Classification of topological quantum matter with reflection symmetries

Transcription

1 Classification of topological quantum matter with reflection symmetries Andreas P. Schnyder Max Planck Institute for Solid State Research, Stuttgart June 14th, 2016 SPICE Workshop on New Paradigms in Dirac-Weyl Nanoelectronics

2 Outline 0. Introduction: Topological band theory 1. Topological insulators with reflection symmetry - Ca3PbO, Sr3PbO, Ba3PbO arxiv: Mirror plane 2. Topological nodal line semi-metals - Ca3P2, ZrSiS PRB 93, (2016) 3. Nodal non-centrosymmetric superconductors - CePt3Si (c) E 4. Conclusions & Outlook k x k z Review articles: arxiv: ; J. Phys.: Condens. Matter 27, (2015) 2

3 Topological band theory Consider band structure: H(k) u n (k) = E n (k) u n (k) (i) Topological equivalence for insulators (superconductors): Energy gap Energy gap π/a π/a crystal momentum k x π/a π/a crystal momentum k x (ii) Topological equivalence for band crossings (nodes in SCs): Energy Energy π/a π/a crystal momentum π/a. symmetries to consider: time-reversal symmetry, particle-hole, reflection. top. equivalence classes distinguished by: n Z = i Z F dk 2 Z 2 filled Bulk-boundary correspondence: states n Z = # gapless edge states (or surface states) k x π/a crystal momentum k x topological invariant

4 Reflection symmetry Consider reflection R: R 1 H( k x,k y,k z )R = H(k x,k y,k z ) with R = s x x! x w.l.o.g.: eigenvalues of R 2 { 1, +1} z y x mirror Chern number: k x =0 =) H(0,k y,k z )R RH(0,k y,k z )=0 H(0,k y,k z ) project onto eigenspaces of : n ± M = 1 Z 4 F ± d 2 k 2D BZ Berry curvature in R H ± (k y,k z ) ± eigenspace Teo, Fu, Kane PRB 08 Mirror plane total Chern number: n M = n + M + n M mirror Chern number: n M = n + M n M Bulk-boundary correspondence: zero-energy states on surfaces that are left invariant under the mirror symmetry

5 Classification of topological materials with reflection symmetry R + : R commutes with T (C or S) R : R anti-commutes with T (C or S) Reflection sym. class d=1 d=2 d=3 d=4 d=5 d=6 d=7 d=8 Bott cube R A MZ 0 MZ 0 MZ 0 MZ 0 R + AIII 0 MZ 0 MZ 0 MZ 0 MZ R AIII MZ Z 0 MZ Z 0 MZ Z 0 MZ Z 0 AI MZ t = MZ t =3 0 MZ 2 MZ 2 BDI MZ 2 MZ MZ 0 MZ 2 D MZ 2 MZ 2 MZ MZ 0 R +,R ++ DIII 0 MZ 2 MZ 2 MZ MZ AII 2MZ 0 MZ 2 MZ 2 MZ CII 0 2MZ 0 MZ 2 MZ 2 MZ 0 0 C 0 0 2MZ 0 MZ 2 MZ 2 MZ 0 CI MZ 0 MZ 2 MZ 2 MZ AI 0 0 2MZ 0 T Z 2 Z 2 MZ 0 BDI MZ 0 T Z Z MZ t D MZ t = MZ = T Z 2 Z 2 R,R DIII Z 2 MZ MZ 0 T Z 2 AII T Z 2 Z 2 MZ MZ 0 CII 0 T Z 2 Z 2 MZ MZ C 2MZ 0 T Z 2 Z 2 MZ CI 0 2MZ 0 T Z 2 Z 2 MZ 0 0 R + BDI, CII 2Z 0 2MZ 0 2Z 0 2MZ 0 R + DIII, CI 2MZ 0 2Z 0 2MZ 0 2Z 0 R + BDI MZ Z MZ 2Z 0 MZ 2 Z 2 MZ 2 Z 2 R + DIII MZ 2 Z 2 MZ 2 Z 2 MZ Z MZ 2Z 0 R + CII 2MZ 2Z 0 MZ 2 Z 2 MZ 2 Z 2 MZ Z R + CI 0 0 2MZ 2Z 0 MZ 2 Z 2 MZ 2 Z 2 MZ Z 0 a Z and Z invariants only protect Fermi surfaces of dimension zero ( ) at high-symmetry points of the Brillouin Morimoto, Furusaki PRB 2013; Chiu, Schnyder PRB 2014;

