Continuous-mixture Autoregressive Networks for efficient variational calculation of many-body systems

TL;DR

This paper introduces continuous-mixture autoregressive neural networks capable of directly modeling many-body systems with continuous spins, successfully detecting phase transitions and vortices in the 2D XY model.

Contribution

The development of deep autoregressive networks with multi channels for continuous degrees of freedom and their application to the XY model to identify phase transitions and vortices.

Findings

Successfully detected vortices and phase transition in the XY model.

Computed free energy and vortex emergence directly from learned distributions.

Training time remains stable around the critical temperature despite larger lattices.

Abstract

We develop deep autoregressive networks with multi channels to compute many-body systems with \emph{continuous} spin degrees of freedom directly. As a concrete example, we embed the two-dimensional XY model into the continuous-mixture networks and rediscover the Kosterlitz-Thouless (KT) phase transition on a periodic square lattice. Vortices characterizing the quasi-long range order are accurately detected by the autoregressive neural networks. By learning the microscopic probability distributions from the macroscopic thermal distribution, the neural networks compute the free energy directly and find that free vortices and anti-vortices emerge in the high-temperature regime. As a more precise evaluation, we compute the helicity modulus to determine the KT transition temperature. Although the training process becomes more time-consuming with larger lattice sizes, the training time remains unchanged around the KT transition temperature. The continuous-mixture autoregressive networks we developed thus can be potentially used to study other many-body systems with continuous degrees of freedom.