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.
Highlights
From the point of view of thermodynamics, thermal states describe the equilibrium properties of a system
Since the eigenbasis of a many-body system is not always accessible, we propose an optimization strategy based on uniform matrix product states (MPSs) to approximate the purification of ρR directly in the thermodynamic limit
We wish to remark that, at least in the case of α = 2, we find a correspondence βR → β which holds in the thermodynamic limit
Summary
From the point of view of thermodynamics, thermal states describe the equilibrium properties of a system. Thermal states arise from the principle of maximum entropy [1,2]: For a given energy, the thermal ensemble is the one that maximizes the von Neumann entropy SG(ρ) = −Tr(ρ log ρ). Since the eigenbasis of a many-body system is not always accessible, we propose an optimization strategy based on uniform MPSs to approximate the purification of ρR directly in the thermodynamic limit. This nonlinear optimization can be accelerated using state-of-the-art techniques [23] by restricting it to the Grassmann manifold.
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