Topological Physics in Band Insulators II

Transcription

1 Topological Physics in Band Insulators II Gene Mele University of Pennsylvania Topological Insulators in Two and Three Dimensions

2 The canonical list of electric forms of matter is actually incomplete Conductor Insulator Superconductor 18 th century 20 th century Topological Insulator

3 Electronic States of Matter Topological Defects in (CH) x Self conjugate state from Dirac mass inversion

4 Summary of First Lecture: The unsual spin charge relation appears in the strong coupling limit, where it is a property of atoms and decoupled dimers. This is adiabatically connected to a continuum limit where it arises as a transition in the ground state topology. Summary of Second Lecture: This transition occurs at the boundary between a topological insulator and an ordinary insulator.

5 Electronic States of Matter Topological Insulators This novel electronic state of matter is gapped in the bulk and supports the transport of spin and charge in gapless edge states that propagate at the sample boundaries. The edge states are insensitive to disorder because their directionality is correlated with spin Charlie Kane and GM University of Pennsylvania Electron spin admits a topologically distinct insulating state

6 Electronic States of Matter Topological Insulators This state is realized in three dimensional materials where spin orbit coupling produces a bandgap inversion. It has boundary modes (surface states) with a 2D Dirac singularity protected by time reversal symmetry. Bi 2 Se 3 is a prototype.. Hasan/Cava (2009)

7 Graphene: : the Parent Phase

8 .. it has a critical electronic state The dispersion of a free particle in 2D.. is replaced by an unconventional E(k) relation on the graphene lattice

9 The low energy theory is described by an effective mass theory for massless electrons Bloch Wavefunction Wavefunction( s) at K ( r ) H ( r ) iv ( r ) eff It is a massless Dirac Theory in 2+1 Dimensions NOTE: Here the spin degree of freedom describes the sublattice polarization of the state, called pseudospin. In addition electrons carry a physical spin ½ and an isospin ½ describing the valley degeneracy. F D.P. DiVincenzo and GM (1984)

10 Gapping the Dirac Point Gapping the Dirac Point Valley mixing from broken translational symmetry A continuum of structures all with 3 x 3 period hybridizes the two valleys

11 Gapping the Dirac Point Valley mixing from broken translational symmetry H ' 0 i x Kekule i e x 0 e

12 Gapping the Dirac Point Charge transfer from broken inversion symmetry H ' 0 z BN 0 z

13 Gapping the Dirac Point Orbital currents from modulated flux (Broken T-symmetry) H ' 0 z FDMH 0 z Gauged second neighbor hopping breaks T. Chern insulator with Hall conductance e 2 /h FDM Haldane Quantum Hall Effect without Landau Levels (1988)

14 Topological Classification 1 (, ) 0 2 n d k d k1 k2 k d 1 k d 2 4 S

15 Topological Classification H ' 0 z FDMH 0 z 1 (, ) 1 e Chern Insulator with xy h 2 n d k d k1 k2 k d 1 k d 2 4 S 2 (has equal contributions from two valleys)

16 a n: b n' : Orthodoxy: Spectrum Gapped only for Broken Symmetry States H ( k) t cos k a sin k a M n n x n y z 2t cos cos k b sin sin k b 2 n' 0 n' z n' Breaks e-h symmetry triad of nearest neighbor bond vectors triad of directed left turn second neighbor bond vectors Breaks T Breaks P Crucially, this ignores the electron spin

17 Microscopic Coupling orbital motion to the electron spin H s V p SO V ( r ) V ( r T ) H i ( r r ) Lattice model SO m n n m m n Spin orbit field Bond vector Intersite hopping with spin precession

18 Coupling orbital motion to the electron spin Breaking mirror symmetry with a perpendicular spin orbit field xy 0, 0 H s nˆ p nˆ s p z R it n s d ˆ 1 n mn m Modifies first neighbor coupling by spin dependent potential s s R R x z y y x Renormalizes Fermi velocity and can fission the Dirac point

19 xy Coupling orbital motion to the electron spin Preserve mirror symmetry with a parallel spin orbit field 0, 0 H s nˆ p p ( s nˆ ) p a z SO z z eff / 2 a a eff eff d 0 d 0 i i t2 e cncm e cmcn t 2 cos cncm cmcn isin c c c c Generates a spin-dependent Haldane-type mass (two copies) SO SO z zsz n m m n

