Modern Topics in Solid-State Theory: Topological insulators and superconductors

Transcription

1 Modern Topics in Solid-State Theory: Topological insulators and superconductors Andreas P. Schnyder Max-Planck-Institut für Festkörperforschung, Stuttgart Universität Stuttgart January 2016

2 Lecture Four: Classification schemes 1. Topological superconductors - Topological superconductors w/ TRS in 2D - Topological superconductors w/ TRS in 3D 2. Symmetries & ten-fold classification - Symmetry classes of ten-fold way - Dirac Hamiltonians and Dirac mass gaps - Periodic table of topological insulators and superconductors

3 Helical superconductors (w/ time-reversal symmetry) TRI topological SC

4 Time-reversal-invariant topological superconductor Superconducting pairing with spin: Cooper pair H MF = X kc k c k + X h i 0(k)c k, c k, 0 + 0(k)c k, 0c k, k k 0 2 x 2 Gap matrix: (k) =[ s(k) 0 + d(k) ]i y Time-reversal symmetry: y (k) y = T ( k) Different spin-pairing symmetries: (anti-symmetry of wavefunction) spin-singlet: s(k) : 1 2 ( ) even parity: s(k) = s ( k) d x (k) id y (k) : spin-triplet: d x (k)+id y (k) : odd parity: d(k) = d( k) d z (k) : 1 2 ( + )

5 2D time-reversal-invariant topological superconductor (also known as helical superconductor ) Square lattice BdG Hamiltonian in the presence of time-reversal symmetry: Simplest model: Hp+ip (k) 0 H BdG (k) = 0 H p ip (k) (spinless chiral p-wave SC) 2 dz(k) =0 (k) =2t(cos k x + cos k y ) µ d x (k) =sink x d y (k) =sink y TRS: PHS: T H BdG (k)t 1 =+H BdG ( k) CH BdG (k)c 1 = H BdG ( k) T = i y 0 K T 2 = 1 C = 0 x K C 2 =+1 Combination of time-reversal and particle-hole symmetry: 9 = class DIII ; (chiral symmetry) U S =(i y 0 )( 0 x ) U S H BdG (k)+h BdG (k)u S =0 H BdG can be brought into block-off diagonal form: 0 D(k) eh BdG (k) = D(k) =(i y ) { k 0 + i t[d k ]} D (k) 0 (transform to basis in which S is diagonal) TRS acts on D(k) as follows: D T ( k) = D(k)

6 2D time-reversal-invariant topological superconductor (also known as helical superconductor ) Square lattice BdG Hamiltonian in the presence of time-reversal symmetry: Simplest model: (k) H BdG (k) = 0 t [d(k) ](i y ) (spinless chiral p-wave SC) 2 t( i y )[d(k) ] (k) 0 (k) =2t(cos k x + cos k y ) µ d x (k) =sink x d y (k) =sink y TRS: T H BdG (k)t 1 =+H BdG ( k) T = i y 0 K T 2 = 1 PHS: CH BdG (k)c 1 = H BdG ( k) C = 0 x K C 2 =+1 d z (k) =0 9 = class DIII ; Combination of time-reversal and particle-hole symmetry: (chiral symmetry) U S =(i y 0 )( 0 x ) U S H BdG (k)+h BdG (k)u S =0 H BdG can be brought into block-off diagonal form: 0 D(k) eh BdG (k) = D(k) =(i y ) { k 0 + i t[d k ]} D (k) 0 (transform to basis in which S is diagonal) TRS acts on D(k) as follows: D T ( k) = D(k)

7 ( ) ( 2D time-reversal-invariant topological superconductor ( ) eh BdG (k) = 0 D(k) D (k) 0 Spectrum flattening: Q = TRS acts on 4N 2P where: Projector onto filled Bloch bands Q(k) = 0 q(k) q (k) 0 q(k) as follows: q(k) = q T ( k) D(k) =(i y ) { k 0 + i t[d k ]} Λ 4 k y Λ 3 Λ 1 Λ 2 k x The eigenfunctions of u ± a (k) N = 1 2 ( Q(k) ) n a ±q (k)n a are: where: (n a ) b = ab are globally defined. Z 2 topological invariant: ( 1) = 4Y a=1 Pf [!( a )] p det [!( a )] = ±1 ) ( 1) = 4Y a=1 (k) = N u a ( k) u b (k) N Pf q T ( a ) p det [q( a )] = ±1 sewing matrix q(k) = q T ( k) q (k) =q 1 (k) (same symmetries as sewing matrix)