6 Classification of topological materials with reflection symmetry R + : R commutes with T (C or S) R : R anti-commutes with T (C or S) TI/TSC d=1 d=2 d=3 d=4 d=5 d=6 d=7 d=8 Reflection FS1 p=8 p=1 p=2 p=3 p=4 p=5 p=6 p=7 FS2 p=2 p=3 p=4 p=5 p=6 p=7 p=8 p=1 R A MZ 0 MZ 0 MZ 0 MZ 0 R + AIII 0 MZ 0 MZ 0 MZ 0 MZ R AIII MZ Z 0 MZ Z 0 MZ Z 0 MZ Z 0 R +,R ++ R,R AI MZ MZ 0 MZ 2 MZ 2 BDI MZ 2 MZ MZ 0 MZ 2 D MZ 2 MZ 2 MZ MZ 0 DIII 0 MZ 2 MZ 2 MZ MZ AII 2MZ 0 MZ 2 MZ 2 MZ CII 0 2MZ 0 MZ 2 MZ 2 MZ 0 0 C 0 0 2MZ 0 MZ 2 MZ 2 MZ 0 there a topological insulator or topological CI MZ 0 MZ 2 MZ 2 MZ?For which symmetry class and dimension is semi-metal protected by reflection symmetry? AI 0 0 2MZ 0 T Z 2 Z 2 MZ 0 BDI MZ 0 T Z 2 Z 2 MZ D MZ MZ 0 T Z 2 Z 2 DIII Z 2 MZ MZ 0 T Z 2 AII T Z 2 Z 2 MZ MZ 0 CII 0 T Z 2 Z 2 MZ MZ C 2MZ 0 T Z 2 Z 2 MZ CI 0 2MZ 0 T Z 2 Z 2 MZ 0 0 R + BDI, CII 2Z 0 2MZ 0 2Z 0 2MZ 0 R + DIII, CI 2MZ 0 2Z 0 2MZ 0 2Z 0 R + BDI MZ Z MZ 2Z 0 MZ 2 Z 2 MZ 2 Z 2 R + DIII MZ 2 Z 2 MZ 2 Z 2 MZ Z MZ 2Z 0 R + CII 2MZ 2Z 0 MZ 2 Z 2 MZ 2 Z 2 MZ Z R + CI 0 0 2MZ 2Z 0 MZ 2 Z 2 MZ 2 Z 2 MZ Z 0

7 Classification of topological materials with reflection symmetry R + : R commutes with T (C or S) R : R anti-commutes with T (C or S) TI/TSC d=1 d=2 d=3 d=4 d=5 d=6 d=7 d=8 Reflection FS1 p=8 p=1 p=2 p=3 p=4 p=5 p=6 p=7 FS2 p=2 p=3 p=4 p=5 p=6 p=7 p=8 p=1 R A MZ 0 MZ 0 MZ 0 MZ 0 R + AIII 0 MZ 0 MZ 0 MZ 0 MZ R AIII MZ Z 0 MZ Z 0 MZ Z 0 MZ Z 0 R +,R ++ R,R AI MZ MZ 0 MZ 2 MZ 2 BDI MZ 2 MZ MZ 0 MZ 2 D MZ 2 MZ 2 MZ MZ 0 DIII 0 MZ 2 MZ 2 MZ MZ AII 2MZ 0 MZ 2 MZ 2 MZ CII 0 2MZ 0 MZ 2 MZ 2 MZ 0 0 C 0 0 2MZ 0 MZ 2 MZ 2 MZ 0 CI MZ 0 MZ 2 MZ 2 MZ AI 0 0 2MZ 0 T Z 2 Z 2 MZ 0 BDI MZ 0 T Z 2 Z 2 MZ D MZ MZ 0 T Z 2 Z 2 DIII Z 2 MZ MZ 0 T Z 2 AII T Z 2 Z 2 MZ MZ 0 CII 0 T Z 2 Z 2 MZ MZ C 2MZ 0 T Z 2 Z 2 MZ CI 0 2MZ 0 T Z 2 Z 2 MZ 0 0 R + BDI, CII 2Z 0 2MZ 0 2Z 0 2MZ 0 R + DIII, CI 2MZ 0 2Z 0 2MZ 0 2Z 0 R + BDI MZ Z MZ 2Z 0 MZ 2 Z 2 MZ 2 Z 2 R + DIII MZ 2 Z 2 MZ 2 Z 2 MZ Z MZ 2Z 0 R + CII 2MZ 2Z 0 MZ 2 Z 2 MZ 2 Z 2 MZ Z R + CI 0 0 2MZ 2Z 0 MZ 2 Z 2 MZ 2 Z 2 MZ Z 0 Chiu, Schnyder PRB 2014

