Topological Field Theory and Matrix Product States

TL;DR

This paper connects 2D topological quantum field theories with matrix product states, providing a classification framework for 1+1D gapped phases, including symmetry-protected topological phases, through algebraic and topological methods.

Contribution

It demonstrates that 2D TQFT state-sum constructions naturally produce MPS and classifies 1+1D gapped phases using Morita-equivalence of G-equivariant algebras.

Findings

MPS can be derived from 2D TQFT constructions.

Classification of gapped phases via Morita-equivalence classes.

Recovery of group cohomology classification for SPT phases.

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. Non-uniqueness 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.