8 2D time-reversal-invariant topological superconductor Effective low-energy continuum theory: (expand around k =0) (k) = tk 2 +4t µ H BdG (k) = (k) 0 t [d(k) ](i y ) t( i y )[d(k) ] (k) 0 d x (k) =k x d y (k) =k y d z (k) =0 q Energy spectrum: E ± = ± (k) =± 2 (k)+ t (kx 2 + ky) 2 TRIM: k =0, k =+1 Z 2 topological invariant : ( 1) = sgn(4t µ)sgn(t) µ>4t : µ<4t : trivial superconductor Bulk-boundary correspondence: TRI topological superconductor Energy By analogy to chiral p-wave SC: (for µ < 4t) two counter-propagating Majorana edge modes protected by TRS and PHS possible condensed matter realization: thin film of CePt3Si? helical Majorana edge states: TRI topological SC k x

9 3D time-reversal-invariant topological superconductor Cubic lattice BdG Hamiltonian in the presence of time-reversal symmetry: (k) H BdG (k) = 0 t [d(k) ](i y ) t( i y )[d(k) ] (k) 0 "(k) =2t(cos k x + cos k y + cos k z ) µ equivalent to B phase of 3 He d x (k) =sink x d y (k) =sink y d z (k) =sink z TRS PHS T H BdG (k)t 1 =+H BdG ( k) CH BdG (k)c 1 = H BdG ( k) T = i y 0 K T 2 = 1 C = 0 x K C 2 =+1 9 = class DIII ; Chiral symmetry (TRS x PHS): U S H BdG (k)+h BdG (k)u S =0 H BdG can be brought into block-off diagonal form: 0 D(k) eh BdG (k) = D(k) =(i y ) { k 0 + i t[d k ]} D (k) 0 (transform to basis in which S is diagonal) TRS acts on D(k) as follows: D T ( k) = D(k)

10 3D time-reversal-invariant topological superconductor Lattice BdG Hamiltonian: =) Off-diagonal block: Mapping Spectrum flattening: Mapping D(k) : Brillouin zone D(k) TRS: D(k) = D T (k) q(k) : 0 D(k) eh BdG (k) = D (k) 0 D(k) =(i y ) { k 0 + i t[d k ]} q(k) = X a Brillouin zone TRS: q(k) = q T ( k) classified by winding number: W = Bulk-boundary correspondence: 1 a(k) u a(k)u a(k)d(k) Z BZ q(k) U(2) u a (k) : eigenvectors of 2[U(2)] = 0 π 3 [U(2)] = DD. d 3 k µ Tr (q 1 µ q)(q 1 q)(q 1 q) W =# Kramers-degenerate Majorana states Possible condensed matter realization: CePt3Si, Li2Pt3B, CeRhSi3, CeIrSi3, etc.

11 Classification schemes Symmetry dim Class T P S A Z 0 AIII Z 0 Z AI BDI Z 0 0 D Z 2 Z 0 DIII Z 2 Z 2 Z AII Z 2 Z 2 CII Z 0 Z 2 C Z 0 CI Z

12 Symmetry classes: Ten-fold way (originally introduced in the context of random Hamiltonians / matrices) time-reversal invariance: T : 0 no time reversal invariance +1 time reversal invariance and -1 time reversal invariance and particle-hole symmetry ( ): C : T 1 H( k)t =+H(k) C 1 H( k)c = H(k) T = U T K C = U C K 0 no particle-hole symmetry +1 particle-hole symmetry and -1 particle-hole symmetry and (is antiunitary) T 2 =+1 T 2 = 1 C 2 =+1 C 2 = 1 complex conjugation In addition we can also consider the sublattice symmetry S : S / TC SH(k)+H(k)S =0 Note: SLS is often also called chiral symmetry

13 Ten-fold classification of topological insulators and superconductors Ten-fold classification: classifies fully gapped topological materials in terms of non-spatial symmetries (i.e., symmetries that act locally in space) non-spatial symmetries: - time-reversal: - particle-hole: - sublattice: T H(k)T 1 =+H( k); T 2 = ±1 CH(k)C 1 = H( k); C 2 = ±1 SH(k)S 1 = H(k); S / TP ten symmetry classes Altland-Zirnbauer Random Matrix Classes Symmetry dim Class T CP S A Z 0 AIII Z 0 Z AI BDI Z 0 0 D Z 2 Z 0 DIII Z 2 Z 2 Z AII Z 2 Z 2 CII Z 0 Z 2 C Z 0 CI Z CI C CII Bott clock T 2 AI BDI A AIII D AII DIII C 2

