Higher Berry Connection for Matrix Product States

TL;DR

This paper introduces a higher Berry connection for parameterized Matrix Product States in one-dimensional quantum systems, enabling the calculation of topological invariants related to higher Berry phases.

Contribution

It extends the mathematical framework of higher Berry phases by defining a higher Berry connection specifically for Matrix Product States, facilitating topological classification.

Findings

Successfully applied the formula to simple non-trivial models

Provides a new tool for topological analysis of quantum states

Builds on gerbe structure and multi-wavefunction overlaps

Abstract

In one spatial dimension, families of short-range entangled many-body quantum states, parameterized over some parameter space, can be topologically distinguished and classified by topological invariants built from the higher Berry phase -- a many-body generalization of the Berry phase. Previous works identified the underlying mathematical structure (the gerbe structure) and introduced a multi-wavefunction overlap, a generalization of the inner product in quantum mechanics, which allows for the extraction of the higher Berry phase and topological invariants. In this paper, building on these works, we introduce a connection, the higher Berry connection, for a family of parameterized Matrix Product States (MPS) over a parameter space. We demonstrate the use of our formula for simple non-trivial models.