# Nonequilibrium-relaxation approach to quantum phase transitions: Nontrivial critical relaxation in cluster-update quantum Monte Carlo

## TL;DR

This paper demonstrates that the nonequilibrium relaxation (NER) method reveals nontrivial critical relaxation behavior in quantum phase transitions analyzed with cluster-update quantum Monte Carlo, challenging previous assumptions about relaxation speed.

## Contribution

The study shows that NER in quantum systems with cluster-update QMC exhibits stretched-exponential relaxation, enabling new analysis of quantum critical dynamics.

## Key findings

- NER exhibits stretched-exponential relaxation in quantum phase transitions.
- Cluster-update QMC allows effective NER analysis despite rapid relaxation.
- Application to the 2D Heisenberg model confirms the method's utility.

## Abstract

Although the nonequilibrium relaxation (NER) method has been widely used in Monte Carlo studies on phase transitions in classical spin systems, such studies have been quite limited in quantum phase transitions. The reason is that relaxation process based on cluster-update quantum Monte Carlo (QMC) algorithms, which are now standards in Monte Carlo studies on quantum systems, has been considered "too fast" for such analyses. Recently the present authors revealed that the NER process in classical spin systems based on cluster-update algorithms is characterized by the stretched-exponential critical relaxation, rather than the conventional power-law one in local-update algorithms. In the present article we show that this is also the case in quantum phase transitions analyzed with the cluster-update QMC, and that advantages of NER analyses are available. As the simplest example of isotropic quantum spin models which exhibit quantum phase transitions, we investigate the N\'eel-dimer quantum phase transition in the two-dimensional $S=1/2$ columnar-dimerized antiferromagnetic Heisenberg model with the continuous-time loop algorithm.