Ten-fold classification of topological insulators and - PowerPoint PPT Presentation

Ten-fold classification of topological insulators and superconductors Shinsei Ryu Univ. of California, Berkeley

Table of contents t - What are topological insulators and SCs ? - T able of topological insulators and superdoncutors t - Search for topological materials

band theory and band insulator Bloch (1928), Wilson (1931) metal insulator conduction band fermi "surface" gap valence band fermi energy crystal momentum in 1st Brillouin Zone, BZ Role of electric wavefucntions in insulators ???

topological insulator: preview distinction of insulators by their wavefunctions (or: entanglement) --> "topological insulators" topological insulator boring insulator more entangled less entangled

topological insulator: preview physical consequence of entangled wavefunctions boundary of ordinary insulator = insulator boundary of topological insulator = perfect metal topological insulator perfect metal boundary is completely immune to disorder (evades Anderson localization). A consequence: ordinary and topological insulators cannot be connected adiabatically.

Examples of topoloigcal insulators/SCs - IQHE (1980) SiMOS,GaAs K.v.Klitzing, G. Dorda, M. Pepper (1980) time-resersal symmetry --> new topological insulators HgT e - QSHE (2007) B.Bernevig, T.Hughes, S.C.Zhang (06) M. Konig et al. (07) Kane and Mele et al. (05-06) - 3D Z2 topological insulator (2008) BiT e, BiSe, BiSb Moore-Balents (07) Fu-Kane-Mele (07) Roy (07) D. Hsieh et al (08) - chiral p-wave SC (topological superconductor) SrRuO nu=5/2 FQHE Y . Maeno et al (94) R.L.Willett et al (87) G. Moore and N. Read et al (91) more topological insulators/superconductors ? never ending story or happy ending ?

topological superconductor superconductor = Cooper pair (boson) + BdG quasi-particle (fermion) fully gapped superconductor = band insulator for BdG quasi-particle topological superconductor = full quasiparticle gap in bulk, but gapless Andreev boundary state at boundaries topological superconductor perfect thermal or spin metal boundary completely is immune to disorder (evades Anderson localization). vortex/edge/surface often supports Majorana fermion modes

quantum spin Hall effect (QSHE) in d=2 spatial dimensions, with good T TRS - time-reversal invariant band insulator - gapless Kramers pair of edge modes - strong spin-orbit interaction B IQHE for spin up bulk states bulk states IQHE for spin down -B

quantum spin Hall effect (QSHE) in d=2 spatial dimensions, with good T quantum spin Hall insulator is characterized by a binary ( ) topological quantity. Kane-Mele (05) - odd number of Kramers pairs at edge --> stable even number of Kramers pairs at edge --> unstable ordinary QSHE vac vac 1+1=0 3 = 1 experimental realization: Bernevig-Hughes-Zhang (2006) HgTe quantum well M. Koenig et al. Science (2007)

quantum spin Hall effect (QSHE) experimental realization: HgTe quantum well strong spin-orbit interaction QSHE Bernevig-Hughes-Zhang (2006) M. Koenig et al. Science (2007)

Z2 topological insulator in d=3 spatial dimensions Fu-Kane-Mele, Moore-Balents, Roy (06) d=3 dimensions time-reversal invariant characterized by a Z2 quantity non-trivial trivial when surface states = odd number of Dirac fermions surface bulk (insulator)

surface of top. insulator = "1/4 of graphene" ! surface bulk (insulator) two valleys (and spin) condensed matter realization of Theorem (by Nielsen-Ninomiya): domain-wall fermion For any 2D lattice with TRS # of Dirac cones must be even.

ARPES experiments on Z2 topological insulators 5 Dirac cones BiSb D. Hsieh et al. Nature (08) BiSe Y . Xia et al. Nature Phys. (2009) BiT e Y . L. Chen et al. Science (2009)

Table of contents - What are topological insulators and SCs ? - T able of topological insulators and superdoncutors - Search for topological materials

symmetry classification of Hamiltonians - "ten-fold way" exhaustive classification of discrete symmetries - Wigner-Dyson (1951 -1963) : "three-fold way" complex nuclei - Verbaarschot (92 -93) chiral phase transition in QCD - Altland-Zirnbauer (96-97) : "ten-fold way" mesoscopic SC systems BdG Hamiltonians realize 6 out of 10 symmetry classes.