14 Ten-fold classification of topological insulators and superconductors Ten-fold classification: classifies fully gapped topological materials in terms of non-spatial symmetries (i.e., symmetries that act locally in space) non-spatial symmetries: - time-reversal: - particle-hole: - sublattice: T H(k)T 1 =+H( k); T 2 = ±1 CH(k)C 1 = H( k); C 2 = ±1 SH(k)S 1 = H(k); S / TP ten symmetry classes Altland-Zirnbauer Random Matrix Classes Symmetry dim Class T CP S A Z 0 AIII Z 0 Z AI ?For which symmetry class and BDI Z 0 0 D Z 2 Z 0 dimension is there a topological DIII Z 2 Z 2 Z insulator/superconductor? AII Z 2 Z 2 CII Z 0 Z 2 C Z 0 CI Z

15 Symmetries and Dirac Hamiltonians Dirac Hamiltonian in spatial dimension d : H(k) = dx k i i + m 0 i=1 v u E ± = ± t m2 + dx i=1 k d i Gamma matrices i obey: { i, j} =2 ij i =0, 1,...,d TRS, PHS and chiral symmetry lead to the conditions: [ 0,T]=0 { i6=0,t} =0 { i,s} =0 { 0,C} =0 [ i6=0,c]=0 Topological phase transition as a function of mass term m 0? n=1 n=0 m<0 m>0 are there extra symmetry preserving mass terms that connect the two phases without gap closing? { d+1, i} =0 i =0, 1, 2 NO: topologically non-trivial YES: M d+1 v u E ± = ± t m2 + M 2 + dx i=1 k d i topologically trivial

16 Symmetries and Dirac Hamiltonians Dirac Hamiltonian in spatial dimension d : H(k) = dx k i i + m 0 E ± = ± i=1 v u t m2 + dx i=1 k d i Gapless surface states (interface Xd 1 H = emi i 0 d + k 0 i d surface state : i=1 i 0 d =± Xd 1 surface Hamiltonian: H surf = k i P i P i=1 v gapless surface spectrum: E ± surf = ± ux t d 1 k d! i@/@r d n=1 n=0 r d < 0 r d > 0 P =(I i 0 d ) /2 Presence of extra symmetry preserving mass term implies gapped surface states extra mass term projected onto surface is non-vanishing i=1 k 2 i m<0 m>0 MP d+1 P anti-commutes with P i P i =1,...,d 1 =) gapped surface spectrum

17 Dirac Hamiltonian in symmetry class AIII Topological phase transition as a function of mass term m 0 n1 n2 m<0 m>0 S = 1 SH(k)+H(k)S =0 One-dimensional Dirac Hamiltonian with rank 2: H(k) =k 3 + m 2 no extra symmetry preserving mass term exists ) class AIII in 1D is topologically non-trivial space of normalized mass matrices V AIII d=1,r=2 = {± 2 }

18 One-dimensional Dirac Hamiltonian in symmetry class AII T 1 H( k)t =+H(k) T 2 = 1 Dirac matrices with rank 2: H(k) =k 3 T = i 2 K no symmetry-allowed mass term exists impossible to localize ( 1 and 2 violate TRS) describes edge state of 2D topological insulator in class AII ) Dirac matrices with rank 4: H(k) =k m 0 3 extra symmetry preserving mass term: T = i 2 0 K M 3 2 =) class AII in 1D is topologically trivial space of normalized mass matrices V AII d=1,r=4 = {M X M 2 =1 = S 1 M =(m, M), X =( 0 3, 3 2 ) R 3 : U(2N)/Sp(N) connectedness of space of normalized Dirac masses: 0 (R 3 )=0

19 Two-dimensional Dirac Hamiltonian in symmetry class AII T 1 H( k)t =+H(k) T 2 = 1 Dirac matrices with rank 4: T = i 2 0 K H(k) =k k m 0 3 no symmetry-allowed mass term exists ) topologically non-trivial ( 1 1, 2 1 violate TRS) Doubled Dirac Hamiltonian: H 2 (k) = H(k) 0 0 Ĥ µ (k) µ,, 2{+1, 1} Ĥ µ (k) =µk k m 0 3 extra symmetry preserving mass terms: e.g. for µ =+, =+, =+: 2 1 s 1, 1 2 s 2 =) =) gapped surface spectrum class AII in 2D has Z2 classification space of normalized mass matrices: R 2 = O(2N)/U(N) 0 (R 2 )=Z 2

