Tensor Network Markov Chain Monte Carlo: Efficient Sampling of Three-Dimensional Spin Glasses and Beyond

Abstract

Sampling the three-dimensional (3D) spin glass -- i.e., generating equilibrium configurations of a 3D lattice with quenched random couplings -- is widely regarded as one of the central and long-standing open problems in statistical physics. The rugged energy landscape, pronounced critical slowing down, and intrinsic ergodicity breaking render standard Monte Carlo methods severely inefficient, particularly for large systems at low temperatures. In this work, we introduce the Tensor Network Markov Chain Monte Carlo (TNMCMC) approach to address the issue. It generates large-scale collective updates in MCMC using tensor networks on the 2D slices of the 3D lattice, greatly improving the autocorrelation time and offering orders-of-magnitude speed-ups over conventional MCMC in generating unbiased samples of the Boltzmann distribution. We conduct numerical experiments on 3D spin glasses up to system size $64\times 64\times 64$ using a single CPU, and show that TNMCMC dramatically suppresses critical slowing down in large disordered systems, which usually require a supercomputer to perform MCMC simulations. Furthermore, we apply our approach to the 3-state Potts model up to system size $64\times 64\times 64$ using a single CPU, and show that the TNMCMC approach efficiently traverses the exponential barriers of the strong first-order transition, whereas conventional MCMC fails. Our results reveal that TNMCMC opens a promising path toward tackling long-standing, formidable three-dimensional problems in statistical physics.