symmetry classification of Hamiltonians - "ten-fold way" "nick name" of presence/absence of discrete symmetries symmetry classes and their type - Wigner-Dyson (1951 -1963) : "three-fold way" complex nuclei - Verbaarschot (92 -93) chiral phase transition in QCD - Altland-Zirnbauer (96-97) : "ten-fold way" mesoscopic SC systems BdG Hamiltonians realize 6 out of 10 symmetry classes.

main result: classification of topological insulators and superconductors integer classification binary classification no top. ins./SC

main result: classification of topological insulators and superconductors spatial dimensions presence/absence of topological state integer classification binary classification no top. ins./SC symmetry classes of quadratic fermionic Hamiltonians (Altland-Zirnbauer)

main result: classification of topological insulators and superconductors IQHE p+ip wave SC polyacetylene TMTSF Z2 topological QSHE insulator d+id wave SC

classification of topological insulators and superconductors 10 = 8 + 2 - periodicity 8 both in spatial dimension and symmetry class - always 5 kinds of topological states for each dimension. - Z followed by two Z2 - d>3 can characterize ("dimensional reduction") adiabatic processes, rather than states themselves - Schnyder, SR, Furusaki, Ludwig (for d=1,2,3, 2008) - Kitaev (all d and periodicity "Periodic Table", 2009) - SR and Takayanagi (construction by D-branes, 2010)

underlying strategies for classification - discover a topological invariant - obtained 3D analogue of TKNN integer: - "boundary-to-bulk" approach Anderson delocalization non-linear sigma model on G/H + (discrete) topological term topological insulators/SC

Table of contents - What are topological insulators and SCs ? - T able of topological insulators and superdoncutors - Search for topological materials

IQHE p+ip wave SC polyacetylene 3He B TMTSF Z2 topological insulator QSHE d+id wave SC some outcomes of classification in d=3: - 3He B is newly identified as a topological SC (superfluid) in d=3. - topological singlet SC in d=3 is predicted.

3He B is a topological "superconductor" in class DIII Schnyder, SR, Furusaki, Ludwig (08) topologically protected Roy (08) surface Majorana fermion Qi, Hughes, Raghu, Zhang (08) 3d analogue of Moore-Read state

Majorana mode detected by surface acoustic impedance Y. Aoki et al. PRL (05) Y. Wada et al. PRB (08) S. Murakawa et al. PRL (09) Salomma and Volovik, 80s Y. Nagato et al. JLTP (07) M. Saitoh et al. PRB(R) (06)

non-centro symmetric superconductors key ingridient: antisymmetric spin orbit coupling mixture of "singlet" and "triplet" pairing Ce based heavy fermion conpound CePt3Si Bauer et al (2004)

single-band model non-centro symmetric superconductors BdG equation : Cubic system (e.g. LiPdPtB) : T etragonal system (e.g. CePtSi):

Cubic system (Li compound) strength of 2nd order SO coupling singlet SC

- The winding number can take a large value even for a simple single-band model. - T opological SC phases are surrounded by gapless SC phase with line nodes. - "Subphases" in the gapless regions distinguished by different node topology. - Nodal lines are stable and carry "a topological charge".

Cubic system (Li compound) - Different nodal SC phases can be distinguised by the surface flat band. Peak in surface DOS Completely flat surface Andreev band ! experiments: LiPdB : gapped LiPtB : nodal SC Yuan et al PRL (06)

Tetragonal system (Ce compound) NMR T_1 --> nodal SC - No topological SC - Nodal phase with a surface flat band Yogi et al. PRL (04)

- Complete classification of topological phases in fermion systems in all dimensions and symmetry classes Introduced 3D analogue of TKNN integer D-brane / topological insulator correspondence some predictions: - surface of 3d Z2 topological insulator: perfect metal stable gapless Majorana surface mode - 3He B is a topological SC: - there are topological singlet with good T and in d=3 spatial dimensions SC - topological SC and flat surface band in non-centrosymmetric SC systems LiPdPtB and CePtSi collaborators: Andreas Schnyder (Max Planck) Ashvin Vishwanath (Berkeley) Akira Furusaki (RIKEN) Joel Moore (Berkeley) Andreas Ludwig (Santa Barbara) Pavan Hosur (Berkeley)