Tensor-network algorithm for nonequilibrium relaxation in the thermodynamic limit

TL;DR

This paper introduces a tensor-network algorithm that efficiently simulates nonequilibrium relaxation in infinite systems, enabling precise analysis of critical dynamics in models like the Ising model.

Contribution

The paper presents a novel tensor-network method for simulating discrete-time stochastic dynamics directly in the thermodynamic limit, combining advantages of relaxation methods and tensor networks.

Findings

Estimated dynamical critical exponent z=2.16(5) for 2D Ising model

Accurately computes magnetization time evolution in large systems

Analyzes nonequilibrium relaxation directly in the thermodynamic limit

Abstract

We propose a tensor-network algorithm for discrete-time stochastic dynamics of a homogeneous system in the thermodynamic limit. We map a $d$-dimensional nonequilibrium Markov process to a $(d+1)$-dimensional infinite tensor network by using a higher-order singular-value decomposition. As an application of the algorithm, we compute the nonequilibrium relaxation from a fully magnetized state to equilibrium of the one- and two- dimensional Ising models with periodic boundary conditions. Utilizing the translational invariance of the systems, we analyze the behavior in the thermodynamic limit directly. We estimated the dynamical critical exponent $z=2.16(5)$ for the two-dimensional Ising model. Our approach fits well with the framework of the nonequilibrium-relaxation method. Our algorithm can compute time evolution of the magnetization of a large system precisely for a relatively short period. In the nonequilibrium-relaxation method, on the other hand, one needs to simulate dynamics of a large system for a short time. The combination of the two provides a new approach to the study of critical phenomena.