Entanglement renormalization of thermofield double states

TL;DR

This paper develops an entanglement renormalization method for finite-temperature quantum states using the thermofield double approach, enabling analysis of topological order and critical behavior in thermal systems.

Contribution

It introduces a novel scheme applying MERA to thermofield double states for finite-temperature systems, including an exact circuit for the toric code and analysis of thermal critical states.

Findings

Exact renormalization circuit for 2D toric code at finite temperature

Demonstration of topological order loss after renormalization

Identification of Lifshitz theory description for critical thermal states

Abstract

Entanglement renormalization is a method for coarse-graining a quantum state in real space, with the multi-scale entanglement renormalization ansatz (MERA) as a notable example. We obtain an entanglement renormalization scheme for finite-temperature (Gibbs) states by applying MERA to their canonical purification, the thermofield double state. As an example, we find an analytically exact renormalization circuit for finite temperature two-dimensional toric code which maps it to a coarse-grained system with a renormalized higher temperature, thus explicitly demonstrating its lack of topological order. Furthermore, we apply this scheme to one-dimensional free boson models at a finite temperature and find that the thermofield double corresponding to the critical thermal state is described by a Lifshitz theory. We numerically demonstrate the relevance and irrelevance of various perturbations under real space renormalization.