20 Dirac Hamiltonian in symmetry class A One-dimensional Dirac Hamiltonian with rank 2: H(k) =k 1 + m 2 + µ 0 extra symmetry preserving mass term: M 3 =) class A in 1D is topologically trivial space of normalized mass matrices V A d=1,r=2 = { 2 cos + 3 sin 0 apple <2 } = S 1 C 1 : U(N) connectedness of space of normalized Dirac masses: 0 (C 1 )=0 Two-dimensional Dirac Hamiltonian with rank 2: H(k) =k x x + k y y + m z + µ 0 no extra mass term exists ) class A in 2D is topologically non-trivial describes two-dimensional Chern insulator Two-dimensional doubled Dirac Hamiltonian: H 2 (k) =H(k) 0 no extra gap opening mass term exists ) topologically non-trivial ) indicates Z classification

21 Homotopy classification of Dirac mass gaps s belongs to di erent The space of mass matrices Vd,r=N classifying spaces Cs d (for complex class ) or Rs d (for real class ) the relation between AZ symmetry class and classifying space is as follows: classifying space 0 ( ) C0 [N n)]} Z n=0 {U (N )/[U (n) U (N C1 U (N ) 0 R0 [N n)]} Z n=0 {O(N )/[O(n) O(N ) Z2 1 Path R connectedness of theo(n normalized Dirac masses R2 O(2N )/U (N ) Z2 Path connectedness of the R3 U (N )/Sp(N ) 0 Case (a): 0 (V ) = {0} Trivial phase R4 [N n)]} Z n=0 {Sp(N )/[Sp(n) Sp(N R5 Sp(N 0 Case (a): 0)(V ) = {0} R6 Sp(2N )/U (N ) 0 R7 U (N )/O(N ) ν= 1/2 0 ν=+1/2 Odd N Case (b): 0The (V ) = 0th Z 1D AZ class 2D AZ class AIII A A AIII BDI D D DIII DIII Dirac masses AII normalized AII CII CII C C CI Trivial phase CI AI AI BDI homotopy group indexes the disconnected parts TABLE spaces for complex (Cmatrices p ) and real (Rp ) Altland-Zirnbauer classes. of IV the Classifying space of normalized mass ν= 1 ν=0 ν=+1 ν= 1/2 ν=+1/2 Odd N Even N e normalized Dirac masses... 0 (V ) =Case 0 (b): (V )0 (V ) = Z2 = Z 0... language of K-theory similar to the ones we have diseven N cussed 2010b). For a fixed Case (c): 0 (V previously ) Trivial = Z2 phase (Teo and Kane,... ν=0 ν=1 AZ symmetry class and dimensions (d, D), the collection of defect Hamiltonians forms a commutative monoid (V ) = Z always assume a Hamiltonian has an eq cupied and unoccupied bands. Conside... ν= 1 ν=0 ν=+1 H ( H), where H inverts the occupi cupied ones. This sum is topological tri

22 Ten-fold classification of topological insulators and superconductors Ten-fold classification: classifies fully gapped topological materials in terms of non-spatial symmetries (i.e., symmetries that act locally in space) non-spatial symmetries: - time-reversal: - particle-hole: - sublattice: T H(k)T 1 =+H( k); T 2 = ±1 CH(k)C 1 = H( k); C 2 = ±1 SH(k)S 1 = H(k); S / TP ten symmetry classes Altland-Zirnbauer Random Matrix Classes Symmetry dim Class T CP S A Z 0 AIII Z 0 Z AI BDI Z 0 0 D Z 2 Z 0 DIII Z 2 Z 2 Z AII Z 2 Z 2 CII Z 0 Z 2 C Z 0 CI Z Chern insulator : integer classification 2: binary classification 0 : no topological state polyacetylene 2D topological insulator w/ SOC 3D topological insulator w/ SOC Schnyder, Ryu, Furusaki, Ludwig, PRB (2008) A. Kitaev, AIP (2009)

23 Ten-fold classification of topological insulators and superconductors Ten-fold classification: classifies fully gapped topological materials in terms of non-spatial symmetries (i.e., symmetries that act locally in space) non-spatial symmetries: - time-reversal: - particle-hole: - sublattice: T H(k)T 1 =+H( k); T 2 = ±1 CH(k)C 1 = H( k); C 2 = ±1 SH(k)S 1 = H(k); S / TP ten symmetry classes Altland-Zirnbauer Random Matrix Classes Symmetry dim Class T CP S A Z 0 AIII Z 0 Z AI BDI Z 0 0 D Z 2 Z 0 DIII Z 2 Z 2 Z AII Z 2 Z 2 CII Z 0 Z 2 C Z 0 CI Z : integer classification 2: binary classification 0 : no topological state chiral p-wave superconductor (Sr2RuO4) TRI topological triplet SC ( 3 He B) chiral d-wave superconductor

