Statistical mechanics and machine learning of the α-Rényi ensemble

We study the statistical physics of the classical Ising model in the so-called alpha-R & eacute;nyi ensemble, a finitetemperature thermal state approximation that minimizes a modified free energy based on the alpha-R & eacute;nyi entropy. We begin by characterizing its critical behavior in mean-field theory in different regimes of the R & eacute;nyi index alpha. 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 alpha similar to 1.303 and a first-order transition above it, the Monte Carlo results in two dimensions point to a continuous transition at all alpha. We conclude by performing a variational minimization of the alpha-R & eacute;nyi free energy using a recurrent neural network (RNN) Ansatz 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 alpha-R & eacute;nyi ensemble formalism in probing the thermodynamic equilibrium properties of classical and quantum systems.