Dynamic Models for Two Nonreciprocally Coupled Fields: A Microscopic Derivation for Zero, One, and Two Conservation Laws

TL;DR

The paper addresses how two nonreciprocally coupled scalar fields, arising from two coupled Ising lattices, can be described by continuum field theories with zero, one, or two conservation laws. By deriving mean-field equations and performing hydrodynamic coarse-graining, it connects microscopic spin dynamics to nonreciprocal Allen–Cahn, Cahn–Hilliard, and reactive CH models, and, in doing so, establishes a microscopic foundation for nonreciprocal pattern formation. The authors map out linear instabilities, including Hopf, exceptional-point, and Allen–Cahn–type instabilities, and explain the absence of Turing instabilities at the MF level, while also identifying spurious-gradient structures and potential regularization via higher-gradient terms. They also show that combining different kinetic rules yields sixteen possible two-field models, clarifying how conservation laws shape the emergent dynamics and pattern formation. Overall, the work provides a systematic framework linking nonequilibrium statistical mechanics to macroscopic, nonreciprocal continuum descriptions with broad implications for multi-component active and social systems.

Abstract

We construct dynamic models governing two nonreciprocally coupled fields for several cases with zero, one, and two conservation laws. Starting from two microscopic nonreciprocally coupled Ising models, and using the mean-field approximation, we obtain closed-form evolution equations for the spatially resolved magnetization in each lattice. Only allowing for single spin-flip dynamics, the macroscopic equations in the thermodynamic limit are closely related to the nonreciprocal Allen-Cahn equations, i.e. conservation laws are absent. Likewise, only accounting for spin-exchange dynamics within each lattice, the thermodynamic limit yields equations similar to the nonreciprocal Cahn-Hilliard model, i.e. with two conservation laws. In the case of spin-exchange dynamics within and between the two lattices, we obtain two nonreciprocally coupled equations that add up to one conservation law. For each of these cases, we systematically map out the linear instabilities that can arise. Moreover, combining the different dynamics gives a large number of further models. Our results provide a microscopic foundation for a broad class of nonreciprocal field theories, establishing a direct link between nonequilibrium statistical mechanics and macroscopic continuum descriptions.

Figures (5)