8 Classification of topological materials with reflection symmetry R + : R commutes with T (C or S) R : R anti-commutes with T (C or S) TI/TSC d=1 d=2 d=3 d=4 d=5 d=6 d=7 d=8 Reflection FS1 p=8 p=1 p=2 p=3 p=4 p=5 p=6 p=7 FS2 p=2 p=3 p=4 p=5 p=6 p=7 p=8 p=1 R A MZ 0 MZ 0 MZ 0 MZ 0 R + AIII 0 MZ 0 MZ 0 MZ 0 MZ R AIII MZ Z 0 MZ Z 0 MZ Z 0 MZ Z 0 R +,R ++ R,R Ca3P2 AI MZ MZ 0 MZ 2 MZ 2 BDI MZ 2 MZ CePt3Si MZ 0 MZ 2 D MZ 2 MZ 2 MZ MZ 0 DIII 0 MZ 2 MZ 2 MZ MZ AII 2MZ 0 MZ 2 MZ 2 MZ CII 0 2MZ 0 MZ 2 MZ 2 MZ 0 0 C 0 0 2MZ 0 MZ 2 MZ 2 MZ 0 CI MZ 0 MZ 2 MZ 2 MZ AI 0 0 2MZ 0 T Z 2 Z 2 MZ 0 BDI MZ 0 T Z 2 Z 2 MZ D MZ Ca3PbO, 2MZ Sr3PbO 0 T Z 2 Z 2 DIII Z 2 MZ MZ 0 T Z 2 AII T Z 2 Z 2 MZ MZ 0 CII 0 T Z 2 Z 2 MZ MZ C 2MZ 0 T Z 2 Z 2 MZ CI 0 2MZ 0 T Z 2 Z 2 MZ 0 0 R + BDI, CII 2Z 0 2MZ 0 2Z 0 2MZ 0 R + DIII, CI 2MZ 0 2Z 0 2MZ 0 2Z 0 R + BDI MZ Z MZ 2Z 0 MZ 2 Z 2 MZ 2 Z 2 R + DIII MZ 2 Z 2 MZ 2 Z 2 MZ Z MZ 2Z 0 R + CII 2MZ 2Z 0 MZ 2 Z 2 MZ 2 Z 2 MZ Z R + CI 0 0 2MZ 2Z 0 MZ 2 Z 2 MZ 2 Z 2 MZ Z 0 Chiu, Schnyder PRB 2014

9 1. Topological insulators with reflection symmetry Y. Nohara (MPI-FKF) Yang-Hao Chan (A. Sinica) Ching-Kai Chiu (UMD) Ca3PbO, Sr3PbO

