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

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 $F(m)=a{m}^2+b{m}^4-hm$, the correct conjugate field is the entire even-Fourier component part of the applied field. The correct order parameter is $z_k=\sqrt{\mid m_k^2-\mathrm{\mid }{m}_{k,c}{\mathrm{\mid }}^2\mid }$, 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 $z_k\propto {\epsilon }^{1\mathrm{/}2}$ (computationally tested for modes $k\le 30$), where $\epsilon =(P_c-P)\mathrm{/}{P}_c$. 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 $z_k\propto h_{mult}^{1\mathrm{/}3}$ across many $k$. (2) Mode-resolved deviations obey a parity rule: $\mathrm{\mid }\delta m_{2n}\mathrm{\mid }\propto h_{mult}^{1\mathrm{/}3}$ and $\mathrm{\mid }\delta m_{2n+1}\mathrm{\mid }\propto h_{mult}^{2\mathrm{/}3}$. (3) These scalings persist in two MFGL models with higher-order nonlinearities.