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arXiv:2012.12848 · Physical review. B./Physical review. B

Rényi free energy and variational approximations to thermal states

Giacomo Giudice, Aslı Çakan, J. I. Cirac, Mari Carmen Bañuls

Published May 17, 2021• 8 citations
arXivHTMLPDF

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 nonlinear 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.

DOI
10.1103/physrevb.103.205128

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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 nonlinear 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.
We study the statistical physics of the classical Ising model in the so-called α -Rényi ensemble, a finite-temperature thermal state approximation that minimizes a modified free energy based on the α -Rényi entropy. We begin by characterizing its critical behavior in mean-field theory in different regimes of the Rényi index α . Next, we re-introduce correlations and consider the model in one and two dimensions, presenting analytical arguments for the former and devising a Monte Carlo approach to the study of the latter. Remarkably, we find that while mean-field predicts a continuous phase transition below a threshold index value of α ∼ 1.303 and a first-order transition above it, the Monte Carlo results in two dimensions point to a continuous transition at all α . We conclude by performing a variational minimization of the α -Rényi free energy using a recurrent neural network (RNN) where we find that the RNN performs well in two dimensions when compared to the Monte Carlo simulations. Our work highlights the potential opportunities and limitations associated with the use of the α -Rényi ensemble formalism in probing the thermodynamic equilibrium properties of classical and quantum systems.
We introduce a variational Monte Carlo algorithm for approximating finite-temperature quantum many-body systems, based on the minimization of a modified free energy. This approach directly approximates the state at a fixed temperature, allowing for systematic improvement of the expressiveness without accumulating errors from iterative imaginary-time evolution. We employ a variety of trial states—both tensor networks as well as neural networks—as variational for our numerical optimization. We benchmark and compare different constructions in the above classes, both for one- and two-dimensional problems, with systems made of up to N = 100 spins. Our results demonstrate that while restricted Boltzmann machines show limitations, string bond tensor network states exhibit systematic improvements with increasing bond dimensions and the number of strings.
We introduce a variational algorithm based on Matrix Product States that is trained by minimizing a generalized free energy defined using Tsallis entropy instead of the standard Gibbs entropy. As a result, our model can generate the probability distributions associated with generalized statistical mechanics. The resulting model can be efficiently trained, since the resulting free energy and its gradient can be calculated exactly through tensor network contractions, as opposed to standard methods which require estimating the Gibbs entropy by sampling. We devise a variational annealing scheme by ramping up the inverse temperature, which allows us to train the model while avoiding getting trapped in local minima. We show the validity of our approach in Ising spin-glass problems by comparing it to exact numerical results and quasi-exact analytical approximations. Our work opens up new possibilities for studying generalized statistical physics and solving combinatorial optimization problems with tensor networks.
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We introduce a tensor network method for approximating thermal equilibrium states of quantum many-body systems at low temperatures. Whereas the usual approach starts from infinite temperature and evolves the state in imaginary time (toward lower temperature), our ansatz is constructed from the zero-temperature limit, the ground state, which can be found with a standard tensor network approach. Motivated by properties of the ground state for conformal field theories, our ansatz is especially well suited near criticality. Moreover, it allows an efficient computation of thermodynamic quantities and entanglement properties. We demonstrate the performance of our approach with a tree tensor network ansatz, although it can be extended to other tensor networks, and present results illustrating its effectiveness in capturing the finite-temperature properties in one- and two-dimensional scenarios. In particular, in the critical one-dimensional case we show how the ansatz reproduces the finite temperature scaling of entanglement in a conformal field theory.