10 Ca3PbO is a reflection symmetry protected TI Anti-perovskites: Ca3PbO, Sr3PbO in collaboration with A. Rost, H. Takagi Ca Band structure (without SOC): Ca d Ca 3d [Thesis] January, 2012 (a) Opening of bulk gap: Pb 6p O Λ2p Δ Pb 6s X e band structure (e) of Ca 3 PbO obtained in the p ) shows the total Figure and partial 2.5: DOS (a) Momentum for each atom, path in the cubic Brillouin zone on which the x band OS for eachhybridisation atomstructure decomposed is calculated. w/ intoca components d(b) xy i Positions, d xz i of, d the yz Dirac i points in the entire Brillouin p symmetry. In (e), zone. the position of Dirac point y te that the + energy SOC is measured opens up frombulk the Fermi gap of ~10 mev p z n the p-bands and d-bands, two bands cross the Fermi Γ Pb p O p Σ Pb s R M (a) (b) Γ Δ Λ Σ X R M Pb O Orbital character of bands: at the Fermi energy, the Fermi energy must be at the Dirac [after point Kariyado in order and toogata, maintain JPSJ the 12] Pb: Ca: p x i, p y i, p z i d x 2 y 2, dx 2 z 2i, d y 2 z 2 (a) ( Ca1 Ca2 Ca3 z y x

11 Ca3PbO is a reflection symmetry protected TI Symmetries: Time-reversal: T 1 H( k)t =+H(k) two reflection symmetries : R 1 and R 2 T = is y K R R j anti-commutes with T : TR j T 1 = R j Ca Pb O =) two mirror Chern numbers: n M1,n M2 R 1 reflection R 2 reflection

12 Ca3PbO is a reflection symmetry protected TI Symmetries: Time-reversal: T 1 H( k)t =+H(k) two reflection symmetries : R 1 and R 2 R j anti-commutes with T : TR j T 1 = R j =) two mirror Chern numbers: T = is Ca y K R Pb n M1,n M2 O Effective low-energy Hamiltonian for one Dirac cone within R1 mirror plane: H ± (k y,k z )=± sin k z x ± sin k y y ± " k z = m ± (k) ~ E = ± m ± (k) trivial phase ˆm ± = m ±(k) m ± (k) non-trivial phase R 1 reflection R 2 reflection µ > 6t m z ± µ < 6t m z ± n ± =0 n ± = ±1 m x ± m x ± m y ± =) n ± = 1 8 Z 2D BZ m y ± d 2 k µ ˆm kµ ˆm k ˆm ±

13 Energy Energy Ca3PbO is a reflection symmetry protected TI Mirror Chern numbers: k y for Ca3PbO: n M1 = 2, n M2 =+2 Bulk-boundary correspondence: n M =# Dirac cone surface states E =5.6 ev Dirac cone surface states on (001) surface: k x n M1 = 2 n M2 =+2 X X M M Chiu, Chan, Nohara, Schnyder, arxiv:

14 Ca3PbO is a reflection symmetry protected TI Type-II Dirac states on (111) surface: H surf TCI (k x, k y ) = Ak y 0 + k y x k x y A>0 : type-ii Dirac state Mirror symmetry: R x = x R x HTCI surf ( k ar x, k y )Rx 1 term = HTCI surf A (k x, k y ) NB: Ak y 0 is forbidden by TRS ) dense Landau level spectrum Chiu, Chan, Nohara, Schnyder, arxiv:

15 2. Topological nodal line semi-metals Yang-Hao Chan (A. Sinica) Ching-Kai Chiu (UMD) Ca3P2

