Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model
Three-dimensional (3D) topological superconductors (TScs) protected by time-reversal (T) symmetry are characterized by gapless Majorana cones on their surface. Free-fermion phases with this symmetry (class DIII) are indexed by an integer ν, of which ν=1 is realized by the B phase of superfluid ^{3}H...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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American Physical Society
2013-11-01
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| Series: | Physical Review X |
| Online Access: | http://doi.org/10.1103/PhysRevX.3.041016 |
| _version_ | 1818984155716780032 |
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| author | Lukasz Fidkowski Xie Chen Ashvin Vishwanath |
| author_facet | Lukasz Fidkowski Xie Chen Ashvin Vishwanath |
| author_sort | Lukasz Fidkowski |
| collection | DOAJ |
| description | Three-dimensional (3D) topological superconductors (TScs) protected by time-reversal (T) symmetry are characterized by gapless Majorana cones on their surface. Free-fermion phases with this symmetry (class DIII) are indexed by an integer ν, of which ν=1 is realized by the B phase of superfluid ^{3}He. Previously, it was believed that the surface must be gapless unless time-reversal symmetry is broken. Here, we argue that a fully symmetric and gapped surface is possible in the presence of strong interactions, if a special type of topological order appears on the surface. The topological order realizes T symmetry in an anomalous way, one that is impossible to achieve in purely two dimensions. For odd ν TScs, the surface topological order must be non-Abelian. We propose the simplest non-Abelian topological order that contains electronlike excitations, SO(3)_{6}, with four quasiparticles, as a candidate surface state. Remarkably, this theory has a hidden T invariance that, however, is broken in any two-dimensional realization. By explicitly constructing an exactly soluble Walker-Wang model, we show that it can be realized at the surface of a short-ranged entangled 3D fermionic phase protected by T symmetry, with bulk electrons transforming as Kramers pairs, i.e. T^{2}=-1 under time reversal. We also propose an Abelian theory, the semion-fermion topological order, to realize an even ν TSc surface, for which an explicit model is derived using a coupled-layer construction. We argue that this is related to the ν=2 TSc, and we use this to build candidate surface topological orders for ν=4 and ν=8 TScs. The latter is equivalent to the three-fermion state, which is the surface topological order of a Z_{2} bosonic topological phase protected by T invariance. One particular consequence of this equivalence is that a ν=16 TSc admits a trivially gapped T-symmetric surface. |
| first_indexed | 2024-12-20T18:14:31Z |
| format | Article |
| id | doaj.art-fdd3aa8ad1a74906bcc01893f176f3a5 |
| institution | Directory Open Access Journal |
| issn | 2160-3308 |
| language | English |
| last_indexed | 2024-12-20T18:14:31Z |
| publishDate | 2013-11-01 |
| publisher | American Physical Society |
| record_format | Article |
| series | Physical Review X |
| spelling | doaj.art-fdd3aa8ad1a74906bcc01893f176f3a52022-12-21T19:30:24ZengAmerican Physical SocietyPhysical Review X2160-33082013-11-013404101610.1103/PhysRevX.3.041016Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved ModelLukasz FidkowskiXie ChenAshvin VishwanathThree-dimensional (3D) topological superconductors (TScs) protected by time-reversal (T) symmetry are characterized by gapless Majorana cones on their surface. Free-fermion phases with this symmetry (class DIII) are indexed by an integer ν, of which ν=1 is realized by the B phase of superfluid ^{3}He. Previously, it was believed that the surface must be gapless unless time-reversal symmetry is broken. Here, we argue that a fully symmetric and gapped surface is possible in the presence of strong interactions, if a special type of topological order appears on the surface. The topological order realizes T symmetry in an anomalous way, one that is impossible to achieve in purely two dimensions. For odd ν TScs, the surface topological order must be non-Abelian. We propose the simplest non-Abelian topological order that contains electronlike excitations, SO(3)_{6}, with four quasiparticles, as a candidate surface state. Remarkably, this theory has a hidden T invariance that, however, is broken in any two-dimensional realization. By explicitly constructing an exactly soluble Walker-Wang model, we show that it can be realized at the surface of a short-ranged entangled 3D fermionic phase protected by T symmetry, with bulk electrons transforming as Kramers pairs, i.e. T^{2}=-1 under time reversal. We also propose an Abelian theory, the semion-fermion topological order, to realize an even ν TSc surface, for which an explicit model is derived using a coupled-layer construction. We argue that this is related to the ν=2 TSc, and we use this to build candidate surface topological orders for ν=4 and ν=8 TScs. The latter is equivalent to the three-fermion state, which is the surface topological order of a Z_{2} bosonic topological phase protected by T invariance. One particular consequence of this equivalence is that a ν=16 TSc admits a trivially gapped T-symmetric surface.http://doi.org/10.1103/PhysRevX.3.041016 |
| spellingShingle | Lukasz Fidkowski Xie Chen Ashvin Vishwanath Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model Physical Review X |
| title | Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model |
| title_full | Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model |
| title_fullStr | Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model |
| title_full_unstemmed | Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model |
| title_short | Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model |
| title_sort | non abelian topological order on the surface of a 3d topological superconductor from an exactly solved model |
| url | http://doi.org/10.1103/PhysRevX.3.041016 |
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