Nonequilibrium dynamics of the Ising chain in a fluctuating transverse field

TL;DR

This paper investigates the nonequilibrium quantum dynamics of the Ising chain under stochastic transverse field variations, revealing diffusive behavior, entanglement growth, and conditions for coherence preservation in a noisy quantum environment.

Contribution

It introduces a fermionic representation of the stochastic dynamics, deriving a quantum random walk model that captures decoherence and coherence preservation mechanisms.

Findings

Entanglement entropy grows as t^{1/2} under continuous noise.

Magnetization decays logarithmically as t^{1/2}.

Coherent modes can persist under dichotomous noise with specific energy conditions.

Abstract

We study nonequilibrium dynamics of the quantum Ising chain at zero temperature when the transverse field is varied stochastically. In the equivalent fermion representation, the equation of motion of Majorana operators is derived in the form of a one-dimensional, continuous-time quantum random walk with stochastic, time-dependent transition amplitudes. This type of external noise gives rise to decoherence in the associated quantum walk and the semiclassical wave-packet generally has a diffusive behavior. As a consequence, in the quantum Ising chain, the average entanglement entropy grows in time as $t^{1/2}$ and the logarithmic average magnetization decays in the same form. In the case of a dichotomous noise, when the transverse-field is changed in discrete time-steps, $\tau$, there can be excitation modes, for which coherence is maintained, provided their energy satisfies $\epsilon_k \tau\approx n\pi$ with a positive integer $n$. If the dispersion of $\epsilon_k$ is quadratic, the long-time behavior of the entanglement entropy and the logarithmic magnetization is dominated by these ballistically traveling coherent modes and both will have a $t^{3/4}$ time-dependence.