Symmetry, Topology And Phases Of Matter

Symmetry, Topology andPhases of MatterEEk Λak Λbk Λak Λb

Topological Phases of MatterMany 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 experimentsTopological superconductivity (BCS mean field theory)- Majorana bound states- Quantum informationClassical analogues: topological wave phenomena- photonic bands- phononic bands- isostatic latticesBeyond Band Theory: Strongly correlated statesState with intrinsic topological order (ie fractional quantum Hall effect)- fractional quantum numbers- topological ground state degeneracy- quantum information- Symmetry protected topological states- Surface topological order Much recent conceptualprogress, but theory isstill far from the real electrons

Topological Band TheoryTopological Band Theory I:IntroductionTopologically protected gapless states (without symmetry)Topological Band Theory II:Time Reversal symmetryCrystal symmetryTopological superconductivity10 fold wayTopological MechanicsGeneral References :“Colloquium: Topological Insulators”M.Z. Hasan and C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010).“Topological Band Theory and the Z2 Invariant,”C. L. Kane in “Topological insulators”edited by M. Franz and L. Molenkamp, Elsevier, 2013.

Topology and Band Theory II.Introduction- Insulating state, topology and band theoryII.Band Topology in One Dimension- Berry phase and electric polarization- Su Schrieffer Heeger model- Domain walls, Jackiw Rebbi problem- Thouless charge cumpIII. Band Topology in Two Dimensions- Integer quantum Hall effect- TKNN invariant- Edge states, chiral Dirac fermionsIV. Generalizations- Higher dimensions- Topological defects- Weyl semimetal

Insulator vs Quantum Hall stateThe Insulating Stateatomic insulator4satomic energylevelsEg3pE3s π/aThe Integer Quantum Hall State2D Cyclotron Motion,π/aσxy e2/h3Landaulevelsa0Eg hωc2E1Φ h/e π/a0π/aWhat’s the difference? Distinguished by Topological Invariant

TopologyThe study of geometrical properties that are insensitive to smooth deformationsExample: 2D surfaces in 3DA closed surface is characterized by its genus, g # holesg 0g 1g is an integer topological invariant that can be expressed in terms of thegaussian curvature κ that characterizes the local radii of curvature1κ r1r2κ κ 01 02rGauss Bonnet Theorem :κ 0 κ dA 4π (1 g )SA good math book : Nakahara, ‘Geometry, Topology and Physics’

Band Theory of SolidsBloch Theorem :Lattice translation symmetryT (R) ψ eik R ψBloch HamiltonianH (k ) e ik r Ηeik rψ eik r u(k )H (k ) un (k ) En (k ) un (k )kyπ/ak Brillouin Zone Torus, Tdkx π/a π/aπ/a BZBand Structure :A mappingEgapk a H (k )(or equivalently to En (k ) and un (k ) )E π/akxπ/a

Topology and Quantum PhasesTopological Equivalence :Principle of Adiabatic ContinuityQuantum phases with an energy gap are topologicallyequivalent if they can be smoothly deformed into oneanother without closing the gap.Eexcited statesTopologically distinct phases are separated byquantum phase transition.GapEGtopological quantumcritical pointGround state E0adiabatic deformationTopological Band TheoryDescribe states that are adiabatically connected tonon interacting fermionsEEg 1 eVClassify single particle Bloch band structuresH (k ) : Brillouin zone (torus) aBloch Hamiltonanswith energy gapBand Theory of Solidse.g. Siliconk

Berry PhasePhase ambiguity of quantum mechanical wave functionu(k ) eiφ (k ) u(k )A i u(k ) k u(k )Berry connection : like a vector potentialA A kφ (k )Berry phase : change in phase on a closed loop CγC — A dkF k Aγ C Fd 2 kBerry curvature :CSFamous example : eigenstates of 2 level Hamiltonian dzH (k ) d(k ) σ d x id yrH (k ) u(k ) d(k ) u(k )C Sd x id y d z d̂γC 1Solid Angle swept out by dˆ (k )2()

Topology in one dimension : Berry phase and electric polarizationsee, e.g. Resta, RMP 66, 899 (1994)Classical electric polarization :dipole momentP lengthBound charge density-Q1D insulator Qρbound PQend P nˆEnd chargeProposition: The quantum polarization is a Berry phaseeP 2π— A(k )dkBZA i u(k ) k u(k )BZ 1D Brillouin Zone π/a-π/aS1k0

Circumstantial evidence #1 :The polarization and the Berry phase share the same ambiguity:They are both only defined modulo an integer. The end charge is not completely determined by the bulkpolarization P because integer charges can be added orremoved from the ends : The Berry phase is gauge invariant under continuous gauge transformations,but is not gauge invariant under “large” gauge transformations.P P enwhenu(k ) eiφ ( k ) u(k )withQend P mod eφ (π / a) φ ( π / a) 2π nChanges in P, due to adiabatic variation are well defined and gauge invariant1u(k ) u(k , λ (t ))ΔP Pλ 1 Pλ 0e 2πe— C Adk 2πgauge invariant Berry curvature Fdkd λSλCSk0-π/aπ/a

r : i kCircumstantial evidence #2 :“dkieP e— BZ 2π u(k ) r u(k ) 2π— u (k ) BZku (k )”A more rigorous argument:Construct Localized Wannier Orbitals :dk ik ( R r )ϕ ( R) —u (k ) BZ 2π eϕ R (r )RWannier states are gauge dependent, but for a sufficiently smooth gauge,they are localized states associated with a Bravais Lattice point Rϕ R (r )P e ϕ ( R) r R ϕ ( R)ie 2π— u (k ) BZku (k )Rr

