Geometry of Matrix Product States: metric, parallel transport and curvature

TL;DR

This paper explores the geometric structure of matrix product states, modeling them as a fiber bundle, and introduces tools like metrics, parallel transport, and curvature to analyze their properties in quantum many-body physics.

Contribution

It provides a geometric framework for matrix product states, including the definition of metrics, parallel transport, and curvature, enhancing understanding of their tangent space structure.

Findings

Defined the fiber bundle structure of MPS

Introduced an inverse metric for tangent vectors

Analyzed the curvature and parallel transport in MPS manifold

Abstract

We study the geometric properties of the manifold of states described as (uniform) matrix product states. Due to the parameter redundancy in the matrix product state representation, matrix product states have the mathematical structure of a (principal) fiber bundle. The total space or bundle space corresponds to the parameter space, i.e. the space of tensors associated to every physical site. The base manifold is embedded in Hilbert space and can be given the structure of a K\"ahler manifold by inducing the Hilbert space metric. Our main interest is in the states living in the tangent space to the base manifold, which have recently been shown to be interesting in relation to time dependence and elementary excitations. By lifting these tangent vectors to the (tangent space) of the bundle space using a well-chosen prescription (a principal bundle connection), we can define and efficiently compute an inverse metric, and introduce differential geometric concepts such as parallel transport (related to the Levi-Civita connection) and the Riemann curvature tensor.