Extending the nonequilibrium Landau theory of the transition to spontaneous oscillations in the kinetic mean-field spin-field model beyond the perturbative near-critical regime

The kinetic mean-field Ising model with dynamical field variables coupled nonreciprocally to the spins (Guislain and Bertin, Phys. Rev. Lett. 130, 207102 (2023)) exhibits paramagnetic, ferromagnetic, and spontaneously oscillating phases. Its transition to temporal oscillations is described by a nonequilibrium generalization of the Landau free energy: a rate function $\phi(m,\dot m)$ for the joint distribution of the magnetization $m$ and a smoothed stochastic derivative $\dot m$, which close to the continuous transition lines takes the form of a function $f(H)$ of an effective Hamiltonian $H=\dot m^2/2+V(m)$. This framework is perturbative, near-critical, and strictly mean-field, and its entropy-production evaluation rests on further small-asymmetry and gradient-truncation approximations. Later work by the same authors closed part of the program (the detailed study of the ferromagnetic-to-oscillating transition, and confirmation of the oscillatory transition in a three-dimensional short-range version of the model), but three connected questions remain open: the definitive universality class and renormalization-group description of the finite-dimensional transition; the validity of the entropy-production approximations away from the near-critical, weakly irreversible regime; and a nonperturbative determination of the nonequilibrium Landau free energy far from criticality.

Background

Spontaneous collective oscillations - macroscopic periodic dynamics emerging in many-body systems whose microscopic units do not oscillate - occur in biochemical clocks, active particles with nonreciprocal interactions, and driven spin systems whenever detailed balance is broken. Detailed balance is the condition $W(C'|C)P(C)=W(C|C')P(C')$ on the transition rates $W$ between microscopic configurations $C$, which guarantees an equilibrium steady state; nonreciprocity refers to interactions that act differently on the two species involved (species A responds to B differently from how B responds to A), which is incompatible with any such equilibrium potential.

Guislain and Bertin [Phys. Rev. Lett. 130, 207102 (2023)] studied an explicit mean-field model with $N$ Ising spins $s_i=\pm1$ and $N$ auxiliary binary field variables $h_i=\pm1$, with magnetization $m=N^{-1}\sum_i s_i$ and mean field $h=N^{-1}\sum_i h_i$. Flips of a spin or a field occur with heat-bath (Glauber) rates $W_{s,h}=[1+\exp(\beta\Delta E_{s,h})]^{-1}$ derived from the energies $E_s=-N(\frac{J_1}{2}m^2+\frac{J_2}{2}h^2+mh)$ and $E_h=E_s+\mu Nhm$ at inverse temperature $\beta=1/T$. For $\mu=0$ one has $E_h=E_s$ and the dynamics satisfies detailed balance; $\mu\neq0$ makes the spin-field coupling nonreciprocal and drives the system out of equilibrium. Depending on $(T,\mu)$ the model displays a paramagnetic phase, a ferromagnetic phase, and an oscillating phase in which $m(t)$ follows a limit cycle (a closed, isolated periodic trajectory of the deterministic large-$N$ dynamics; the onset of such a cycle at mean-field level is a Hopf bifurcation).

The theory is built on the joint distribution of $m$ and a smoothed stochastic time derivative $\dot m$, defined for a Markov jump dynamics as $\dot m(C)=\sum_{C'\neq C}(m(C')-m(C))W(C'|C)$; unlike the raw time derivative of $m(t)$, $\dot m$ has fluctuations of the same scale as $m$. For large $N$ the joint distribution takes a large-deviation form $P_N(m,\dot m)\sim\exp[-N\phi(m,\dot m)]$, where the rate function $\phi$ (obtained by a WKB-type expansion of the master equation, that is, an expansion in the gradient $\nabla\phi$ truncated at second order) plays the role of a nonequilibrium generalization of the Landau free energy. Close to the continuous transition lines, an expansion in a small parameter $\epsilon\propto(T_c-T)/T_c$ yields $\phi(m,\dot m)=f(H)+f_0$ with $H=\dot m^2/2+V(m)$ an effective Hamiltonian whose level sets are the deterministic trajectories. The function $f(H)$ is the nonequilibrium Landau free energy: its minimum sits at $H^*=0$ in static phases and at $H^*>0$ in the oscillating phase, so $H$ (practically, the observable $\langle\dot m^2\rangle$) is the order parameter of the transition. Away from the tricritical point where paramagnetic, ferromagnetic, and oscillating phases meet, $f(H)=-\epsilon aH+bH^2$ (continuous transition, elliptic limit cycle); near the tricritical point $f(H)=-\epsilon aH+cH^{3/2}$ is nonanalytic (flattened cycle, diverging period); and beyond $\mu_d$ the transition is discontinuous, with coexistence of static and oscillating solutions whose relative stability is set by the global minimum of $f(H)$.

