Analytical derivation and extension of the anti-Kibble-Zurek scaling in the transverse field Ising model

TL;DR

This paper analytically explores how Gaussian white noise affects defect density scaling during phase transitions in the transverse field Ising model, revealing conditions under which anti-Kibble-Zurek scaling applies and simplifying its derivation.

Contribution

It provides an analytical approximation of defect density under noise, extending understanding of anti-Kibble-Zurek scaling without solving differential equations.

Findings

Anti-Kibble-Zurek scaling holds for small noise levels.

Scaling can be derived using adiabatic approximation at higher noise levels.

Identifies parameters that minimize defect density based on new scaling.

Abstract

A defect density which quantifies the deviation from the spin ground state characterizes non-equilibrium dynamics during phase transitions. The widely recognized Kibble-Zurek scaling predicts how the defect density evolves during phase transitions. However, it can be perturbed by a noise, leading to the anti-Kibble-Zurek scaling. In this research, we analytically investigate the effect of Gaussian white noise on the transition probabilities of the Landau-Zener model. We apply this analysis to the one-dimensional transverse field Ising model and obtain an analytical approximate solution of the defect density. Our analysis reveals that when the introduced noise is small, the model follows the previously known anti-Kibble-Zurek scaling. Conversely, when the noise increases, the scaling can be obtained by using the adiabatic approximation. This result indicates that deriving the anti-Kibble-Zurek scaling does not require solving differential equations, instead, it can be achieved simply by applying the adiabatic approximation. Furthermore, we identify the parameter that minimizes the defect density based on the new scaling, which allows us to verify how effective the already known scaling of the optimized parameter is.