Classification and surface anomaly of glide symmetry protected topological phases in three dimensions. (English) Zbl 1516.81204
Summary: We study glide protected topological (GSPT) phases of interacting bosons and fermions in three spatial dimensions with certain on-site symmetries. They are crystalline SPT phases, which are distinguished from a trivial product state only in the presence of non-symmorphic glide symmetry. We classify these GSPT phases with various on-site symmetries such as \(U(1)\) and time reversal, and show that they can all be understood by stacking and coupling two-dimensional (2D) short-range-entangled phases in a glide-invariant way. Using such a coupled layer construction we study the anomalous surface topological orders of these GSPT phases, which gap out the 2D surface states without breaking any symmetries. While this framework can be applied to any non-symmorphic SPT phase, we demonstrate it in many examples of GSPT phases including the non-symmorphic topological insulator with ‘hourglass fermion’ surface states.
MSC:
| 81V70 | Many-body theory; quantum Hall effect |
| 81T60 | Supersymmetric field theories in quantum mechanics |
| 82B23 | Exactly solvable models; Bethe ansatz |
| 81S05 | Commutation relations and statistics as related to quantum mechanics (general) |
References:
| [1] | Hasan M Z and Kane C L 2010 Colloquium: topological insulators Rev. Mod. Phys.82 3045 · doi:10.1103/RevModPhys.82.3045 |
| [2] | Hasan M Z and Moore J E 2011 Three-dimensional topological insulators Annu. Rev Condens. Matter Phys.2 55 · doi:10.1146/annurev-conmatphys-062910-140432 |
| [3] | Qi X-L and Zhang S-C 2011 Topological insulators and superconductors Rev. Mod. Phys.83 1057 · doi:10.1103/RevModPhys.83.1057 |
| [4] | Chen X, Gu Z-C, Liu Z-X and Wen X-G 2012 Symmetry-protected topological orders in interacting bosonic systems Science338 1604 · Zbl 1355.81189 · doi:10.1126/science.1227224 |
| [5] | Senthil T 2015 Symmetry-protected topological phases of quantum matter Annu. Rev. Condens. Matter Phys.6 299 · doi:10.1146/annurev-conmatphys-031214-014740 |
| [6] | Vishwanath A and Senthil T 2013 Physics of three-dimensional bosonic topological insulators: surface-deconfined criticality and quantized magnetoelectric effect Phys. Rev. X 3 011016 · doi:10.1103/PhysRevX.3.011016 |
| [7] | Fidkowski L, Chen X and Vishwanath A 2013 Non-abelian topological order on the surface of a 3D topological superconductor from an exactly solved model Phys. Rev. X 3 041016 · doi:10.1103/PhysRevX.3.041016 |
| [8] | Wang C and Senthil T 2013 Boson topological insulators: a window into highly entangled quantum phases Phys. Rev. B 87 235122 · doi:10.1103/PhysRevB.87.235122 |
| [9] | Metlitski M A, Kane C L and Fisher M P A 2013 Bosonic topological insulator in three dimensions and the statistical witten effect Phys. Rev. B 88 035131 · doi:10.1103/PhysRevB.88.035131 |
| [10] | Bonderson P, Nayak C and Qi X-L 2013 A time-reversal invariant topological phase at the surface of a 3D topological insulator J. Stat. Mech. P09016 · Zbl 1456.82576 · doi:10.1088/1742-5468/2013/09/P09016 |
| [11] | Wang C, Potter A C and Senthil T 2013 Gapped symmetry preserving surface state for the electron topological insulator Phys. Rev. B 88 115137 · doi:10.1103/PhysRevB.88.115137 |
| [12] | Chen X, Fidkowski L and Vishwanath A 2014 Symmetry enforced non-abelian topological order at the surface of a topological insulator Phys. Rev. B 89 165132 · doi:10.1103/PhysRevB.89.165132 |
| [13] | Cho G Y, Teo J C Y and Ryu S 2014 Conflicting symmetries in topologically ordered surface states of three-dimensional bosonic symmetry protected topological phases Phys. Rev. B 89 235103 · doi:10.1103/PhysRevB.89.235103 |
