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

TL;DR

The study addresses dynamic phase transitions in periodically driven mean-field Ginzburg-Landau systems by decomposing the magnetization response into Fourier modes. Using limit-cycle simulations and Fourier analysis, it identifies the correct conjugate field as the even-Fourier content and defines a mode-resolved order parameter $z_k$. The results reveal robust scaling: below $P_c$, $z_k \propto \varepsilon^{1/2}$ for odd modes; at $P_c$, even perturbations induce $z_k \propto h_{mult}^{1/3}$ with parity-specific scaling of $\delta m_k$, and these findings extend to $m^6$-type models via simple one-period integral criteria. The work provides a concrete, mode-resolved framework for locating $P_c$ and predicting the dynamic response across Fourier modes, with potential relevance to experimental DPTs in periodically forced ferromagnets.

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: $|δm_{2n}| \propto h_{mult}^{1/3}$ and $|δm_{2n+1}| \propto h_{mult}^{2/3}$. (3) The same findings persist in MFGL models where an $m^6$ replaces the $m^4$ term and come with simple one-period integral criteria to locate $P_c$.

Figures (5)

• Figure 1: Free energy $F(m)$ vs magnetization $m(t)$, for varying values of the material-specific constant $a$.
• Figure 2: Free energy $F(m)$ vs magnetization $m(t)$, for varying values of $h$, where $a<0$.
• Figure 3: Hysteresis behavior in dynamic systems.
• Figure 4: Scaling of the terms $T_1, T_2, T_3$ and their sum vs $h_0$ (re-created from Robb 2014 2014).
• Figure 7: Scaling of $\delta m_n$ for even and odd $n$ with $h_{\mathrm{mult}}$.