Symmetry, Topology and Phases of Matter E E k=λ a k=λ b k=λ a k=λ b
Topological Phases of Matter Many examples of topological band phenomena States adiabatically connected to independent electrons: - Quantum Hall (Chern) insulators - Topological insulators - Weak topological insulators - Topological crystalline insulators - Topological (Fermi, Weyl and Dirac) semimetals.. Topological superconductivity (BCS mean field theory) - Majorana bound states - Quantum information Many real materials and experiments Classical analogues: topological wave phenomena - photonic bands - phononic bands - isostatic lattices Beyond Band Theory: Strongly correlated states State 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 conceptual progress, but theory is still far from the real electrons
Topological Band Theory Topological Band Theory I: Introduction Topologically protected gapless states (without symmetry) Topological Band Theory II: Time Reversal symmetry Crystal symmetry Topological superconductivity 1 fold way Topological Mechanics General References : Colloquium: Topological Insulators M.Z. Hasan and C.L. Kane, Rev. Mod. Phys. 8, 345 (1). Topological Band Theory and the Z Invariant, C. L. Kane in Topological insulators edited by M. Franz and L. Molenkamp, Elsevier, 13.
Topology and Band Theory I I. Introduction II. - Insulating state, topology and band theory Band Topology in One Dimension - Berry phase and electric polarization - Su Schrieffer Heeger model - Domain walls, Jackiw Rebbi problem - Thouless charge cump III. Band Topology in Two Dimensions - Integer quantum Hall effect - TKNN invariant - Edge states, chiral Dirac fermions IV. Generalizations - Higher dimensions - Topological defects - Weyl semimetal
Insulator vs Quantum Hall state The Insulating State atomic insulator 4s atomic energy levels 3p E g E 3s The Integer Quantum Hall State π/a π/a D Cyclotron Motion, σ xy = e /h a Landau levels 3 E g = hω c E 1 Φ=h/ e π/a π/a What s the difference? Distinguished by Topological Invariant
Topology The study of geometrical properties that are insensitive to smooth deformations Example: D surfaces in 3D A closed surface is characterized by its genus, g = # holes g= g=1 g is an integer topological invariant that can be expressed in terms of the gaussian curvature κ that characterizes the local radii of curvature κ = 1 rr 1 Gauss Bonnet Theorem : 1 κ = > κ = r S κ < κda = 4 π(1 g) A good math book : Nakahara, Geometry, Topology and Physics
Band Theory of Solids Bloch Theorem : Lattice translation symmetry i T( R) ψ = e ψ i ψ = e u( k) ikr Bloch Hamiltonian H( k) = e Ηe ikr H( k) u ( k) = E ( k) u ( k) n n n k y k = Brillouin Zone Torus, T d π/a π/a π/a π/a BZ k x = Band Structure : A mapping k a H ( k) (or equivalently to E ( k) and u ( k) ) n n E π/a k x π/a E gap
Topology and Quantum Phases Topological Equivalence : Principle of Adiabatic Continuity Quantum phases with an energy gap are topologically equivalent if they can be smoothly deformed into one another without closing the gap. E excited states Topologically distinct phases are separated by quantum phase transition. Gap E G topological quantum critical point Ground state E adiabatic deformation Topological Band Theory Describe states that are adiabatically connected to non interacting fermions Classify single particle Bloch band structures E E g ~ 1 ev H( k) : Brillouin zone (torus) a Bloch Hamiltonans with energy gap Band Theory of Solids e.g. Silicon k
Berry Phase Phase ambiguity of quantum mechanical wave function u e u iφ ( ) ( k) k ( k) Berry connection : like a vector potential A= iu( k) k u( k) A A+ k φ( k) Berry phase : change in phase on a closed loop C Berry curvature : F= k Famous example : eigenstates of level Hamiltonian A dz dx id y H( k) = d( k) r σ = dx + id y dz H( ) u( ) =+ ( ) u( ) γ C = A d k C γ C = Fdk S k k d k k 1 γ ( Solid Angle swept out by ˆ( ) ) C = dk C S ˆd
Topology in one dimension : Berry phase and electric polarization Classical electric polarization : see, e.g. Resta, RMP 66, 899 (1994) P = dipole moment length -Q 1D insulator +Q Bound charge density End charge ρ bound Qend = P = P nˆ Proposition: The quantum polarization is a Berry phase P e = π BZ A( k) dk A= iu( k) k u( k) BZ = 1D Brillouin Zone = S 1 π/a -π/a k
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 bulk polarization P because integer charges can be added or removed from the ends : Qend = P mod e The Berry phase is gauge invariant under continuous gauge transformations, but is not gauge invariant under large gauge transformations. P P en i ( k) + when uk ( ) e φ uk ( ) with ( / ) ( / ) = φπ a φ π a πn Changes in P, due to adiabatic variation are well defined and gauge invariant λ uk ( ) uk (, λ( t)) 1 e e Δ P= P P = dk dkd π A = π F λ= 1 λ= λ C S gauge invariant Berry curvature S -π/a C π/a k
Circumstantial evidence # : r : i k dk ie P= e u( k) r u( k) = u( k) k u( k) BZ π π BZ A more rigorous argument: Construct Localized Wannier Orbitals : ϕ dk π ik ( R r) ( R) = e u( k) BZ R ϕ R () r Wannier states are gauge dependent, but for a sufficiently smooth gauge, they are localized states associated with a Bravais Lattice point R P= e ϕ( R) r R ϕ( R) ie = uk ( ) k uk ( ) π BZ R r ϕ R () r
Su Schrieffer Heeger Model model for polyacetylene simplest two band model δ t > δ t < H = t+ t c c + t t c c + hc i ( δ ) Ai Bi ( δ ) Ai+ 1 Bi.. r Hk ( ) = d( k) σ d ( k) = ( t+ δt) + ( t δt)cos ka x d ( k) = ( t δt)sin ka d y z ( k) = A,i B,i A,i+1 a d y d(k) d y d(k) d x d x Gap 4 δt E(k) π/a π/a Peierls instability δt δt> : Berry phase P = δt< : Berry phase π P = e/ k Provided symmetry requires d z (k)=, the states with δt> and δt< are distinguished by an integer winding number. Without extra symmetry, all 1D band structures are topologically equivalent.