20 Mass Terms (amended) x x, x y Kekule: valley mixing z Heteropolar (breaks P) spinless z z s x z y y x z zsz s Modulated flux (breaks T) Spin orbit (Rashba, broken z -z) Spin orbit (parallel)* *This term respects all symmetries and is therefore present, though possibly weak For carbon definitely weak, but still important

21 Topologically different states Charge transfer insulator Spin orbit coupled insulator 1 n 0 n 1 ( 1) 0 2 n d k d ( k1, k2) k d 1 k d 2 4 S Topology of Chern insulator in a T-invariant state

22 Boundary Modes Ballistic propagation through one-way edge state Counter Intrinsic propagating SO-Graphene spin polarized model on edge a ribbon states

23 Quantum Spin Hall Effect Its boundary modes are spin filtered propagating surface states (edge states)

24 Comments The H 2 model conserves S z and is oversimplified. Spin, unlike charge, is not conserved. But the edge state picture is robust! Boundary modes: Kramers pair (a) Band crossing protected by T-reversal symmetry (b) Elastic backscattering eliminated by T-symmetry QSHE: quantum but not quantized

25 More comments Counter-propagating surface modes reflect the bulk topological order. They can only be eliminated by a phase transition to a non-topological phase. weak sublattice symmetry breaking strong sublattice symmetry breaking

26 Symmetry Classification Conductors: unbroken state 1 Insulators: broken translational symmetry: bandgap from Bragg reflection 2 Superconductor: broken gauge symmetry Topological Insulator? 1 possibly with mass anisotropy 2 band insulators

27 Symmetry Classification Ordinary insulators and topological insulators are distinguished by a two-valued (even-odd) surface index. Kramers Theorem: T-symmetry requires E(k,) =E(-k,) But at special points k and -k are identified (TRIM) even: ordinary (trivial) odd: topological Kane and GM (2005)

28 Bulk Signature The surface modes reflect bulk topological order distinguished by a bulk symmetry e.g. TKKN invariant = Chern number = Hall conductance 1 2 n d k d ( k1, k2) k d 1 k d 2 4 S T-reversal symmetry requires n=0 Spin Chern number in S z conserving model is nontopological TI index is defined mod 2

29 Bulk time-reversal invariant momenta G G H H 2 2 Symmetry-protected twofold degeneracy at opposing points (d and d) on Bloch sphere Comparison of T reversal pairs allows topological classification of ground state

30 Diagnostic for Topological Order: ik r Periodic part of Bloch state: un( k ) e n( k; r ) Q. How different are un( k ) and un( k )? A. For a trivial atomic insulator they are the same A. For N bands quantify by w ( k ) u ( k ) u ( k ) N mn m n Antisymmetric: periodic complex-valued P( k ) Pf (w) P( k ) 0 points (vortices) at k but never at TRIM (k=-k) N Kane and GM (2005)

31 Count the zeroes of Pfaffian Test P( k ) in one half of Brillouin zone Zero: Trivial, like an atomic insulator Even: Adiabatically connected to atomic insulator by pairwise annihilation of its zeroes Odd: Can t be adiabatically connected to atomic insulator since P( k ) 0 is forbidden at TRIM. Direct integration requires a smooth gauge and is awkward

32 Pointwise Integration Rules ( 1) a Atomic insulator: all a a a 0 (or 0) Pf( w( )) a det w( ) Track sign changes of s between TRIM a 1 a b 0 : exchange Kramers partners Gauge Invariant Products: a a a a a 0 : a 0 : a "conventional" a "topological" Fu, Kane and GM (2007)

33 With inversion symmetry Ordinary insulators and topological insulators are distinguished by a two-valued ( = 0,1) bulk index. N ( 1) a m (parity eigenvalues, 1) a1 a m Fu, Kane and GM (2007)

34 Example: one orbital diamond lattice

35 Fu Kane (2007) Example: Bi x Sb 1-x

36 Some References: Review Article: M.Z. Hasan and C.L. Kane Rev. Mod. Phys. 82, 3045 (2010) QSH in Graphene: C.L. Kane and E.J. Mele Phys. Rev. Lett. 95, (2005) Z2 insulators: C.L. Kane and E.J. Mele Phys. Rev. Lett. 95, (2005) Three Dimensional TI s. L. Fu, C.L. Kane and E.J. Mele Phys. Rev. Lett. 98, (2007) Inversion symmetric TI s. L. Fu and C.L. Kane Phys. Rev. B 76, (2007)