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arXiv:2005.04857 · Chinese Physics Letters

Continuous-Mixture Autoregressive Networks Learning the Kosterlitz-Thouless Transition

Lingxiao Wang, Yin Jiang, Lianyi He, Kai Zhou

Published Nov 21, 2022• 19 citations
arXivHTMLPDF

Abstract

We develop deep autoregressive networks with multi channels to compute many-body systems with continuous spin degrees of freedom directly. As a concrete example, we demonstrate the two-dimensional XY model with 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 generative model. By learning the microscopic probability distributions from the macroscopic thermal distribution, the networks are trained as an efficient physical sampler which can approximate the free energy and estimate thermodynamic observables unbiasedly with importance sampling. 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.

DOI
10.1088/0256-307x/39/12/120502

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We develop deep autoregressive networks with multi channels to compute many-body systems with continuous spin degrees of freedom directly. As a concrete example, we demonstrate the two-dimensional XY model with 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 generative model. By learning the microscopic probability distributions from the macroscopic thermal distribution, the networks are trained as an efficient physical sampler which can approximate the free energy and estimate thermodynamic observables unbiasedly with importance sampling. 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.
We develop the deep autoregressive networks with multi-channel to compute continuous spin system directly. As a concrete example, the 2-D XY model is embedded into the continuous-mixture networks, in which the Kosterlitz-Thouless phase transition emerges on a periodic square lattice. Vortices as the quasi-Long Range Order(LRO) are accurately detected by the autoregressive neural networks. By learning the microscopic possibility distributions from macroscopic thermal distribution, the free energy is directly calculated and vortices (anti-vortices) pair emerge with temperature rise. As a more precise estimation, the helicity modulus is evaluated to anchor the transition temperature to be at βc=1.101\beta_\text{c}=1.101βc​=1.101. Although the time-consuming of the training process inevitably increases with the lattice size, the training time remains unchanged around the transition temperature.
A generative model, the Variational Autoregressive Networks(VANs) is introduced to recognize the Kosterlitz-Thouless phase transition on a periodic square lattice. Vortices as the quasi-Long Range Order(LRO) are accurately detected by the ordering neural networks. By learning the microscopic possibility distributions from macroscopic thermal distribution, the free energy is directly calculated and vortices(anti-vortices) pair emerge with temperature rising. As a more precise estimation, the helicity modulus is evaluated to anchor the transition temperature to be at βc=1.101\beta_\text{c}=1.101βc​=1.101. Although time-consuming of the training process is inevitably increasing with the lattice size, the training time remains unchanged around the transition temperature.
We present a transformer-based variational autoregressive network for sampling lattice systems with continuous degrees of freedom and apply it to the two-dimensional XY model near the Berezinskii-Kosterlitz-Thouless transition. The architecture is combined with physics-informed ingredients: locality-informed attention and approximate probabilities. The resulting sampler significantly outperforms existing neural samplers. We train the model on lattices up to 96×9696\times9696×96, verify with a mode-dropping estimator that no parts of the configuration space are missed, and reproduce the energy, magnetization and helicity modulus obtained with the traditional Markov Chain Monte Carlo algorithm.
In several physical systems, important properties characterizing the system itself are theoretically related with specific degrees of freedom. Although standard Monte Carlo simulations provide an effective tool to accurately reconstruct the physical configurations of the system, they are unable to isolate the different contributions corresponding to different degrees of freedom. Here we show that unsupervised deep learning can become a valid support to MC simulation, coupling useful insights in the phases detection task with good reconstruction performance. As a testbed we consider the 2D XY model, showing that a deep neural network based on variational autoencoders can detect the continuous Kosterlitz-Thouless (KT) transitions, and that, if endowed with the appropriate constrains, they generate configurations with meaningful physical content.
The Berezinskii-Kosterlitz-Thouless transition is a very specific phase transition where all thermodynamic quantities are smooth.Therefore, it is difficult to determine the critical temperature in a precise way.In this paper we demonstrate how neural networks can be used to perform this task.In particular, we study how the accuracy of the transition identification depends on the way the neural networks are trained.We apply our approach to three different systems: (i) the classical XY model, (ii) the phase-fermion model, where classical and quantum degrees of freedom are coupled and (iii) the quantum XY model.
Detecting the topological Kosterlitz-Thouless (KT) transition in the prototypical two-dimensional (2D) XY model using unsupervised machine learning methods has long been a challenging problem due to the lack of suitable order parameters. To address this issue, we begin with a conventional real-valued restricted Boltzmann machine (RBM) (RBM-xy), which uses exponential conditional probabilities to generate visible units. We then develop a novel real-valued RBM (RBM-CosSin) featuring nonlinear cos/sincos/sincos/sin activation, whose visible units follow the von Mises distribution. Our findings reveal that RBM-CosSin effectively learns the underlying Boltzmann distribution of 2D XY systems and generates authentic XY configurations that accurately capture both thermodynamics and topological order (vortex). Furthermore, we demonstrate that it is possible to extract phase transition information, including the KT transition, from the weight matrices without relying on prior physics knowledge.