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Topological Field Theory and Matrix Product States

Kapustin, Anton and Turzillo, Alex and You, Minyoung (2017) Topological Field Theory and Matrix Product States. Physical Review B, 96 (7). Art. No. 075125. ISSN 2469-9950.

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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.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper
Kapustin, Anton0000-0003-3903-5158
Turzillo, Alex0000-0003-4293-4293
Alternate Title:Spin Topological Field Theory and Fermionic Matrix Product States
Additional Information:© 2017 American Physical Society. Received 6 June 2017; published 14 August 2017. A.K. would like to thank P. Etingof and V. Ostrik for discussions. A.T. is grateful to I. Saberi and D. Williamson for helpful conversations. While this paper was nearing completion, we learned that closely related results have been obtained by K. Shiozaki and S. Ryu. This paper was supported in part by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC0011632.
Funding AgencyGrant Number
Department of Energy (DOE)DE-SC0011632
Issue or Number:7
Record Number:CaltechAUTHORS:20161107-091619292
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:71762
Deposited By: Tony Diaz
Deposited On:07 Nov 2016 19:16
Last Modified:03 Oct 2019 16:10

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