16 Classification of topological materials with reflection symmetry R + : R commutes with T (C or S) R : R anti-commutes with T (C or S) TI/TSC d=1 d=2 d=3 d=4 d=5 d=6 d=7 d=8 Reflection FS1 p=8 p=1 p=2 p=3 p=4 p=5 p=6 p=7 FS2 p=2 p=3 p=4 p=5 p=6 p=7 p=8 p=1 R A MZ 0 MZ 0 MZ 0 MZ 0 R + AIII 0 MZ 0 MZ 0 MZ 0 MZ R AIII MZ Z 0 MZ Z 0 MZ Z 0 MZ Z 0 R +,R ++ R,R Ca3P2 AI MZ MZ 0 MZ 2 MZ 2 BDI MZ 2 MZ MZ 0 MZ 2 D MZ 2 MZ 2 MZ MZ 0 DIII 0 MZ 2 MZ 2 MZ MZ AII 2MZ 0 MZ 2 MZ 2 MZ CII 0 2MZ 0 MZ 2 MZ 2 MZ 0 0 C 0 0 2MZ 0 MZ 2 MZ 2 MZ 0 CI MZ 0 MZ 2 MZ 2 MZ AI 0 0 2MZ 0 T Z 2 Z 2 MZ 0 BDI MZ 0 T Z 2 Z 2 MZ D MZ MZ 0 T Z 2 Z 2 DIII Z 2 MZ MZ 0 T Z 2 AII T Z 2 Z 2 MZ MZ 0 CII 0 T Z 2 Z 2 MZ MZ C 2MZ 0 T Z 2 Z 2 MZ CI 0 2MZ 0 T Z 2 Z 2 MZ 0 0 R + BDI, CII 2Z 0 2MZ 0 2Z 0 2MZ 0 R + DIII, CI 2MZ 0 2Z 0 2MZ 0 2Z 0 R + BDI MZ Z MZ 2Z 0 MZ 2 Z 2 MZ 2 Z 2 R + DIII MZ 2 Z 2 MZ 2 Z 2 MZ Z MZ 2Z 0 R + CII 2MZ 2Z 0 MZ 2 Z 2 MZ 2 Z 2 MZ Z R + CI 0 0 2MZ 2Z 0 MZ 2 Z 2 MZ 2 Z 2 MZ Z 0 Chiu, Schnyder PRB 2014

17 Topological nodal lines in Ca3P2 see talk by Leslie Schoop Band structure: Crystal structure P63/mcm Ca P mirror plane Dirac ring within reflection plane charge balanced: Ca 2+ P 3- M A kz K K' Orbital character of bands near EF: (6 Ca atoms, 6 P atoms) Ca: d z 2 orbitals from 6 Ca atoms M kx K K' ky P: p x orbitals from 6 P atoms Chan, Chiu, Chou, Schnyder, Phys. Rev. B 93, (2016)

18 Topological nodal line: Mirror invariant Reflection ( z! z ): P4 Ca4 Ca6 P6 R 1 H(k x,k y, k z )R = H(k x,k y,k z ) R(k) = mirror plane B e ik z 0 (lower plane) C e ik z C A mirror plane P4 P2 Ca2 Ca4 Ca6 Ca1 P6 P1 px dz2 dz2 Mirror invariant: number of occupied states with R =+1 N 0 MZ = n +,0 occ ( k >k 0 ) n +,0 occ ( k <k 0 ) R =+1 n +,0 occ (k) = 1 k <k0 0 k >k 0 R = 1 Chan, Chiu, Chou, Schnyder, Phys. Rev. B 93, (2016) 0.4 +k 0 M k K k 0

19 Drumhead surface state and Berry phase Berry phase & charge polarization: P(k k )= i X j2filled Z D u (j) u (j) k? E dk? Berry phase P(k k ) quantized to ) stable line node In Ca3P2 Berry phase is quantized due to: (i) reflection symmetry z! z (ii) inversion + time-reversal symmetry ( 1) n+,0 occ (k)+n+, occ (k) e i@r = e ip(k) M kz K K' A M kx K K' ky Chan, Chiu, Chou, Schnyder, Phys. Rev. B 93, (2016)

20 Drumhead surface state and Berry phase Berry phase & charge polarization: P(k k )= i X j2filled Z D u (j) u (j) k? E dk? Berry phase P(k k ) quantized to ) stable line node In Ca3P2 Berry phase is quantized due to: (i) reflection symmetry z! z (ii) inversion + time-reversal symmetry ( 1) n+,0 occ (k)+n+, occ (k) e i@r = e ip(k) (b) Surface spectrum Bulk-boundary correspondence: surface charge: surf = e 2 P mod e ) Nearly flat 2D surface states connecting Dirac ring Chan, Chiu, Chou, Schnyder, Phys. Rev. B 93, (2016)

21 Drumhead surface state and Berry phase Nearly flat surface states connecting Dirac ring M kz K K' A M kx K K' ky Drumhead surface state E k y k x Chan, Chiu, Chou, Schnyder, arxiv:

