Classification of Topological Insulators and Superconductors

Classification of topological insulators and superconductors

Shinsei Ryu (Berkeley)

in collaboration with

Andreas Schnyder (KITP, UCSB) Akira Furusaki (RIKEN, Japan) Andreas Ludwig (UCSB) Christopher Mudry (PSI, Switzerland) Hideaki Obuse (RIKEN, Japan) Kentaro Nomura (Tohoku, Japan) Mikito Koshino (Titech, Japan) question

How many different topoloigcal insulators and superconductors are there in nature ? question

How many different topoloigcal insulators and superconductors are there in nature ? topological:

- support stable gapless modes at boundaries, possibly topologcial non-topological in the presence of general discrete symmetries insulator (vacuum) - states with and without boundary modes are not adiabatically connected

- may be characterized by a bulk topological invariant of some sort quantum phase transition

space of all ground states

···-2-1 0 1 2 ··· 0 1 topological insulators; examples

(i) IQHE in 2D, strong T breaking by B

a) quantized Hall conductance b) stable edge states

e2 TKNN (82) Halperin (82) IQHE σ ∈ Z × xy h Laughlin (81)

(ii) Z2 topological insulator (QSHE) in 2D TRI

T iσyH (−iσy)= H (iii) Z2 topological insulator in 3D )

- characterized by Z2 topological number∆=0,1 - stable edge/surface states classification of discrete symmetries

-natural framework: random matrix theory (RMT) Wigner-Dyson Zirnbauer (96), Altland &Zirnbauer (97) two types of anti-unitary symmetries

integer spin particle Time-Reversal Symmetry (TRS) 0 no TRS TRS = +1 TRS with T T =+T TH∗T −1 = H ( -1 TRS with T T = −T

half-odd integer spin particle Particle-Hole Symmetry (PHS) 0 no PHS PHS = +1 PHS with CT =+C CHT C−1 = −H ( -1 PHS with CT = −C

PHS + TRS = chiral symmetry

T H∗T −1 = H TCH(TC)−1 = −H CH∗C−1 = −H ) classification of discrete symmetries

-natural framework: random matrix theory (RMT)

Wigner-Dyson Zirnbauer (96), Altland &Zirnbauer (97)

TRS PHS SLS description RM ensembles

Wigner-Dyson A 0 0 0 unitary U(N) (standard) AI +1 0 0 orthogonal U(N)/O(N) AII -1 0 0 symplectic (spin-orbit) U(2N)/Sp(N)

chiral AIII 0 0 1 chiral unitary U(2N)/U(N) × U(N) (sublattice) BDI +1 +1 1 chiral orthogonal O(2N)/O(2N) × O(2N) CII -1 -1 1 chiral symplectic Sp(4N)/Sp(2N) × Sp(2N) D 0 +1 0 singlet/triplet SC O(N) BdG C 0 -1 0 singlet SC Sp(N) DIII -1 +1 1 singlet/triplet SC with TRS O(2N)/U(N) CI +1 -1 1 singlet SC with TRS Sp(N)/U(N) classification of discrete symmetries

-natural framework: random matrix theory (RMT)

Wigner-Dyson Zirnbauer (96), Altland &Zirnbauer (97)

TRS PHS SLS description

Wigner-Dyson A 0 0 0 unitary (standard) AI +1 0 0 orthogonal AII -1 0 0 symplectic (spin-orbit)

chiral AIII 0 0 1 chiral unitary (sublattice) BDI +1 +1 1 chiral orthogonal CII -1 -1 1 chiral symplectic D 0 +1 0 singlet/triplet SC BdG C 0 -1 0 singlet SC DIII -1 +1 1 singlet/triplet SC with TRS CI +1 -1 1 singlet SC with TRS

-IQHE is a topological insulator in unitary class (A). -Z2 toplological insulator is a topological insulator in symplectic class (AII). -Is there a topological insulator in other symmetry classes ? BdG symmetry classes

- S^z non-conserving SC c ↑ c 1 ξ ∆ † T H = c† , c† , c , c H  ↓  H = ξ = ξ , ∆= −∆ ↑ ↓ ↑ ↓ c† −∆∗ −ξT 2  ↑  !  †   c   ↓    TR SU(2) examples in 2D T D τxH τx = −H spinless chiral p-wave T T DIII τxH τx = −H, σyH σy = H p-wave

- S^z conserving SC c c† c ↑ ξ↑ ∆ † H = ( ↑, ↓)H c† H = † T ξσ = ξσ ↓ ! ∆ −ξ↓ !

