# Spin topological field theory and fermionic matrix product states

## Abstract

Here, we study state-sum constructions of G-equivariant spin topological quantum field theory (TQFTs) and their relationship to matrix product states. In the Neveu-Schwarz, Ramond, and twisted sectors, states of the TQFT are generalized matrix product states. Our findings are applied to the classification of fermionic short-range-entangled phases with a unitary symmetry G to determine the group law on the set of such phases. Interesting subtleties appear when the total symmetry group is a nontrivial extension of G by fermion parity.

- Authors:

- California Inst. of Technology (CalTech), Pasadena, CA (United States)

- Publication Date:

- Research Org.:
- California Inst. of Technology (CalTech), Pasadena, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

- OSTI Identifier:
- 1598281

- Alternate Identifier(s):
- OSTI ID: 1468856

- Grant/Contract Number:
- [SC0011632]

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physical Review B

- Additional Journal Information:
- [ Journal Volume: 98; Journal Issue: 12]; Journal ID: ISSN 2469-9950

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 36 MATERIALS SCIENCE; 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY

### Citation Formats

```
Kapustin, Anton, Turzillo, Alex, and You, Minyoung. Spin topological field theory and fermionic matrix product states. United States: N. p., 2018.
Web. doi:10.1103/PhysRevB.98.125101.
```

```
Kapustin, Anton, Turzillo, Alex, & You, Minyoung. Spin topological field theory and fermionic matrix product states. United States. doi:10.1103/PhysRevB.98.125101.
```

```
Kapustin, Anton, Turzillo, Alex, and You, Minyoung. Tue .
"Spin topological field theory and fermionic matrix product states". United States. doi:10.1103/PhysRevB.98.125101. https://www.osti.gov/servlets/purl/1598281.
```

```
@article{osti_1598281,
```

title = {Spin topological field theory and fermionic matrix product states},

author = {Kapustin, Anton and Turzillo, Alex and You, Minyoung},

abstractNote = {Here, we study state-sum constructions of G-equivariant spin topological quantum field theory (TQFTs) and their relationship to matrix product states. In the Neveu-Schwarz, Ramond, and twisted sectors, states of the TQFT are generalized matrix product states. Our findings are applied to the classification of fermionic short-range-entangled phases with a unitary symmetry G to determine the group law on the set of such phases. Interesting subtleties appear when the total symmetry group is a nontrivial extension of G by fermion parity.},

doi = {10.1103/PhysRevB.98.125101},

journal = {Physical Review B},

number = [12],

volume = [98],

place = {United States},

year = {2018},

month = {9}

}

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