantDF = f0 = h/e DQ = N e Flux f0  Quantized change in Electron Number at the end. n=N IQHE on cylinder: Laughlin ArgumentQuantum Spin Hall Effect on cylinderDF = f0 / 2Flux f0 /2  Change in Electron Number Parityat the end, signaling changein Kramers degeneracy.KramersDegeneracyNo Kramers DegeneracySensitivity to boundary conditions in a multiply connected geometry

24. Formula for the 2 invariant Bloch wavefunctions : T - Reversal Matrix : Antisymmetry property : T - invariant momenta : L4L1L2L3kxkyBulk 2D Brillouin Zone Pfaffian : Z2 invariant :Gauge invariant, but requires continuous gauge(N occupied bands) Fixed point parity : Gauge dependent product : “time reversal polarization” analogous to

25. Sz conserved : independent spin Chern integers : (due to time reversal) n is easier to determine if there is extra symmetry:Inversion (P) Symmetry : determined by Parity of occupied 2D Bloch statesQuantum spin Hall Effect :J↑J↓EAllows a straightforward determination of n from band structurecalculations.In a special gauge:

26. Quantum Spin Hall Effect in HgTe quantum wellsTheory: Bernevig, Hughes and Zhang, Science ‘06HgTeHgxCd1-xTeHgxCd1-xTedd < 6.3 nm : Normal band orderd > 6.3 nm : Inverted band orderConventional InsulatorQuantum spin Hall Insulatorwith topological edge statesG6 ~ sG8 ~ pkEG6 ~ sG8 ~ pkEEgap~10 meVBand inversion transition:Switch parity at k=0BHZ Model : 4 band T-invariant band inversion model

27. 3D Topological InsulatorsThere are 4 surface Dirac Points due to Kramers degeneracySurface Brillouin Zone2D Dirac PointEk=Lak=LbEk=Lak=Lbn0 = 1 : Strong Topological InsulatorFermi circle encloses odd number of Dirac pointsTopological Metal : 1/4 graphene Berry’s phase p Robust to disorder: impossible to localizen0 = 0 : Weak Topological InsulatorRelated to layered 2D QSHI ; (n1n2n3) ~ Miller indicesFermi surface encloses even number of Dirac pointsORL4L1L2L3EFHow do the Dirac points connect? Determined by 4 bulk Z2 topological invariants n0 ; (n1n2n3) kxkykxkykxky

28. Topological Invariants in 3D1. 2D → 3D : Time reversal invariant planesThe 2D invariant kxkykzWeak Topological Invariants (vector):ki=0planeStrong Topological Invariant (scalar)Lap/ap/ap/aEach of the time reversal invariant planes in the 3D Brillouin zone is characterized by a 2D invariant.“mod 2” reciprocal lattice vector indexes lattice planes for layered 2D QSHIGn

29. Topological Invariants in 3D2. 4D → 3D : Dimensional ReductionAdd an extra parameter, k4, that smoothly connects the topological insulatorto a trivial insulator (while breaking time reversal symmetry)H(k,k4) is characterized by its second Chern numbern depends on how H(k) is connected to H0, butdue to time reversal, the difference must be even. (Trivial insulator)k4Express in terms of Chern Simons 3-form : Gauge invariant up to an even integer.