# Topological field theory and matrix product states

## Abstract

It is believed that most (perhaps all) gapped phases of matter can be described at long distances by topological quantum field theory (TQFT). On the other hand, it has been rigorously established that in 1+1d ground states of gapped Hamiltonians can be approximated by matrix product states (MPS). We show that the state-sum construction of 2d TQFT naturally leads to MPS in their standard form. In the case of systems with a global symmetry G, this leads to a classification of gapped phases in 1+1d in terms of Morita-equivalence classes of G-equivariant algebras. Nonuniqueness of the MPS representation is traced to the freedom of choosing an algebra in a particular Morita class. In the case of short-range entangled phases, we recover the group cohomology classification of SPT phases

- 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:
- 1425379

- Alternate Identifier(s):
- OSTI ID: 1374935

- Grant/Contract Number:
- SC0011632

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physical Review B

- Additional Journal Information:
- Journal Volume: 96; Journal Issue: 7; Journal ID: ISSN 2469-9950

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Topological field theories; topological phases of matter; strongly correlated systems; topological materials; particles and fields; condensed matter and materials physics

### Citation Formats

```
Kapustin, Anton, Turzillo, Alex, and You, Minyoung. Topological field theory and matrix product states. United States: N. p., 2017.
Web. doi:10.1103/PhysRevB.96.075125.
```

```
Kapustin, Anton, Turzillo, Alex, & You, Minyoung. Topological field theory and matrix product states. United States. doi:10.1103/PhysRevB.96.075125.
```

```
Kapustin, Anton, Turzillo, Alex, and You, Minyoung. Mon .
"Topological field theory and matrix product states". United States. doi:10.1103/PhysRevB.96.075125. https://www.osti.gov/servlets/purl/1425379.
```

```
@article{osti_1425379,
```

title = {Topological field theory and matrix product states},

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

abstractNote = {It is believed that most (perhaps all) gapped phases of matter can be described at long distances by topological quantum field theory (TQFT). On the other hand, it has been rigorously established that in 1+1d ground states of gapped Hamiltonians can be approximated by matrix product states (MPS). We show that the state-sum construction of 2d TQFT naturally leads to MPS in their standard form. In the case of systems with a global symmetry G, this leads to a classification of gapped phases in 1+1d in terms of Morita-equivalence classes of G-equivariant algebras. Nonuniqueness of the MPS representation is traced to the freedom of choosing an algebra in a particular Morita class. In the case of short-range entangled phases, we recover the group cohomology classification of SPT phases},

doi = {10.1103/PhysRevB.96.075125},

journal = {Physical Review B},

number = 7,

volume = 96,

place = {United States},

year = {2017},

month = {8}

}

*Citation information provided by*

Web of Science

Web of Science

#### Figures / Tables:

Works referenced in this record:

##
Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order

journal, October 2010

- Chen, Xie; Gu, Zheng-Cheng; Wen, Xiao-Gang
- Physical Review B, Vol. 82, Issue 15

##
P.L. Homeomorphic Manifolds are Equivalent by Elementary 5hellingst

journal, March 1991

- Pachner, Udo
- European Journal of Combinatorics, Vol. 12, Issue 2

##
Topological quantum field theories

journal, January 1988

- Atiyah, Michael
- Publications mathématiques de l'IHÉS, Vol. 68, Issue 1

##
Module categories, weak Hopf algebras and modular invariants

journal, June 2003

- Ostrik, Victor
- Transformation Groups, Vol. 8, Issue 2

##
State sum Construction of Two-Dimensional Open-Closed Topological Quantum Field Theories

journal, November 2007

- Lauda, Aaron D.; Pfeiffer, Hendryk
- Journal of Knot Theory and Its Ramifications, Vol. 16, Issue 09

##
Complete classification of one-dimensional gapped quantum phases in interacting spin systems

journal, December 2011

- Chen, Xie; Gu, Zheng-Cheng; Wen, Xiao-Gang
- Physical Review B, Vol. 84, Issue 23

##
Classification of gapped symmetric phases in one-dimensional spin systems

journal, January 2011

- Chen, Xie; Gu, Zheng-Cheng; Wen, Xiao-Gang
- Physical Review B, Vol. 83, Issue 3

##
Classifying quantum phases using matrix product states and projected entangled pair states

journal, October 2011

- Schuch, Norbert; Pérez-García, David; Cirac, Ignacio
- Physical Review B, Vol. 84, Issue 16

##
Finitely correlated states on quantum spin chains

journal, March 1992

- Fannes, M.; Nachtergaele, B.; Werner, R. F.
- Communications in Mathematical Physics, Vol. 144, Issue 3

##
Two-Dimensional Topological Quantum Field Theories and Frobenius Algebras

journal, October 1996

- Abrams, Lowell
- Journal of Knot Theory and Its Ramifications, Vol. 05, Issue 05

##
Topological phases of fermions in one dimension

journal, February 2011

- Fidkowski, Lukasz; Kitaev, Alexei
- Physical Review B, Vol. 83, Issue 7

##
Renormalization-Group Transformations on Quantum States

journal, April 2005

- Verstraete, F.; Cirac, J. I.; Latorre, J. I.
- Physical Review Letters, Vol. 94, Issue 14

##
Entropy and entanglement in quantum ground states

journal, July 2007

- Hastings, M. B.
- Physical Review B, Vol. 76, Issue 3

##
Lattice topological field theory in two dimensions

journal, March 1994

- Fukuma, M.; Hosono, S.; Kawai, H.
- Communications in Mathematical Physics, Vol. 161, Issue 1

*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*