Diverging conditional correlation lengths in the approach to high temperature

TL;DR

This paper demonstrates that the Markov length diverges exponentially during heating in classical systems, indicating a non-Gibbsian state and a divergence in the effective Hamiltonian range, with implications for understanding thermalization.

Contribution

It introduces a numerical method using matrix-product states to compute the Markov length and analyzes its divergence during classical thermal quenches.

Findings

Markov length diverges exponentially during heating

State becomes increasingly non-Gibbsian over time

Range of the parent Hamiltonian diverges with Markov length

Abstract

The Markov length was recently proposed as an information-theoretic diagnostic for quantum mixed-state phase transitions [Sang & Hsieh, Phys. Rev. Lett. 134, 070403 (2025)]. Here, we show that the Markov length diverges even under classical stochastic dynamics, when a low-temperature ordered state is quenched into the high temperature phase. Conventional observables do not exhibit growing length scales upon quenching into the high-temperature phase; however, the Markov length grows exponentially in time. Consequently, the state of a system as it heats becomes increasingly non-Gibbsian, and the range of its putative "parent Hamiltonian" must diverge with the Markov length. From this information-theoretic point of view the late-time limit of thermalization is singular. We introduce a numerical technique for computing the Markov length based on matrix-product states, and explore its dynamics under general thermal quenches in the one-dimensional classical Ising model. For all cases, we provide simple information-theoretic arguments that explain our results.