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11/10/2011 @ NSYSU Topology and solid state physics Ming-Che Chang Department of Physics 1 A brief review of topology extrinsic curvature K vs intrinsic (Gaussian) curvature G Positive and negative Gaussian curvature G>0 K≠0 G=0 G=0 G<0 K≠0 G≠0 2 The most beautiful theorem in differential topology • Gauss-Bonnet theorem (for a 2-dim closed surface) M da G 2 ( M ), 2(1 g ) g 0 Euler characteristic g 1 g2 • Gauss-Bonnet theorem (for a surface with boundary) M da G ds k g 2 M , M M Marder, Phys Today, Feb 2007 3 A brief review of Berry phase Adiabatic evolution of a quantum system • Energy spectrum: H (r , p; ) • After a cyclic evolution E(λ(t)) n+1 x n x (T ) (0) n, (T ) n-1 0 i T dt ' En (t ') 0 e n, ( 0 ) dynamical phase λ(t) • Phases of the snapshot states at different λ’s are independent and can be arbitrarily assigned n , (t ) ei n ( ) n , ( t ) • Do we need to worry about this phase? 4 Fock, Z. Phys 1928 Schiff, Quantum Mechanics (3rd ed.) p.290 Pf : No! Consider the n-th level, i n ( ) (t ) e H (t ) i n i n , redefine the phase, i t dt ' En ( t ') 0 e n , (t ) t Stationary, snapshot state H n, En n, n , 0 ≡An(λ) 'n , ei n ( ) An’(λ) An(λ) n , n Choose a (λ) such that, An’(λ)=0 Thus removing the extra phase! 5 One problem: A( ) does not always have a well-defined (global) solution A Vector flow Vector flow Contour of A is not defined here Contour of C C (T ) (0) C A d 0 C A d 0 M. Berry, 1984 : Parameter-dependent phase NOT always removable! 6 (T ) T dt ' E ( t ') 0 e e (0) i C i Index n neglected • Berry phase (path dependent, but gauge indep.) C C A( ) d • Berry connection A( ) i 3 • Berry curvature (t ) ( ) A( ) i S • Stokes theorem (3-dim here, can be higher) C C A d da 1 2 C S 7 • Aharonov-Bohm (AB) phase (1959) • Berry phase (1984) Φ solenoid z magnetic flux y B(t ) x AB =2 0 Independent of precise path S C 1 C = (C ) 2 solid angle Independent of the speed (as long as it’s slow) 8 • AB phase and persistent charge current (in a metal ring) After one circle, the electron gets a phase Φ kL kL 2 0 AB phase → Persistent current confirmed in 1990 (Levy et al, PRL) • Berry phase and persistent spin current (Loss et al, PRL 1990) B textured B field S e- Ω(C) S After circling once, an electron gets z kL kL 2 (C ) → persistent charge and spin current 0 2 9 Berry curvature and Bloch state p k 2m 2 U (r ) unk (r ) nk unk (r ) n (k ) i k unk k unk For scalar Bloch state (non-degenerate band): • Space inversion symmetry n (k ) n (k ) both symmetries • Time reversal symmetry n (k ) n (k ) n (k ) 0, k When do we expect to have it? • SI symmetry is broken ← electric polarization • TR symmetry is broken ← QHE • spinor Bloch state (degenerate band) ← SHE Also, • band crossing ← monopole 10 Topology and Bloch state = Brillouin zone 1 2 d 2 k z (k ) C1 ~ Gauss-Bonnet theorem BZ 1st Chern ~ Euler characteristic number → Quantization of Hall conductance (Thouless et al 1982) e2 H C1 h Remains quantized even with disorder, e-e interaction (Niu, Thouless, Wu, PRB, 1985) 11 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. 12 Edge states in quantum Hall system (Semiclassical picture) Gapless excitations at the edge LLs number of edge modes = Hall conductance 13 The topology in “topological insulator” First, a brief history of insulators Band insulator (Wilson, Bloch) Mott insulator Anderson insulator Peierls transition 1930 1940 1950 Hubbard model 1960 1970 Quantum Hall insulator Topological insulator Scaling theory of localization 1980 1990 2000 2010 2D TI is also called QSHI 14 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” 15 • 2D TI characterized by a Z2 number (Fu and Kane 2006 ) 1 2 mod 2 d k dk A ( EBZ ) 2 EBZ ~ Gauss-Bonnet theorem with edge How can one get a TI? : band inversion due to SO coupling 0-type 1-type 16 Bulk-edge correspondence in TI Bulk states Dirac point Edge states Fermi level TRIM (2-fold degeneracy at TRIM due to Kramer’s degeneracy) helical edge states robust backscattering by non-magnetic impurity forbidden 17 A stack of 2D TI 2D TI z fragile SS Helical Helical edge state y x 3 TI indices 18 3 weak TI indices: (ν1, ν2, ν3 ) Eg., (x0, y0, z0 ) z+ z0 x0 x+ y0 y+ 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 19 Topological insulators in real life • Graphene (Kane and Mele, PRLs 2005) 2D SO coupling only 10-3 meV • 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) 3D • thallium-based III-V-VI2 chalcogenides (TlBiSe2 …) (Lin, PRL 2010) • GenBi2mTe3m+n family (GeBi2Te4 …) •… • strong spin-orbit coupling • band inversion 21 Band inversion, parity change, and spin-momentum locking (helical Dirac cone) S.Y. Xu et al Science 2011 22 Physics related to the Z2 invariant Topological insulator graphene Quantum SHE Spin pump • (Generalization of) Gauss-Bonnett theorem 4D QHE Helical surface state Magneto-electric response Interplay with SC, magnetism Majorana fermion in TI-SC interface QC … Time reversal inv, spin-orbit coupling, band inversion 23 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) … 24 Thank you! 25