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arXiv:1610.10075 · Physical review. B./Physical review. B

Spin topological field theory and fermionic matrix product states

Anton Kapustin, Alex Turzillo, Minyoung You

Published Sep 4, 2018• 33 citations
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Abstract

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 results 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

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We study state-sum constructions of GGG-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 results are applied to the classification of fermionic short-range-entangled phases with a unitary symmetry GGG to determine the group law on the set of such phases. Interesting subtleties appear when the total symmetry group is a nontrivial extension of GGG by fermion parity.
This thesis is an attempt to understand the physics of short-range entangled phases of fermions through several related approaches. The first angle is topological quantum field theory. We discuss the classification of interacting fermionic short-range entangled phases by spin cobordism and give an algebraic characterization of unoriented equivariant bosonic topological quantum field theories in one spatial dimension. A second tool is tensor network representation. We develop the formalism of fermionic matrix product states and use it to derive the stacking group law for one dimensional symmetry-enriched fermionic short-range entangled phases. We also study its relationship with state sum constructions of topological quantum field theories and develop a state sum construction for pin-minus theories in one spatial dimension. The third approach is topological band theory. We classify free fermionic phases enriched by a unitary symmetry in any dimension and determine the map into the interacting classification.
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 GGG, this leads to a classification of gapped phases in 1+1d in terms of Morita-equivalence classes of GGG-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.
Short-Range Entangled topological phases of matter are closely related to Topological Quantum Field Theory. We use this connection to classify Symmetry Protected Topological phases in low dimensions, including the case when the symmetry involves time-reversal. To accomplish this, we generalize Turaev’s description of equivariant TQFT to the unoriented case. We show that invertible unoriented equivariant TQFTs in one or fewer spatial dimensions are classified by twisted group cohomology, in agreement with the proposal of Chen, Gu, Liu and Wen. We also show that invertible oriented equivariant TQFTs in spatial dimension two or fewer are classified by ordinary group cohomology.
Matrix Product States (MPSs) provide a powerful framework to study and classify gapped quantum phases — symmetry-protected topological (SPT) phases in particular — defined in one dimensional lattices. On the other hand, it is natural to expect that gapped quantum phases in the limit of zero correlation length are described by topological quantum field theories (TFTs or TQFTs). In this paper, for (1+1)-dimensional bosonic SPT phases protected by symmetry G, we bridge their descriptions in terms of MPSs, and those in terms of G-equivariant TFTs. In particular, for various topological invariants (SPT invariants) constructed previously using MPSs, we provide derivations from the point of view of (1+1) TFTs. We also discuss the connection between boundary degrees of freedom, which appear when one introduces a physical boundary in SPT phases, and “open” TFTs, which are TFTs defined on spacetimes with boundaries.
We study aspects of gapped phases of matter, focusing on their classification, including the group law under stacking, and their relation to topological quantum field theories (TQFT). In one spatial dimension, it is well-known that Matrix Product States (MPS) efficiently approximate ground states of gapped systems; by showing that these states arise naturally in 1 + 1-dimensional lattice TQFT, which in turn are closely related to continuum TQFT, we provide a concrete connection between ground states of lattice systems and TQFT in 1 + 1 dimensions. We generalize this to systems with symmetries and fermions, and obtain a classification and group law for the stacking of 1 + 1-dimensional symmetry-protected topological phases. Further, we study the effect of turning on/off interactions for the classification: the phase classification of a given symmetry class of Hamiltonians can be different depending on whether we allow interactions or not, and in low dimensions we provide some concrete formulas relating the phases under the non-interacting classification and those under the interacting classification. Lastly, we study the phases of the 2 + 1-dimensional topological superconductor, and show that for all 16 phases braiding statistics of vortices, which determine the underlying TQFT, can be obtained by stacking layers of the basic p + ip superconductor.
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