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 Pc of the Mean-Field Ginzburg–Landau (MFGL) dynamics with energy F(m) = am2 + bm4 − hm, the correct conjugate field is the entire even-Fourier component part of the applied field. The correct order parameter is zk=|mk|2−|mk,c|2, where mk is the kth Fourier component of the magnetization m(t), and mk,c is the kth 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 Pc exhibits zk ∼ ɛ1/2 (computationally tested for modes k ≤ 30), where ɛ = (Pc − P)/Pc. This provides strong enough evidence that the 1/2 scaling holds for all Fourier modes. We then find three robust facts: (1) Exactly at Pc, adding a small perturbation composed of even Fourier components with an overall field multiplier hmult yields zk∼hmult1/3 across many k. (2) Mode-resolved deviations obey a parity rule: |δm2n|∝hmult1/3 and |δm2n+1|∝hmult2/3. (3) These scalings persist in two MFGL models with higher-order nonlinearities.