Su Schrieffer Heeger ModelH (t δ t )c†Ai cBi (t δ t )c†Ai 1cBi h.c.iδt 0A,iE(k)Gap 4 δt kB,iA,i 1a π/aδt 0H ( k ) d( k ) σd x (k ) (t δ t ) (t δ t ) cos kad y (k ) (t δ t ) sin kaπ/aPeierls’ instability δtrd z (k ) 0model for polyacetylenesimplest “two band” modeldyd(k)dxδt 0 : Berry phase 0P 0dyd(k)dxδt 0 : Berry phase πP e/2Provided symmetry requires dz (k) 0, the states with δt 0 and δt 0 are distinguished byan integer winding number. Without extra symmetry, all 1D band structures aretopologically equivalent.

Symmetries of the SSH model“Chiral” Symmetry : {H (k ),σ z } 0(or σ z H (k )σ z H (k ) )c ciAArtificial symmetry of polyacetylene. Consequence iAciB ciBof bipartite lattice with only A-B hopping: Requires dz(k) 0 : integer winding number Leads to particle-hole symmetric spectrum:Hσ z ψ E Eσ z ψ EReflection Symmetry : σ z ψ E ψ EH ( k ) σ x H (k )σ x 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 invariantP 0 or e/2 mod e

Domain Wall StatesAn interface between different topological states has topologically protected midgap statesδt 0δt 0Low energy continuum theory :For small δt focus on low energy states with k π/aH iv F σ x x m( x)σ yMassive 1 1 D Dirac Hamiltoniank πa q ; q i xv F ta ; m 2δ tE (q ) ( v F q) 2 m 2“Chiral” Symmetry : {σ z , H } 0 σ z ψ E ψ EAny eigenstate at Ehas a partner at -EZero mode : topologically protected eigenstate at E 0(Jackiw and Rebbi 76, Su Schrieffer, Heeger 79)m 0Domain wallbound state ψ0xEgap 2 m m 0ψ 0 ( x) e m ( x ')dx '/ v F 0 1 0

Thouless Charge PumpThe integer charge pumped across a 1D insulator in one period of an adiabatic cycleis a topological invariant that characterizes the cycle.t 0P 0t TP eH (k , t T ) H (k , t )t TeΔP 2π1n 2π(— A(k , T )dk — A(k , 0)dk ) ne T2t 0-π/aFdkdtk π/aThe integral of the Berry curvature defines the first Chern number, n, an integertopological invariant characterizing the occupied Bloch states, u (k , t )In the 2 band model, the Chern number is related to the solid angle swept out by dˆ ( k , t ),which must wrap around the sphere an integer n times.dˆ ( k , t )n 14π T2dkdt dˆ ( k dˆ t dˆ )

Integer Quantum Hall Effect : Laughlin ArgumentAdiabatically thread a quantum of magnetic flux through cylinder.-QE 1 dΦ2π R dt QI 2π R σ xy ETdΦhΔQ σ xydt σ xydte0Just like a Thouless pump :H (T ) U † H (0)Ue2ΔQ ne σ xy nhΦ (t 0) 0Φ (t T ) h / e

TKNN InvariantThouless, Kohmoto, Nightingale and den Nijs 82 View cylinder as 1D system with subbands labeled by k ym (Φ) 1 m Φ R φ0 Em (kx ) E ( kx , k ym (Φ) )φe 0mΔQ dΦdkFk,k(xxy (Φ ) ) ne m 2π 0TKNN number Chern number1n 2π12 BZ d kF(k ) 2πe2σ xy nh— A dkCkxkyCkxky m(φ0)ky m(0)Distinguishes topologically distinct 2D band structures. Analogous to Gauss-Bonnet thm.Alternative calculation: compute σxy via Kubo formula

TKNN InvariantThouless, Kohmoto,Nightingale and den Nijs 82For a 2D band structure, define Α(k ) i u(k ) k u(k )π/a kyC1 π/akxπ/aC2 π/a11n A dk — C2π 12π12 dkF(k ) 2π BZ— A dk C2e2Physical meaning: Hall conductivity σ xy nhLaughlin Argument: Thread magnetic flux φ0 h/e through a 1D cylinderPolarization changes by σxy φ0E-neIneΦ kyΔP ne

A good math book : Nakahara, ‘Geometry, Topology and Physics’ . Topology and Quantum Phases Topological Equivalence : Principle of Adiabatic Continuity Quantum phases with an energy gap are topologically equivalent if they can be sm