Higher Berry curvature from matrix product states

TL;DR

This paper introduces a new way to compute higher Berry curvature in quantum many-body systems using matrix product states, enabling gauge-invariant and quantized results over parameter space.

Contribution

It proposes an alternative formulation of higher Berry curvature with matrix product states, addressing gauge invariance and quantization in many-body quantum systems.

Findings

Higher Berry curvature varies continuously during adiabatic evolution.

Curvature becomes quantized over closed 3D parameter space.

Method confirms gauge-invariant computation of higher Berry curvature.

Abstract

The higher Berry curvature was introduced by Kapustin and Spodyneiko as an extension of the Berry curvature in quantum mechanical systems with finite degrees of freedom to quantum many-body systems in finite spatial dimensions. In this paper, we propose an alternative formulation of the higher Berry curvature using translationally invariant matrix product states. They are the ground states of a set of gapped Hamiltonians which are evolved adiabatically through a discretized parameter space. Because matrix product states transform under a projective representation, evaluating the Berry curvature on a closed loop through parameter space is not sufficient to fix all the gauge degrees of freedom. To obtain a gauge-invariant real quantity, the higher-dimensional Berry curvature is evaluated on small tetrahedra in parameter space. Our numerical calculations confirm that the higher Berry curvature varies continuously throughout an adiabatic evolution and becomes quantized over a closed 3-dimensional parameter space.