24 Ten-fold classification of topological insulators and superconductors Ten-fold classification: classifies fully gapped topological materials in terms of non-spatial symmetries (i.e., symmetries that act locally in space) non-spatial symmetries: - time-reversal: - particle-hole: - sublattice: T H(k)T 1 =+H( k); T 2 = ±1 CH(k)C 1 = H( k); C 2 = ±1 SH(k)S 1 = H(k); S / TP ten symmetry classes Altland-Zirnbauer Random Matrix Classes Symmetry Spatial Dimension d Class T C S A Z 0 Z 0 Z 0 Z AIII Z 0 Z 0 Z 0 Z 0 AI Z 0 Z 2 Z 2 Z BDI Z Z 0 Z 2 Z 2 D Z 2 Z Z 0 Z 2 DIII Z 2 Z 2 Z Z 0 AII Z 2 Z 2 Z Z CII Z 0 Z 2 Z 2 Z C Z 0 Z 2 Z 2 Z 0 0 CI Z 0 Z 2 Z 2 Z 0

25 Ten-fold classification of topological insulators and superconductors Ten-fold classification: classifies fully gapped topological materials in terms of non-spatial symmetries (i.e., symmetries that act locally in space) Altland-Zirnbauer Random Matrix Classes Symmetry Spatial Dimension d Class T C S A Z 0 Z 0 Z 0 Z AIII Z 0 Z 0 Z 0 Z 0 AI Z 0 Z 2 Z 2 Z BDI Z Z 0 Z 2 Z 2 D Z 2 Z Z 0 Z 2 DIII Z 2 Z 2 Z Z 0 AII Z 2 Z 2 Z Z CII Z 0 Z 2 Z 2 Z C Z 0 Z 2 Z 2 Z 0 0 CI Z 0 Z 2 Z 2 Z 0 Topological invariants: Chern numbers and winding numbers Z n+1 1 if Ch n+1 [F] = tr (n + 1)! BZ d=2n+2 2 2n+1 [q] = ( n+1 Z 1)n n! i 1 2 tr q 1 (2n + 1)! 2 1 q q 1 2 q d 2n+1 k BZ

26 Extension I: Weak topological insulators and supercondutors strong topological insulators (superconductors): not destroyed by positional disorder weak topological insulators (superconductors): only possess topological features when translational symmetry is present weak topological insulators (superconductors) are topologically equivalent to parallel stacks of lowerdimensional strong topological insulator (SCs). co-dimension k=1 co-dimension k=2 AZ A AIII AI BDI D DIII AII CII C CI Symmetry T Dimension C S Z 0 Z Z 0 Z Z Z Z 2 Z 0 0 Z 2 Z 2 Z 0 0 Z 2 Z 2 Z Z 0 Z 2 Z 2 0 Z 0 Z Z 0 I. INTRODUCTION 0 <k d and d-dim.weak topological insulators (SCs) of co-dimension k can occur whenever there exists a strong topological state in same symmetry class but in (d-k) I. dimensions. INTRODUCTION ( d k ) top. invariants 0 <k d and cf. Kitaev, AIP Conf Proc. 1134, 22 (2009) λ 0 / te

27 Extension II: Zero mode localized on topological defect Protected zero modes can also occur at topological defects in D-dim systems Point defect (r=0): Hedgehog (D=3), vortex (D=2), domain wall (D=1) AZ A AIII AI BDI D DIII AII CII C CI Symmetry T Dimension C S Z 0 Z Z 0 Z Z Z Z 2 Z 0 0 Z 2 Z 2 Z 0 0 Z 2 Z 2 Z Z 0 Z 2 Z 2 0 Z 0 Z Z 0 Line defect (r=1): dislocation line (D=3) domain wall (D=2) Two-dim defects (r=2): domain wall (D=3) Freedman, et. al., PRB (2010) Teo & Kane, PRB (2010) Ryu, et al. NJP (2010) Can an r-dimensional topological defect of a given symmetry class bind gapless states or not? look at column d=(r+1) (answer does not depend on D!) line defect in class A: n = 1 Tr[F F] 8π 2 T 3 S 1 (second Chern no = no of zero modes)