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

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) 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 $P_c$.