| [14] | Chen X 2016 Symmetry fractionalization in two dimensional topological phases arXiv:1606.07569 [cond-mat.str-el] |
| [15] | Chen X, Gu Z-C, Liu Z-X and Wen X-G 2013 Symmetry protected topological orders and the group cohomology of their symmetry group Phys. Rev. B 87 155114 · doi:10.1103/PhysRevB.87.155114 |
| [16] | Ando Y and Fu L 2015 Topological crystalline insulators and topological superconductors: from concepts to materials Annu. Rev. Condens. Matter Phys.6 361 · doi:10.1146/annurev-conmatphys-031214-014501 |
| [17] | Isobe H and Fu L 2015 Theory of interacting topological crystalline insulators Phys. Rev. B 92 081304 · doi:10.1103/PhysRevB.92.081304 |
| [18] | Qi Y and Fu L 2015 Anomalous crystal symmetry fractionalization on the surface of topological crystalline insulators Phys. Rev. Lett.115 236801 · doi:10.1103/PhysRevLett.115.236801 |
| [19] | Song H, Huang S-J, Fu L and Hermele M 2017 Topological phase protected by point group symmetry Phys. Rev. X 19 011020 · doi:10.1103/PhysRevX.7.011020 |
| [20] | Thorngren R and Else D V 2016 Gauging spatial symmetries and the classification of topological crystalline phases arXiv:1612.00846 [cond-mat.str-el] |
| [21] | Fang C and Fu L 2015 New classes of three-dimensional topological crystalline insulators: nonsymmorphic and magnetic Phys. Rev. B 91 161105 · doi:10.1103/PhysRevB.91.161105 |
| [22] | Varjas D, de Juan F and Lu Y-M 2015 Bulk invariants and topological response in insulators and superconductors with nonsymmorphic symmetries Phys. Rev. B 92 195116 · doi:10.1103/PhysRevB.92.195116 |
| [23] | Shiozaki K, Sato M and Gomi K 2016 Topology of nonsymmorphic crystalline insulators and superconductors Phys. Rev. B 93 195413 · doi:10.1103/PhysRevB.93.195413 |
| [24] | Alexandradinata A, Wang Z and Bernevig B A 2016 Topological insulators from group cohomology Phys. Rev. X 6 021008 · doi:10.1103/PhysRevX.6.021008 |
| [25] | Wang Z, Alexandradinata A, Cava R J and Bernevig B A 2016 Hourglass fermions Nature532 189 · doi:10.1038/nature17410 |
| [26] | Ezawa M 2016 Hourglass fermion surface states in stacked topological insulators with nonsymmorphic symmetry Phys. Rev. B 94 155148 · doi:10.1103/PhysRevB.94.155148 |
| [27] | Kruthoff J, de Boer J, van Wezel J, Kane C L and Slager R-J 2016 Topological classification of crystalline insulators through band structure combinatorics arXiv:1612.02007 [cond-mat.mes-hall] |
| [28] | Ma J-Z 2016 Experimental discovery of the first nonsymmorphic topological insulator KHgSb arXiv:1605.06824 [cond-mat.mtrl-sci] |
| [29] | Mross D F, Essin A, Alicea J and Stern A 2016 Anomalous quasiparticle symmetries and non-abelian defects on symmetrically gapped surfaces of weak topological insulators Phys. Rev. Lett.116 036803 · doi:10.1103/PhysRevLett.116.036803 |
| [30] | Fulga I C, Avraham N, Beidenkopf H and Stern A 2016 Coupled-layer description of topological crystalline insulators Phys. Rev. B 94 125405 · doi:10.1103/PhysRevB.94.125405 |
| [31] | Hermele M and Chen X 2016 Flux-fusion anomaly test and bosonic topological crystalline insulators Phys. Rev. X 6 041006 · doi:10.1103/PhysRevX.6.041006 |
| [32] | Kapustin A 2014 Symmetry protected topological phases, anomalies, and cobordisms: beyond group cohomology arXiv:1403.1467 [cond-mat.str-el] |
| [33] | Kapustin A, Thorngren R, Turzillo A and Wang Z 2015 Fermionic symmetry protected topological phases and cobordisms J. High Energy Phys. JHEP15(2015)052 · Zbl 1388.81845 · doi:10.1007/JHEP12(2015)052 |
| [34] | Mong R S K, Essin A M and Moore J E 2010 Antiferromagnetic topological insulators Phys. Rev. B 81 245209 · doi:10.1103/PhysRevB.81.245209 |
| [35] | Mross D F, Essin A and Alicea J 2015 Composite dirac liquids: parent states for symmetric surface topological order Phys. Rev. X 5 011011 · doi:10.1103/PhysRevX.5.011011 |
