Organizing Principles for Understanding Matter Symmetry Conceptual simplification Conservation laws Distinguish phases of matter by pattern of broken symmetries Topology Properties insensitive to smooth deformation Quantized topological numbers symmetry group p4 symmetry group p31m Distinguish topological phases of matter genus = 0 genus = 1 Interplay between symmetry and topology has led to a new understanding of electronic phases of matter.
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 0 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) Bloch Hamiltonans with energy gap Band Theory of Solids e.g. Silicon k
Topological Electronic Phases 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.. Many real materials and experiments Beyond Band Theory: Strongly correlated states State 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 is still far from the real electrons Topological Superconductivity Proximity induced topological superconductivity Majorana bound states, quantum information Tantalizing recent experimental progress
Topological Band Theory Lecture #1: Topology and Band Theory Lecture #: Topological Insulators in and 3 dimensions Topological Semimetals Lecture #3: Topological Superconductivity Majorana Fermions Topological quantum computation General References : Colloquium: Topological Insulators M.Z. Hasan and C.L. Kane, Rev. Mod. Phys. 8, 3045 (010) Topological Band Theory and the Z Invariant, C. L. Kane in Topological insulators edited by M. Franz and L. Molenkamp, Elsevier, 013.
Topology and Band Theory I. 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 Pump III. Band Topology in Two Dimensions - Integer quantum Hall effect - TKNN invariant - Edge States, chiral Dirac fermions IV. Generalizations - Bulk-Boundary correspondence - Weyl semimetal - Higher dimensions - Topological Defects
Insulator vs Quantum Hall state The Insulating State atomic insulator 4s atomic energy levels 3p E g E 3s The Integer Quantum Hall State -p/a 0 p/a D Cyclotron Motion, s xy = e /h a Landau levels 3 E g c E 1 h/ e -p/a 0 p/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=0 g=1 g is an integer topological invariant that can be expressed in terms of the gaussian curvature k that characterizes the local radii of curvature k 1 rr 1 Gauss Bonnet Theorem : 1 k 0 k 0 r S k 0 kda 4 p(1 -g) A good math book : Nakahara, Geometry, Topology and Physics
Bloch Theorem : Lattice translation symmetry Band Theory of Solids T( R) e i kr ik r Bloch Hamiltonian H( k) e Ηe - ikr i e kr u( k) H( k) u ( k) E ( k) u ( k) n n n k Brillouin Zone Torus, T Band Structure : A mapping k H( k) d -p/a -p/a (or equivalently to E ( k) and u ( k) ) n n p/a k y E -p/a p/a BZ k x k x = p/a E gap Topological Equivalence : adiabatic continuity Band structures are equivalent if they can be continuously deformed into one another without closing the energy gap
Berry Phase Phase ambiguity of quantum mechanical wave function u e u i ( ) ( k) k ( k) Berry connection : like a vector potential A -i u( k) u( k) k A A k ( k) Berry phase : change in phase on a closed loop C C A d k C Berry curvature : F k A Famous example : eigenstates of level Hamiltonian dz dx -id y H( k) d( k) s dx id y -dz H( ) u( ) ( ) u( ) Fdk k k d k k 1 Solid Angle swept out by ˆ( ) C dk C S ˆd C S
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 bound P End charge Qend Pnˆ Proposition: The quantum polarization is a Berry phase P e p BZ A( k) dk A -i u( k) u( k) k BZ = 1D Brillouin Zone = S 1 p/a -p/a k 0
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) u( k) e u( k) ( / ) - (- / ) when with p a p a pn Changes in P, due to adiabatic variation are well defined and gauge invariant u( k) u( k, ( t)) 1 e e P P - P dk dkd p p 1 0 A F C S gauge invariant Berry curvature S -p/a 0 C p/a k
Circumstantial evidence # : r i k dk ie P e u( k) r u( k) u( k) k u( k) BZ p p BZ A slightly more rigorous argument: Construct Localized Wannier Orbitals : dk p -ik ( R-r) ( R) e u( k) BZ R 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 ( R) r -R ( R) R() r ie p BZ u( k) u( k) k R r
Su Schrieffer Heeger Model model for polyacetylene simplest two band model t t H t t c c t - t c c h c 0 0 i ( ) Ai Bi ( ) Ai1 Bi.. A,i B,i A,i+1 a Gap 4 t E(k) -p/a p/a Peierls instability t k H( k) d( k) s d ( k) ( t t) ( t -t)cos ka x d ( k) ( t -t)sin ka d y z ( k) 0 d y d(k) d y d(k) d x d x t>0 : Berry phase 0 P = 0 t<0 : Berry phase p P = e/ Provided symmetry requires d z (k)=0, the states with t>0 and t<0 are distinguished by an integer winding number. Without extra symmetry, all 1D band structures are topologically equivalent.