Symmetries of the SSH model Chiral Symmetry : { } Hk ( ), σ = (or σ Hk ( ) σ = Hk ( ) ) z z z Artificial symmetry of polyacetylene. Consequence of bipartite lattice with only A-B hopping: Requires d z (k)= : integer winding number Leads to particle-hole symmetric spectrum: Hσ ψ = Eσ ψ σ ψ = ψ z E z E z E E c c ia ib c ia c ib Reflection Symmetry : H( k) =σ H( k) σ x x Real symmetry of polyacetylene. Allows d z (k), but constrains d x (-k)= d x (k), d y,z (-k)= -d y,z (k) No p-h symmetry, but polarization is quantized: Z invariant P = or e/ mod e
Domain Wall States An interface between different topological states has topologically protected midgap states δ t > δ t < Low energy continuum theory : For small δt focus on low energy states with k~π/a H = iv F σ x x + m( x) σ y π k + q ; q i a v F = ta ; m = δ t x Massive 1+1 D Dirac Hamiltonian Eq ( ) =± ( q) + m v F Chiral Symmetry : { σ, H} = σ ψ = ψ z z E E Any eigenstate at +E has a partner at -E Zero mode : topologically protected eigenstate at E= m> Domain wall bound state ψ (Jackiw and Rebbi 76, Su Schrieffer, Heeger 79) E gap = m m< ψ ( x) x mxdx ( ') '/v 1 F = e
Thouless Charge Pump The integer charge pumped across a 1D insulator in one period of an adiabatic cycle is a topological invariant that characterizes the cycle. Hkt (, + T) = Hkt (, ) t= t=t e Δ P= ( A( k, T) dk A( k,) dk) = ne π 1 n= dkdt π F T The integral of the Berry curvature defines the first Chern number, n, an integer topological invariant characterizing the occupied Bloch states, ukt (, ) -π/a t=t t= k π/a = P= P=e In the band model, the Chern number is related to the solid angle swept out by which must wrap around the sphere an integer n times. 1 d ˆ ( kt, ) n= dkdt ˆ ( ˆ ˆ) k t 4π d d d T d ˆ( kt, ),
Integer Quantum Hall Effect : Laughlin Argument Adiabatically thread a quantum of magnetic flux through cylinder. -Q 1 dφ E = π R dt I = R E π σxy +Q Φ ( t = ) = Φ ( t = T) = h/ e T dφ Δ Q = σxy dt = dt Just like a Thouless pump : σ xy Q ne σ xy n h Δ = = e h e HT ( ) = UH() U
TKNN Invariant Thouless, Kohmoto, Nightingale and den Nijs 8 View cylinder as 1D system with subbands labeled by e ( m Δ Q= dφ dk, ( )) x kx ky Φ = ne π F m φ m ky 1 Φ ( Φ ) = m+ R φ E ( k ) = E k, k ( Φ) m ( ) k x m x x y TKNN number = Chern number 1 1 π π C n d k ( ) d = = BZ Fk A k e σ xy = n h C k y k x k ym (φ ) k ym () Distinguishes topologically distinct D band structures. Analogous to Gauss-Bonnet thm. Alternative calculation: compute σ xy via Kubo formula
TKNN Invariant Thouless, Kohmoto, Nightingale and den Nijs 8 π/a For a D band structure, define π/a k y C 1 π/a C k x π/a dk π BZ Physical meaning: Hall conductivity Αk ( ) = iu( k) k u( k) n= 1 1 A π d k A C1 π d k C = 1 Fk ( ) Laughlin Argument: Thread magnetic flux φ = h/e through a 1D cylinder Polarization changes by σ xy φ e σ xy = n h -ne E I ne Φ k Δ P = ne y