TR SU(2) examples in 2D A no constraint spinfull chiral p-wave

AIII τyHτy = −H p-wave T C τyH τy = −H d+id -wave T ∗ CI τyH τy = −H, H = H d-wave, s-wave sublattice symmetry classes

c c† c† A 0 D 1 0 H = ( A, B)H c H = † γH = −Hγ γ = B ! D 0 ! 0 −1 !

TR SU(2) examples

AIII τyHτy = −H random flux model ∗ BDI τyHτy = −H, H = H random hopping model Dyson (53) T Gade (93) CII τyHτy = −H, σyH σy = H

- Classes CI and DIII have an off-diagonal form ! (will be important later)

PHS + TRS = chiral (sublattice) symmetry

T HT T −1 = H TCH(TC)−1 = −H CHT C−1 = −H classification of 3D topological insulators

Schnyder, SR, Furusaki, Ludwig (2008) RESULT:

-3D topological insulators for 5 out of 10 symmetry classes

AIII, DIII, CI : top. insulators labeled by an integer AII, CII: top. insulators of Z2 type

TRS PHS SLS description

Wigner-Dyson A 0 0 0 unitary (standard) AI +1 0 0 orthogonal AII -1 0 0 symplectic (spin-orbit)

Schnyder, SR, Furusaki, Ludwig (2008) underlying strategy

- discover a topological invariant

integer topological invairant for 3 out of 5 classes

d3k ν = ǫµνρtr [(q−1∂ q)(q−1∂ q)(q−1∂ q)] 24π2 µ ν ρ ZBz

q : BZ U(m) spectral projector

- bulk-boundary correspondence

absence of Anderson localization at boundaries topological distinction of ground states

projector: Q(k) = 2 |ua(k)ihua(k)|− 1 ε(k) empty a∈Xﬁlled ε Q2 = 1, Q† = Q, tr Q = m − n F filled filled empty k Q : BZ U(m + n)/U(m) × U(n)

quantum ground state = map from Bz onto Grassmannian

π2[U(m + n)/U(m) × U(n)] = Z IQHE in 2D

π3[U(m + n)/U(m) × U(n)] = 0 no top. insulator in 3D without constraint (Class A) (for large enough m,n) topological distinction of ground states

-projectors in classes AIII

0 q(k) chiral symmetry ΓHΓ= −H Q(k)= † q (k) 0 !

q : BZ U(m)

π3[U(m)] = Z topological insulators labeled by an integer

1 ν = tr [(q−1dq)3] 24π2 ZBz

-discrete symmetries limit possible values of nu

T Z q (−k)= −q(k) DIII AIII & DIII ν ∈ T CI Z q (−k)= q(k) CI ν ∈ 2 CII & BDI q∗(−k)= q(k) BDI ν =0 ∗ iσyq (−k)(−iσy)= −q(k) CII Z2 insulators in CII (later) Anderson delocalization at boundaries topological bulk

TRS PHS SLS fermionic replica NLsM U(2N)/U(N) × U(N) Wigner-Dyson A 0 0 0 Pruisken (standard) AI +1 0 0 Sp(4N)/Sp(2N) × Sp(2N) AII -1 0 0 O(2N)/O(2N) × O(2N) Z2 U(N) chiral AIII 0 0 1 WZW (sublattice) BDI +1 +1 1 U(2N)/Sp(N) CII -1 -1 1 U(N)/O(N) Z2 D 0 +1 0 O(2N)/U(N) Pruisken BdG C 0 -1 0 Sp(N)/U(N) Pruisken newly derived ! DIII -1 +1 1 O(N) WZW CI +1 -1 1 Sp(N) WZW

- Bernard-Le Clair: 13-fold symmetry classifcation of 2d Dirac fermions - AIII, CI, DIII; exact results - "abnormal terms" in NLsM − 1 − Z = D[g]e2πiνΓWZW e S[g] Γ = tr (g 1dg)3 WZW type WZW 2 24π 3 Z ZM − Z2 type Z = D[Q](−1)N[Q]e S[Q] SR, Mudry, Obuse Furusaki (07) Z characterization at boundaries

-classification of 2D Dirac Hamiltonians Bernard-LeClair (2001)

V + V −i∂¯ + A 13 classes (not 10 !) H = + − + +i∂ + A− V+ − V− ! AIII, CI, DIII has an extra class.

