Gauge fixing, canonical forms and optimal truncations in tensor networks with closed loops

TL;DR

This paper introduces a method to fix gauge degrees of freedom in tensor networks with closed loops, enabling canonical forms, measuring internal correlations, and optimizing truncations to improve many-body system simulations.

Contribution

It presents a unified framework for gauge fixing, correlation measurement, and optimal truncation in tensor networks with loops, advancing tensor network manipulation techniques.

Findings

Canonical form for tensor networks with loops achieved.

A measure for internal correlations in tensor networks proposed.

An algorithm for optimal internal index truncation developed.

Abstract

We describe an approach to fix the gauge degrees of freedom in tensor networks, including those with closed loops, which allows a canonical form for arbitrary tensor networks to be realized. Additionally, a measure for the internal correlations present in a tensor network is proposed, which quantifies the extent of resonances around closed loops in the network. Finally we describe an algorithm for the optimal truncation of an internal index from a tensor network, based upon proper removal of the redundant internal correlations. These results, which offer a unified theoretical framework for the manipulation of tensor networks with closed loops, can be applied to improve existing tensor network methods for the study of many-body systems and may also constitute key algorithmic components of sophisticated new tensor methods.