Topology and solid state physics 11/10/2011 @ NSYSU Topology and solid state physics Ming-Che Chang Department of Physics

A brief review of topology extrinsic curvature K vs intrinsic (Gaussian) curvature G Positive and negative Gaussian curvature G>0 G=0 G<0 K≠0 G=0 K≠0 G≠0

The most beautiful theorem in differential topology Gauss-Bonnet theorem (for a 2-dim closed surface) Euler characteristic Gauss-Bonnet theorem (for a surface with boundary) Marder, Phys Today, Feb 2007

x x A brief review of Berry phase n+1 n n-1 Adiabatic evolution of a quantum system Energy spectrum: After a cyclic evolution E(λ(t)) n+1 x n x n-1 dynamical phase λ(t) Phases of the snapshot states at different λ’s are independent and can be arbitrarily assigned Do we need to worry about this phase?

Stationary, snapshot state Fock, Z. Phys 1928 Schiff, Quantum Mechanics (3rd ed.) p.290 No! Pf : Consider the n-th level, Stationary, snapshot state ≡An(λ) redefine the phase, An’(λ) An(λ) Choose a  (λ) such that, An’(λ)=0 Thus removing the extra phase!

One problem: does not always have a well-defined (global) solution  is not defined here Vector flow Vector flow Contour of  Contour of  C M. Berry, 1984 : Parameter-dependent phase NOT always removable!

Stokes theorem (3-dim here, can be higher) Index n neglected Berry phase (path dependent, but gauge indep.) Berry connection Berry curvature C S Stokes theorem (3-dim here, can be higher)

Aharonov-Bohm (AB) phase (1959) Independent of precise path Φ magnetic flux solenoid Berry phase (1984) C Independent of the speed (as long as it’s slow) solid angle

Φ e- AB phase and persistent charge current (in a metal ring) After one circle, the electron gets a phase AB phase → Persistent current confirmed in 1990 (Levy et al, PRL) Berry phase and persistent spin current (Loss et al, PRL 1990) B S e- textured B field Ω(C) S After circling once, an electron gets → persistent charge and spin current

← monopole Berry curvature and Bloch state For scalar Bloch state (non-degenerate band): Space inversion symmetry both symmetries Time reversal symmetry When do we expect to have it? SI symmetry is broken TR symmetry is broken spinor Bloch state (degenerate band) band crossing ← electric polarization ← QHE ← SHE ← monopole Also,

= → Quantization of Hall conductance (Thouless et al 1982) Topology and Bloch state = Brillouin zone ~ Gauss-Bonnet theorem 1st Chern ~ Euler characteristic number → Quantization of Hall conductance (Thouless et al 1982) Remains quantized even with disorder, e-e interaction (Niu, Thouless, Wu, PRB, 1985)

Bulk-edge correspondence Different topological classes Semiclassical (adiabatic) picture: energy levels must cross (otherwise topology won’t change). Eg → gapless states bound to the interface, which are protected by topology.

Edge states in quantum Hall system (Semiclassical picture) Gapless excitations at the edge LLs number of edge modes = Hall conductance

The topology in “topological insulator” First, a brief history of insulators Band insulator (Wilson, Bloch) Mott insulator Anderson insulator Quantum Hall insulator Topological insulator Peierls transition Hubbard model Scaling theory of localization 1930 1940 1950 1960 1970 1980 1990 2000 2010 2D TI is also called QSHI

Lattice fermion with time reversal symmetry 2D Brillouin zone Without B field, Chern number C1= 0 Bloch states at k, -k are not independent EBZ is a cylinder, not a closed torus. ∴ No obvious quantization. Moore and Balents PRB 07 C1 of closed surface may depend on caps C1 of the EBZ (mod 2) is independent of caps (topological insulator, TI) 2 types of insulator, the “0-type”, and the “1-type”

2D TI characterized by a Z2 number (Fu and Kane 2006 ) ~ Gauss-Bonnet theorem with edge How can one get a TI? : band inversion due to SO coupling 0-type 1-type

Bulk-edge correspondence in TI TRIM Dirac point Fermi level Bulk states Edge states (2-fold degeneracy at TRIM due to Kramer’s degeneracy) helical edge states robust backscattering by non-magnetic impurity forbidden

A stack of 2D TI z 2D TI y x 3 TI indices fragile Helical SS Helical edge state y x 3 TI indices

difference between two 2D TI indices 3 weak TI indices: (ν1, ν2, ν3 ) Eg., (x0, y0, z0 ) z+ x+ y+ z0 x0 y0 1 strong TI index: ν0 ν0 = z+-z0 (= y+-y0 = x+-x0 ) difference between two 2D TI indices Fu, Kane, and Mele PRL 07 Moore and Balents PRB 07 Roy, PRB 09

For systems with inversion symmetry, one can calculate the TI indices using parity eigenvalues of Bloch states. (Fu and Kane) For example,

Topological insulators in real life SO coupling only 10-3 meV Graphene (Kane and Mele, PRLs 2005) HgTe/CdTe QW (Bernevig, Hughes, and Zhang, Science 2006) Bi bilayer (Murakami, PRL 2006) … Bi1-xSbx, α-Sn ... (Fu, Kane, Mele, PRL, PRB 2007) Bi2Te3 (0.165 eV), Bi2Se3 (0.3 eV) … (Zhang, Nature Phys 2009) The half Heusler compounds (LuPtBi, YPtBi …) (Lin, Nature Material 2010) thallium-based III-V-VI2 chalcogenides (TlBiSe2 …) (Lin, PRL 2010) GenBi2mTe3m+n family (GeBi2Te4 …) 2D 3D strong spin-orbit coupling band inversion

Band inversion, parity change, and spin-momentum locking (helical Dirac cone) S.Y. Xu et al Science 2011

Physics related to the Z2 invariant Topological insulator Helical surface state graphene Magneto-electric response Quantum SHE (Generalization of) Gauss-Bonnett theorem Interplay with SC, magnetism Spin pump Majorana fermion in TI-SC interface QC 4D QHE … Time reversal inv, spin-orbit coupling, band inversion

Berry phase and topology in condensed matter physics 1982 Quantized Hall conductance (Thouless et al) 1983 Quantized charge transport (Thouless) 1990 Persistent spin current in one-dimensional ring (Loss et al) 1992 Quantum tunneling in magnetic cluster (Loss et al) 1993 Modern theory of electric polarization (King-Smith et al) 1996 Semiclassical dynamics in Bloch band (Chang et al) 2001 Anomalous Hall effect (Taguchi et al) 2003 Spin Hall effect (Murakami et al) 2006 Topological insulator (Kane et al) …

Thank you!