Classification of <math><mo>(</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>)</mo></math>D invertible fermionic topological phases with symmetry - INSPIRE

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Classification of $(2 + 1)$ D invertible fermionic topological phases with symmetry

Maissam Barkeshli
(
- Maryland U., College Park
)
,
Yu-An Chen
(
- Maryland U., College Park
)
,
Po-Shen Hsin
(
- Caltech
)
,
Naren Manjunath
(
- Maryland U., College Park
)

Sep 22, 2021

27 pages

Published in:

Phys.Rev.B 105 (2022) 23, 235143

Published: Jun 15, 2022

e-Print:

2109.11039 [cond-mat.str-el]

DOI:

10.1103/PhysRevB.105.235143 (publication)

View in:

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Abstract: (APS)

We provide a classification of invertible topological phases of interacting fermions with symmetry in two spatial dimensions for general fermionic symmetry groups

G_{f}

and general values of the chiral central charge

c_{-}

. Here

G_{f}

is a central extension of a bosonic symmetry group

G_{b}

by fermion parity,

{(- 1)}^{F}

, specified by a second cohomology class

[ω_{2}] \in H^{2} (G_{b}, Z_{2})

. Our approach proceeds by gauging fermion parity and classifying the resulting

G_{b}

symmetry-enriched topological orders while keeping track of certain additional data and constraints. We perform this analysis through two perspectives, using

G

-crossed braided tensor categories and

Spin {(2 c_{-})}_{1}

Chern-Simons theory coupled to a background

G

gauge field. These results give a way to characterize and classify invertible fermionic topological phases in terms of a concrete set of data and consistency equations, which is more physically transparent and computationally simpler than the more abstract methods using cobordism theory and spectral sequences. Our results also generalize and provide a different approach to the recent classification of fermionic symmetry-protected topological phases by Wang and Gu, which have chiral central charge

c_{-} = 0

. We show how the tenfold way classification of topological insulators and superconductors fits into our scheme, along with general nonperturbative constraints due to certain choices of

c_{-}

and

G_{f}

. Mathematically, our results also suggest an explicit general parametrization of deformation classes of

(2 + 1)

D invertible topological quantum field theories with

G_{f}

symmetry.

Note:

45+27 pages, 5 figures, v2: corrected typos, added Appendix D showing agreement with Wang-Gu results for O_4 obstruction, v3: corrected typos, v4: minor corrections, v5: minor corrections, added Appendix G

phase: topological
fermion: parity
category: tensor
central charge
cobordism
cohomology
topological insulator
Chern-Simons term
space-time: dimension: 3

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