R\'enyi free energy and variational approximations to thermal states

TL;DR

This paper introduces R\'enyi free energy-based ensembles as an alternative to Gibbs states for approximating thermal states, with algorithms for tensor network representations and analysis on spin chains.

Contribution

It develops a novel variational framework using R\'enyi free energy and provides algorithms for tensor network approximations of these ensembles.

Findings

Local properties of R\'enyi ensembles match thermal equilibrium for large systems.

Algorithms successfully find tensor network approximations to the 2-R\'enyi ensemble.

Performance analysis on one-dimensional spin chains demonstrates effectiveness.

Abstract

We propose the construction of thermodynamic ensembles that minimize the R\'enyi free energy, as an alternative to Gibbs states. For large systems, the local properties of these R\'enyi ensembles coincide with those of thermal equilibrium, and they can be used as approximations to thermal states. We provide algorithms to find tensor network approximations to the 2-R\'enyi ensemble. In particular, a matrix-product-state representation can be found by using gradient-based optimization on Riemannian manifolds, or via a non-linear evolution which yields the desired state as a fixed point. We analyze the performance of the algorithms and the properties of the ensembles on one-dimensional spin chains.