TR SU(2) description

Wigner-Dyson A unitary (standard) AI orthogonal AII symplectic (spin-orbit) even/odd effect

chiral AIII chiral unitary (sublattice) AIII chiral unitary extra BDI chiral orthogonal CII chiral symplectic always gapless

BdG C singlet SC D singlet/triplet SC CI singlet SC CI singlet SC extra DIII singlet/triplet SC DIII singlet/triplet SC extra 3He is a 3D topological insulator

- Class DIII top. insulator: B 3He kz |∆| ξ ∆ H = ∗ T −∆ −ξ ! ky

2 k kx ξk = − µ ∆k = |∆|iσ k · σ 2m y isotropic gap

strong pairing weak pairing Z2 classification: µ Roy (2008) ν =0 0 ν =1 Qi-Hughes-Raghu-Zhang (2008)

-stable surface Majorana fermion state Salomma and Volovik (1988) topological singlet superconductor in 3D

- class CI top. insulator: singlet BCS pairing model on the diamond lattice SU(2) symmetric

d3z2−r2-wave like (?) pairing

∆ > 0

∆′ < 0

t nn: hopping Delta: nn d-wave pairing t': nnn hopping mu_s: staggered chemical potential four-nodes, two-fold degenerate = two 8-component Dirac topological singlet superconductor in 3D

staggered chemical potential ν = ±2

nnn hopping nnn hopping t and Delta only

d3k ν = ǫµνρtr [(q−1∂ q)(q−1∂ q)(q−1∂ q)] 24π2 µ ν ρ ZBz surface of 3d top. singlet SC = "1/2 of cuprate" surface Bz

-- stable surf. Dirac fermions -- surface Dirac fermion + disorder s2 σspin 1 N exactly solvable = π × 2 × × h Tsvelik (1995) (irrespective of disorder strength)

-- T-breaking -> half spin quantum Hall effect ("1/2 of d+id SC") summary

- 3D topological insulators for 5 out of 10 symmetry classes.

AIII, DIII, CI : top. insulators labeled by an integer

AII, CII: top. insulators of Z2 type - Topological insulator /Anderson delocalization correspondence

surface of top. insulator is always conducting.

- The same strategy is applicable to other dimensions.

- Transport experiments on Bismuth-Antimony ?

perfectly conducting because of Z2 topological term

- Topological field theory ?

θ 4 µνρλ S d x ǫ F F Aµ ∈ SU(2) = 2 tr [ µν ρλ] 32π Z summary

IQHE QSHE Z2 top. ins polyacetylene

TRS PHS SLS description d=1 d=2 d=3

Wigner-Dyson A 0 0 0 unitary 0 Z 0 (standard) AI +1 0 0 orthogonal 0 0 0

AII -1 0 0 symplectic (spin-orbit) 0 Z2 Z2 chiral AIII 0 0 1 chiral unitary Z 0 Z (sublattice) BDI +1 +1 1 chiral orthogonal Z 0 0 CII -1 -1 1 chiral symplectic Z 0 Z2 D 0 +1 0 singlet/triplet SC Z2 Z 0 BdG C 0 -1 0 singlet SC 0 Z 0 DIII -1 +1 1 singlet/triplet SC with TRS Z2 Z2 Z CI +1 -1 1 singlet SC with TRS 0 0 Z

chiral p-wave SC

- 3D weak topological insulators are also possible d+id wave SC 3He B in classes A, AII, D, C, CI topological field theory description

- generating function for single particle Green's function

† − d3xL † Z = D[ψ , ψ]e L = ψ i(H− iη)ψ (3+0) dim field theory Z R - introduce external gauge fields

L = ψ¯(∂µγµ − iaµγµ − ibµγ0γµ + mγ5)ψ

- integrate over fermions

¯ e−Seﬀ [aµ,bµ] = D[ψ,¯ ψ]e−S[aµ,bµ,ψ,ψ] Z + − ± Seﬀ = ν I[A ] − I[A ] Aµ = aµ ± bµ − i 2i I[A]= d3xǫµνλ A ∂ A + A A A 4π µ ν λ 3 µ ν λ Z non Abelian doubled Chern-Simons