The entropy production density $\sigma=N^{-1}\sum_{C\neq C'}W(C'|C)P(C)\ln[W(C'|C)/W(C|C')]$, the standard stochastic-thermodynamic measure of time-reversal symmetry breaking, was evaluated in the Supplemental Material under two approximations: a small-asymmetry assumption $1-W_k^-/W_k^+\ll1$ on the coarse-grained flip rates of the two variable groups, and the truncation of the gradient expansion at second order in $\nabla\phi$. Within their regime of validity these give $\sigma\to0$ as $N\to\infty$ in static phases and $\sigma\sim\epsilon$ in the oscillating phase, consistent with earlier results on entropy production as a singular thermodynamic potential.

The framework is thus limited in three stated ways: it is mean-field (all-to-all coupling), perturbative in $\epsilon$ and in departures from the transition lines, and its entropy-production evaluation is controlled only near continuous transition lines in the weakly irreversible regime. Establishing what happens beyond these limits requires the standard tools of critical phenomena - finite-size scaling (the analysis of how observables collapse as a function of system size and reduced distance from criticality, yielding critical exponents) and the renormalization group (RG, the coarse-graining procedure that determines universality classes, that is, the sets of transitions sharing the same critical exponents) - applied to a nonequilibrium setting where no free energy exists and the order parameter is an emergent dynamical quantity rather than a static density.

Problem Statement

Extend the nonequilibrium Landau (large-deviation) description of the transition to spontaneous temporal oscillations in the kinetic mean-field spin-field model of Guislain and Bertin beyond the perturbative, near-critical, all-to-all regime in which it was derived. The common objective is to bring this oscillatory phase transition to the level of completeness standard for equilibrium critical phenomena, along the three axes on which the source framework is explicitly incomplete:

(1) Finite-dimensional fate and universality class. Determine the nature and universality class of the transition to the oscillating phase in short-range versions of the spin-field model, by renormalization-group analysis from the spin level of description and/or conclusive critical-exponent determination. This includes the dependence on spatial dimension (in particular whether the oscillating phase survives in two dimensions for this model, given that it is destroyed by defect proliferation in the related nonreciprocal Ising model) and on interaction range, and whether the proposed two-stage scenario - local noisy oscillators emerging through a mesoscopic mean-field transition followed by a synchronization transition - is the generic mechanism. Existing three-dimensional finite-size scaling yields exponents compatible with both the three-dimensional XY and Ising universality classes and cannot discriminate between them.

(2) Entropy production away from criticality. Establish the validity, or provide a controlled replacement, of the approximations used to compute the entropy production density - the small-asymmetry expansion $1-W_k^-/W_k^+\ll1$ and the truncation of the gradient expansion at second order in $\nabla\phi$ - outside the near-critical, weakly irreversible regime where they were justified, and give the resulting form of the entropy production density in the strongly irreversible and far-from-critical regimes.

(3) Nonperturbative regime. Determine the rate function $\phi(m,\dot m)$ and the nonequilibrium Landau free energy $f(H)$ in strongly nonlinear, far-from-critical regimes - deep in the oscillating phase and far from the discontinuous threshold - where the small-$\epsilon$ expansion fails, including the structure of coexistence between static and oscillating solutions away from the threshold.

Answers to the three components count as progress on this one objective and may be delivered separately or jointly; each component is judged against the corresponding regime stated above, not merely against the perturbative near-critical limit already treated in the source.

References

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