[Paper Review] Continuous-mixture Autoregressive Networks for efficient variational calculation of many-body systems

Lingxiao Wang, Yin Jiang+2 more|arXiv (Cornell University)|May 11, 2020

Quantum many-body systems14 references5 citations

TL;DR

This paper introduces Continuous-mixture Autoregressive Networks (CANs), a deep generative model that efficiently computes many-body systems with continuous spin degrees of freedom by modeling conditional probability distributions across lattice sites. The method successfully recovers the Kosterlitz-Thouless phase transition in the 2D XY model, detects vortices unsupervised, and computes the transition temperature at β ≈ 1.10 with helicity modulus, avoiding critical slowing down typical in MCMC simulations.

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.

Motivation & Objective

• To develop a variational method for many-body systems with continuous degrees of freedom that avoids the limitations of traditional MCMC, such as critical slowing down and sign problems.
• To enable unsupervised detection of topological defects like vortices in continuous spin systems using deep generative models.
• To provide a scalable and efficient alternative to Monte Carlo methods for computing thermodynamic quantities and phase transitions in systems like the 2D XY model.
• To demonstrate the capability of autoregressive neural networks to model long-range correlations and topological order in quantum and statistical systems.

Proposed method

• Proposes Continuous-mixture Autoregressive Networks (CANs), which model the joint probability distribution of a many-body system via autoregressive decomposition of conditional probabilities on lattice sites.
• Uses multi-channel mixtures of beta distributions to represent the continuous conditional probability distribution at each lattice site, enabling flexible modeling of continuous spin variables.
• Employs a variational approach to learn the microscopic probability distribution from the macroscopic thermal distribution, directly computing the free energy.
• Leverages GPU-accelerated training to generate configurations efficiently, with training time scaling polynomially with lattice size (t_train ∝ L^1.708).
• Uses the helicity modulus computed from CAN-generated configurations to locate the Kosterlitz-Thouless transition point via crossing with 2kBT/π.
• Trains the network in an unsupervised manner, allowing vortices and anti-vortices to emerge naturally from learned configurations.

Experimental results

Research questions

• RQ1 Can deep autoregressive networks with continuous conditional distributions effectively model and detect topological phase transitions in continuous spin systems?
• RQ2 Can CANs avoid critical slowing down in the vicinity of phase transitions, unlike traditional MCMC methods?
• RQ3 To what extent can CANs reproduce known phase transition temperatures, such as the Kosterlitz-Thouless transition in the 2D XY model?
• RQ4 How does the network’s ability to generate vortices and anti-vortices compare to standard Monte Carlo simulations?
• RQ5 What is the scaling behavior of training time with lattice size in CANs, and can it be efficiently parallelized on GPUs?

Key findings

• The CAN model successfully recovers the Kosterlitz-Thouless transition in the 2D XY model, with the transition temperature estimated at β_KT ≈ 1.10, consistent with standard Monte Carlo results.
• The helicity modulus computed from CAN-generated configurations shows a crossing point with 2kBT/π at β ≈ 1.10 for lattice size L = 16, confirming the phase transition.
• Vortices and anti-vortices emerge naturally in the generated configurations without explicit labeling, demonstrating unsupervised detection of topological defects.
• Training time per step remains nearly constant (~0.2 seconds) across temperatures, indicating effective mitigation of critical slowing down.
• Training time scales as t_train(L) ≈ 0.0026 × L^1.708, showing polynomial dependence on lattice size, which is manageable with GPU acceleration.
• The method enables direct computation of free energy and higher-order statistics like the helicity modulus, which depend on long-range correlations, with high accuracy.