22 Low-energy effective theory for Ca3P2 low-energy effective Hamiltonian: even in k H e (k) =(k 2 k k 2 0) z + k z y + f(k) 0 symmetry operators: reflection: R = z time-reversal: T = 0 K inversion: I = z Gap-opening term x is symmetry forbidden: breaks reflection symmetry: breaks inversion + TRS: Z versus Z 2 classification: R 1 x R = (IT) 1 x IT = x x ) nodal line is stable H e (k) 0 =(k 2 k k 2 0) z 0 + k z y 0 + f(k) 0 0 consider gap opening term ˆm = x y : (IT)-symmetric: ( z 0 K) 1 ˆm( z 0 K)= ˆm but breaks R: ( z 0 ) 1 ˆm( z 0 ) 6= ˆm ) Z 2 classification ) Z classification

23 3. Nodal non-centrosymmetric superconductors R. Queiroz (MPI-FKF) C. Timm (TU Dresden) P. Brydon (U Otago) CePt3Si

24 Nodal non-centrosymmetric superconductors [E. Bauer et al. PRL 04] Lack of inversion causes anti-symmetric SO coupling: Normal state: H = X k k (" k 0 + g k 3) k SO coupling for C4v point group: g k = k y ˆx k x ŷ Lack of inversion allows for admixture of spin-singlet and spin-triplet pairing components k =( s 0 + t d k ~ ) i y ( g k d k ) Gaps on the two Prof. Fermi Prof. M. surfaces: Sigrist, Prof. M. Sigrist, M. WS05/06 Sigrist, ± WS05/06 k = WS05/06 s ± p d k Festkörperphysik II, Musterlösun k z PRL 102, (2009) PRL 102, PHYSICAL (2009) REVIEW PHYL s > s t > s ts > st t st s < s t < t ts < t Festkörperphysik II, Muster Festkörperphysik II, Musterlösung k x negative helicity FS full gap and and line nodes I y = e L 1 x /2 I y = e L x /2 1 I y 2~ = e 1 2~ N y 2~ k y Nn=1 y N y k y full gap n=1 0 0 k y L x /2 de de n=1 {2t sin k y n (E,k y ) 0 for the R cases fro theoretic cos k y x n(e,k {2t sin k {2t y n (E,k sin k y y ) n (E,k cos y ) k y x cos k n(e, de lying co determin

25 Nodal non-centrosymmetric superconductors Symmetries: Time-reversal and particle-hole: T = 0 i 2 T 2 = 1 C = 1 0 C 2 =+1 class DIII 2D surface Brillouin zone 3D Brillouin zone Fermi surface 1D contour in general not centrosymmetric: TRS, PHS S=TRS x PHS Winding number: =) class AIII non trivial projected = +/ 1 Fermi surface trivial = 0 projected gap nodes 1D class AIII Hamiltonian gap nodes (c) E ± k = " k ± g k ± k = s ± t d k Bulk-boundary correspondence: surface flat bands k x k z Surface flat bands have Majorana character: k 1,k (r? ) c k," isgn(k)c k,# + 2,k (r? ) c k,# + isgn(k)c k," Schnyder, Ryu, PRB (2012) Schnyder et al. PRL (2013) Queiroz, Schnyder, PRB (2014) Brydon et al. NJP (2015) Queiroz, Schnyder, PRB (2015)

26 Conclusions & Outlook Conclusions and Outlook Mirror plane Ca 3PbO is a topological insulator with reflection symmetry Two Dirac surface states, type-ii Dirac states arxiv: Topological nodal line semi-metal Ca 3P2 Drumhead surface states Phys. Rev. B 93, (2016) Nodal non-centrosymmetric superconductor CePt 3Si Majorana flat band surface states k x (c) E k z Topological classification schemes: (i) bring order to the growing zoo of topological materials (ii) give guidance for the search and design of new topological states (iii) link the properties of the surface states to the bulk wave function topology Review articles: arxiv: ; J. Phys.: Condens. Matter 27, (2015)