Off-equilibrium scaling behaviors driven by time-dependent external fields in three-dimensional O(N) vector models

TL;DR

This paper investigates the off-equilibrium scaling behavior of three-dimensional O(N) vector models under a slowly-varying external magnetic field, revealing universal scaling laws and hysteresis phenomena near the critical transition.

Contribution

It introduces universal off-equilibrium scaling laws for magnetization in 3D O(N) models driven by time-dependent fields, supported by numerical simulations and analysis of hysteresis.

Findings

Magnetization exhibits off-equilibrium scaling near the transition line.

Scaling variables involve the ratio of time scale to system size and time.

Hysteresis loop area can be scaled to measure deviation from equilibrium.

Abstract

We consider the dynamical off-equilibrium behavior of the three-dimensional O$(N)$ vector model in the presence of a slowly-varying time-dependent spatially-uniform magnetic field ${\bm H}(t) = h(t)\,{\bm e}$, where ${\bm e}$ is a $N$-dimensional constant unit vector, $h(t)=t/t_s$, and $t_s$ is a time scale, at fixed temperature $T\le T_c$, where $T_c$ corresponds to the continuous order-disorder transition. The dynamic evolutions start from equilibrium configurations at $h_i < 0$, correspondingly $t_i < 0$, and end at time $t_f > 0$ with $h(t_f) > 0$, or vice versa. We show that the magnetization displays an off-equilibrium scaling behavior close to the transition line ${\bm H}(t)=0$. It arises from the interplay among the time $t$, the time scale $t_s$, and the finite size $L$. The scaling behavior can be parametrized in terms of the scaling variables $t_s^\kappa/L$ and $t/t_s^{\kappa_t}$, where $\kappa>0$ and $\kappa_t > 0$ are appropriate universal exponents, which differ at the critical point and for $T < T_c$. In the latter case, $\kappa$ and $\kappa_t$ also depend on the shape of the lattice and on the boundary conditions. We present numerical results for the Heisenberg ($N=3$) model under a purely relaxational dynamics. They confirm the predicted off-equilibrium scaling behaviors at and below $T_c$. We also discuss hysteresis phenomena in round-trip protocols for the time dependence of the external field. We define a scaling function for the hysteresis loop area of the magnetization that can be used to quantify how far the system is from equilibrium.