Dynamic Phase Transitions in Mean-Field Ginzburg-Landau Models: Conjugate Fields and Fourier-Mode Scaling

TL;DR

This paper investigates dynamic phase transitions in mean-field Ginzburg-Landau models, revealing that the entire even Fourier component of the applied field acts as the conjugate field at the critical period, with specific scaling laws for order parameters.

Contribution

It identifies the correct conjugate field as the entire even Fourier component and characterizes the Fourier-mode scaling behavior at the critical period in mean-field Ginzburg-Landau models.

Findings

The order parameter scales as  with  for modes 0.

Adding even Fourier components causes the order parameter to scale as h_{mult}^{1/3}.

Scaling laws are consistent across models with higher-order nonlinearities.

Abstract

Dynamic phase transitions of periodically forced mean-field ferromagnets are often described by a single order parameter and a scalar conjugate field. Building from previous work, we show that, at the critical period $P_c$ of the mean-field Ginzburg-Landau (MFGL) dynamics with energy $F(m)=am^2+bm^4-hm$, the correct conjugate field is the entire even-Fourier component part of the applied field. The correct order parameter is $z_k=\sqrt{\bigl|\,m_k^2-|m_{k,c}|^2\,\bigr|}$, where $m_k$ is the $k^{th}$ Fourier component of the magnetization m(t), and $m_{k,c}$ is the $k^{th}$ Fourier component at the critical period. Using high-accuracy limit-cycle integration and Fourier analysis, we first confirm that, for periodic fields that contain only odd components, the symmetry-broken branch below $P_c$ exhibits $z_k \propto \varepsilon^{1/2}$ (computationally tested for modes $k\le30$), where $\varepsilon=(P_c-P)/P_c$. This provides strong evidence that the 1/2 scaling holds for all Fourier modes. We then find three robust facts: (1) Exactly at $P_c$, adding a small perturbation composed of even Fourier components with an overall field multiplier $h_{mult}$ yields $z_k \propto h_{mult}^{1/3}$ across many $k$. (2) Mode-resolved deviations obey a parity rule: $|\delta m_{2n}| \propto h_{mult}^{1/3}$ and $|\delta m_{2n+1}| \propto h_{mult}^{2/3}$. (3) These scalings persist in two MFGL models with higher-order nonlinearities.