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2018•Physical review. B./Physical review. BOpen access

Spin topological field theory and fermionic matrix product states

Anton Kapustin, Alex Turzillo, Minyoung You

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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.

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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.

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Available 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.

Key concepts: Topological quantum field theory, Physics, Unitary state, Topological order, Matrix product state, Symmetry protected topological order, Symmetry group, Quantum mechanics

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