Symmetries of the SSH model Chiral Symmetry : H( k), s 0 (or s H( k) s -H( k) ) z z z Artificial symmetry of polyacetylene. Consequence of bipartite lattice with only A-B hopping: Requires d z (k)=0 : integer winding number Leads to particle-hole symmetric spectrum: Hs -Es s - z E z E z E E c c ia ib c ia -c ib Reflection Symmetry : H( -k) s H( k) s x x Real symmetry of polyacetylene. Allows d z (k) 0, 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 = 0 or e/ mod e
Domain Wall States An interface between different topological states has topologically protected midgap states t 0 t 0 Low energy continuum theory : For small t focus on low energy states with k~p/a H -iv F s x x m( x) s y p k q ; q -i a v F ta ; m t x Massive 1+1 D Dirac Hamiltonian E( q) ( q) m v F Chiral Symmetry : { s, H} 0 s - z z E E Any eigenstate at +E has a partner at -E Zero mode : topologically protected eigenstate at E=0 m>0 Domain wall bound state 0 (Jackiw and Rebbi 76, Su Schrieffer, Heeger 79) E gap = m m<0 ( x) 0 e x - m( x') dx'/ vf 0 1 0
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. H( k, t T) H( k, t) t=0 t=t P=0 P=e e P A( k, T) dk - A( k,0) dk ne p n 1 p T F dkdt The integral of the Berry curvature defines the first Chern number, n, an integer topological invariant characterizing the occupied Bloch states, u( k, t) -p/a t=t t=0 k p/a = 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 4p d d d T d ˆ( kt, ),
Integer Quantum Hall Effect : Laughlin Argument Adiabatically thread a quantum of magnetic flux through cylinder. -Q 1 d E p R dt I R E p sxy +Q ( t 0) 0 ( t T) h / e T d Q sxy dt dt 0 Just like a Thouless pump : s xy Q ne s xy n h e h e H( T) U H(0) U
TKNN Invariant Thouless, Kohmoto, Nightingale and den Nijs 8 View cylinder as 1D system with subbands labeled by 0 e m Q d dkx kx, ky ( ) ne p F m 0 m ky ( ) 1 m R 0 m Em ( kx) E kx, ky ( ) k x TKNN number = Chern number 1 1 F k A k p p C n d k ( ) d BZ e s xy n h C k y k x k ym ( 0 ) k ym (0) Distinguishes topologically distinct D band structures. Analogous to Gauss-Bonnet thm. Alternative calculation: compute s xy via Kubo formula
TKNN Invariant Thouless, Kohmoto, Nightingale and den Nijs 8 -p/a For a D band structure, define p/a k y C 1 C k x p/a Α( k) -i u( k) u( k) n 1 1 A d k A 1 d k p C p C 1 Fk ( ) - dk p BZ -p/a e Physical meaning: Hall conductivity s xy n h Laughlin Argument: Thread magnetic flux 0 = h/e through a 1D cylinder Polarization changes by s xy 0 k -ne E I ne k P ne y
A B Graphene E Novoselov et al. 05 Two band model H( k) d( k) s E( k) d( k) 3 j1 H - t caic ij Inversion and Time reversal symmetry require d ( k) 0 D Dirac points at kk: point vortices in ( d, d ) x z y dˆ( k, k ) H( K q) vs q Massless Dirac Hamiltonian Berry s phase p around Dirac point d( k) -t xˆcos k r yˆsin k r j Bj j x y www.univie.ac.at -K +K k
Topological gapped phases in Graphene Break P or T symmetry : H( K q) vq s m sz E( q) v q m n # times dk ˆ( ) wraps around sphere 1. Broken P : eg Boron Nitride +K & -K dk ˆ( ) S m m - Chern number n=0 : Trivial Insulator. Broken T : Haldane Model 88 m -m - - - - - - - - - Chern number n=1 : Quantum Hall state +K -K dk ˆ( ) S
Edge States Gapless states at the interface between topologically distinct phases IQHE state n=1 Vacuum n=0 Edge states ~ skipping orbits Lead to quantized transport Band inversion transition : Dirac Equation y Domain wall bound state 0 m<0 n=1 n=0 m - = -m + m - = m + x m>0 E gap H v (-is s k ) m( x) s F x x y y z E 0 k y ik y ( ) ~ y 0 x e e x - m( x') dx'/ v 0 F E ( k ) v 0 y F k y Chiral Dirac Fermions Chiral Dirac fermions are unique 1D states : One way ballistic transport, responsible for quantized conductance. Insensitive to disorder, impossible to localize Fermion Doubling Theorem : Chiral Dirac Fermions can not exist in a purely 1D system. in disorder t =1
Bulk - Boundary Correspondence N = N R - N L is a topological invariant characterizing the boundary. E N R (N L ) = # Right (Left) moving chiral fermion branches intersecting E F E F Haldane Model N = 1 0 = 1 E K K k y E F N = 1 = 1 K K k y Bulk Boundary Correspondence : The boundary topological invariant N characterizing the gapless modes = Difference in the topological invariants n characterizing the bulk on either side
Weyl Semimetal Gapless Weyl points in momentum space are topologically protected in 3D A sphere in momentum space can have a Chern number: n S S d k Fk ( ) k S n S =+1: S must enclose a degenerate Weyl point: Magnetic monopole for Berry flux H( k q) q s q s q s ) 0 v( x x y y z z ( or v qs with det[ v ] 0 ) ia i a ia E k x,y,z Total magnetic charge in Brillouin zone must be zero: Weyl points must come in +/- pairs.
Surface Fermi Arc E n 1 =n =0 E F k z =k 1, k k z k k 0 E k y k 1 k y n 0 =1 E F k x k z =k 0 k y k z Surface BZ k y
Generalizations d=4 : 4 dimensional generalization of IQHE Zhang, Hu 01 A u ( k) u ( k) dk ij i k j F da A A 1 n Tr[ ] 4 8p F F T Non-Abelian Berry connection 1-form Non-Abelian Berry curvature -form nd Chern number = integral of 4-form over 4D BZ Boundary states : 3+1D Chiral Dirac fermions Higher Dimensions : Bott periodicity d d+ no symmetry chiral symmetry
Topological Defects Consider insulating Bloch Hamiltonians that vary slowly in real space H H( k, s) 1 parameter family of 3D Bloch Hamiltonians 1 8p nd Chern number : 3 1 n s Tr[ FF] T S Generalized bulk-boundary correspondence : defect line n specifies the number of chiral Dirac fermion modes bound to defect line Teo, Kane 10 Example : dislocation in 3D layered IQHE n 1 G p c B 3D Chern number (vector layers) G c Are there other ways to engineer 1D chiral dirac fermions? Burgers vector