Matrix product states and equivariant topological field theories for bosonic symmetry-protected topological phases in (1+1) dimensions

TL;DR

This paper connects matrix product state descriptions of (1+1)-dimensional bosonic SPT phases with their topological quantum field theory counterparts, providing a unified framework and deriving topological invariants from TFT perspectives.

Contribution

It establishes a bridge between MPS-based classifications and equivariant TFT descriptions for (1+1)D bosonic SPT phases, including derivations of topological invariants.

Findings

Derived SPT invariants from (1+1) TFTs

Linked boundary degrees of freedom with open TFTs

Unified MPS and TFT frameworks for bosonic SPT phases

Abstract

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.