| [36] | Sahoo S, Zhang Z and Teo J C Y 2016 Coupled wire model of symmetric majorana surfaces of topological superconductors Phys. Rev. B 94 165142 · doi:10.1103/PhysRevB.94.165142 |
| [37] | Kong L and Wen X-G 2014 Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions arXiv:1405.5858 [cond-mat.str-el] |
| [38] | Freed D S 2014 Short-range entanglement and invertible field theories arXiv:1406.7278 [cond-mat.str-el] |
| [39] | Kitaev A 2006 Anyons in an exactly solved model and beyond Ann. Phys.321 2 · Zbl 1125.82009 · doi:10.1016/j.aop.2005.10.005 |
| [40] | Read N and Green D 2000 Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum hall effect Phys. Rev. B 61 10267 · doi:10.1103/PhysRevB.61.10267 |
| [41] | Kane C L, Mukhopadhyay R and Lubensky T C 2002 Fractional quantum hall effect in an array of quantum wires Phys. Rev. Lett.88 036401 · doi:10.1103/PhysRevLett.88.036401 |
| [42] | Teo J C Y and Kane C L 2014 From luttinger liquid to non-abelian quantum hall states Phys. Rev. B 89 085101 · doi:10.1103/PhysRevB.89.085101 |
| [43] | Lu Y-M and Vishwanath A 2012 Theory and classification of interacting integer topological phases in two dimensions: a chern-simons approach Phys. Rev. B 86 125119 · doi:10.1103/PhysRevB.86.125119 |
| [44] | Levin M and Stern A 2012 Classification and analysis of two-dimensional abelian fractional topological insulators Phys. Rev. B 86 115131 · doi:10.1103/PhysRevB.86.115131 |
| [45] | Altland A and Zirnbauer M R 1997 Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures Phys. Rev. B 55 1142 · doi:10.1103/PhysRevB.55.1142 |
| [46] | Kane C L and Fisher M P A 1997 Quantized thermal transport in the fractional quantum hall effect Phys. Rev. B 55 15832 · doi:10.1103/PhysRevB.55.15832 |
| [47] | Cano J, Cheng M, Mulligan M, Nayak C, Plamadeala E and Yard J 2014 Bulk-edge correspondence in (2+1)-dimensional abelian topological phases Phys. Rev. B 89 115116 · doi:10.1103/PhysRevB.89.115116 |
| [48] | Senthil T and Levin M 2013 Integer quantum hall effect for bosons Phys. Rev. Lett.110 046801 · doi:10.1103/PhysRevLett.110.046801 |
| [49] | Kitaev A Y 2003 Fault-tolerant quantum computation by anyons Ann. Phys.303 2 · Zbl 1012.81006 · doi:10.1016/S0003-4916(02)00018-0 |
| [50] | Levin M and Gu Z-C 2012 Braiding statistics approach to symmetry-protected topological phases Phys. Rev. B 86 115109 · doi:10.1103/PhysRevB.86.115109 |
| [51] | Ran Y, Vishwanath A and Lee D-H 2008 Spin-charge separated solitons in a topological band insulator Phys. Rev. Lett.101 086801 · doi:10.1103/PhysRevLett.101.086801 |
| [52] | Lu Y-M and Lee D-H 2014 Gapped symmetric edges of symmetry-protected topological phases Phys. Rev. B 89 205117 · doi:10.1103/PhysRevB.89.205117 |
| [53] | Lee S, Hermele M and Parameswaran S A 2016 Fractionalizing glide reflections in two-dimensional Z 2 topologically ordered phases Phys. Rev. B 94 125122 · doi:10.1103/PhysRevB.94.125122 |
| [54] | Lu Y-M and Vishwanath A 2016 Classification and properties of symmetry-enriched topological phases: chern-simons approach with applications to Z 2 spin liquids Phys. Rev. B 93 155121 · doi:10.1103/PhysRevB.93.155121 |
| [55] | Chen X, Lu Y-M and Vishwanath A 2014b Symmetry-protected topological phases from decorated domain walls Nat Commun.5 3507 · doi:10.1038/ncomms4507 |
| [56] | Chiu C-K, Teo J C Y, Schnyder A P and Ryu S 2016 Classification of topological quantum matter with symmetries Rev. Mod. Phys.88 035005 · doi:10.1103/RevModPhys.88.035005 |
| [57] | Chang P-Y, Erten O and Coleman P 2016 Mobius Kondo insulators arXiv:1603.03435 [cond-mat.str-el] |
| [58] | Levin M A and Wen X-G 2005 String-net condensation: a physical mechanism for topological phases Phys. Rev. B 71 045110 · doi:10.1103/PhysRevB.71.045110 |
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