• Figure 1: (a) Schematic of the lattice configuration and interaction structure in the nonreciprocal Ising model. Each square lattice has spacing $\ell$ and dimensions $\{L_{x}=N_x\ell, L_{y}=N_y\ell\}$. The parameter $J_{\mu}$ denotes the coupling between nearest-neighbor spins within lattice $\mu \in \{a, b\}$, $K_{\mu}$ represents the directed interlattice coupling, and $H_{\mu}$ is an external magnetic field applied to lattice $\mu$. When $K_{a} \neq K_{b}$, the interactions become nonreciprocal. (b) In single spin-flip dynamics, individual spins on each lattice flip independently. (c) In intralattice spin-exchange dynamics, two neighboring spins within the same lattice are exchanged. (d) In interlattice spin-exchange dynamics, spins at corresponding positions on opposing lattices are exchanged. (e) Summary table of the kinetic rules and their associated conservation laws for the magnetization $M^{\mu}(t)$ [see Eq. \ref{['M']}], as discussed in Sect. \ref{['sec:2.2']}
• Figure 2: (a)-(f) Dispersion relations for single spin-flip dynamics [see Eq. \ref{['lambdaglauber']}]. Panels (a)-(c) correspond to the net ferromagnetic regime with $\tilde{J}_{a} + \tilde{J}_{b} \geq 1/2$, while panels (d)-(f) correspond to the net antiferromagnetic regime with $\tilde{J}_{a} + \tilde{J}_{b} \leq 0$. In panels (a) and (d), a band of unstable stationary wavenumbers is observed, characterized by ${\rm Re}(\lambda_{\pm}) \geq 0$ and ${\rm Im}(\lambda_{\pm}) = 0$, along with an intermediate band of unstable oscillatory modes where ${\rm Re}(\lambda_{\pm}) \geq 0$ and ${\rm Im}(\lambda_{\pm}) > 0$. In panels (b) and (e), only unstable oscillatory modes are present. Panels (c) and (f) feature a critical exceptional point at $|\mathbf{k}| = k_{\rm R}=k^{\pm}_{\rm I}$, where ${\rm Re}(\lambda_{\pm}) = {\rm Im}(\lambda_{\pm}) = 0$. The auxiliary functions $\mathcal{K}_{1}(\tilde{J}_{a}, \tilde{J}_{b})$ and $\mathcal{K}_{2}(\tilde{J}_{a}, \tilde{J}_{b})$ are defined in Eqs. \ref{['Avar1']}-\ref{['Avar2']}, and the wavenumbers $k^{\pm}_{\rm I}$ and $k_{\rm R}$ are given by Eqs. \ref{['kh']}-\ref{['kE']}. Parameter values $(\tilde{J}_{a},\tilde{J}_{b},\tilde{K}_{a}\tilde{K}_{b})$ used in each panel are given by: (a) $(0.5,0.35,-0.07)$, (b) $(0.5,0.35,-0.19)$, (c) $(0.5,0.35,-0.0311419)$, (d) $(-0.5,-0.35,-0.14)$, (e) $(-0.3,-0.3,-0.14)$, (f) $(-0.5,-0.35,-0.0311419)$.
• Figure 3: (a)-(c) Dispersion relations for single spin-flip dynamics [see Eq. \ref{['lambdaglauber']}] in the presence of unstable stationary modes. Panels (b) and (c) show the emergence of a high-wavenumber band of unstable modes, resulting from effective antiferromagnetic coupling. This instability drives microphase separation, characterized by pattern formation on microscopic length scales. Parameter values $(\tilde{J}_{a},\tilde{J}_{b},\tilde{K}_{a}\tilde{K}_{b})$ used in each panel are given by: (a) $(0.5,0.1,0)$, (b) $(0.5,-0.1,0)$, (c) $(-0.5,-0.1,0)$.
• Figure 4: (a)-(f) Dispersion relations for intralattice spin-exchange dynamics [see Eq. \ref{['lambdakwasaki']}]. Panels (a)-(c) correspond to the net ferromagnetic regime with $\tilde{J}_{a} +\tilde{J}_{b} \geq 1/2$, and panels (d)-(f) correspond to the net antiferromagnetic regime with $\tilde{J}_{a} +\tilde{J}_{b} \leq 0$. In panels (a) and (d), a band of unstable stationary wavenumbers is observed, characterized by ${\rm Re}(\tilde{\lambda}_{\pm}) \geq 0$ and ${\rm Im}(\tilde{\lambda}_{\pm}) = 0$, along with an intermediate band of unstable oscillatory wavenumbers where ${\rm Re}(\tilde{\lambda}_{\pm}) \geq 0$ and ${\rm Im}(\tilde{\lambda}_{\pm}) > 0$. In panels (b) and (e), only unstable oscillatory wavenumbers are present. Panels (c) and (f) feature a critical exceptional point at $|\mathbf{k}|= k_{\rm R}=k^{\pm}_{\rm I}$, where ${\rm Re}(\tilde{\lambda}_{\pm}) = {\rm Im}(\tilde{\lambda}_{\pm}) = 0$. The auxiliary functions $\mathcal{K}_{1}(\tilde{J}_{a}, \tilde{J}_{b})$ and $\mathcal{K}_{2}(\tilde{J}_{a}, \tilde{J}_{b})$ are defined in Eqs. \ref{['Avar1']}-\ref{['Avar2']}, and the wavenumbers $k^{\pm}_{\rm I}$ and $k_{\rm R}$ are given by Eqs. \ref{['kh']}-\ref{['kE']}. Parameter values used in each panel are chosen as in Fig. \ref{['Fig2']}.
• Figure 5: (a)–(c) Reactive nullcline (red line) defined by Eq. \ref{['zeroR']}, shown for parameter values (a) below, (b) at, and (c) above the critical threshold given in Eq. \ref{['specialpoint']}. The background illustrates the flow field for the uniform state, as governed by Eqs. \ref{['RD1']}. In panel (c), the black dashed segment marks parts of the nullcline that are linearly unstable with respect to uniform perturbations. The red points indicate the symmetric uniform steady states [Eq. \ref{['uniformnullcline']}], which is unstable in (c) and therefore evolves along the line $m^{a}_{0}+m^{b}_{0}={\rm const.}$ towards one of two stable states (black points). (d)–(f) Dispersion relations corresponding to linear perturbations around the symmetric uniform steady state, as given by Eq. \ref{['eigenRD']}. In panel (d), both eigenvalues are non-positive at $|\mathbf{k}| = 0$, indicating linear stability. In panel (e), both eigenvalues vanish at $|\mathbf{k}| = 0$, marking the onset of a uniform instability. In panel (f), one eigenvalue is positive at $|\mathbf{k}| = 0$, indicating a stationary instability of a band of harmonic modes. The zero crossing at $k^{+}_{1}$ is given by Eq. \ref{['kAC']}. Parameter values $(J_{a},J_{b},K_{a},K_{b},H_{a},H_{b})$ used in each panel are given by: (a, d) $(0.2,0.25,0.05,-0.05,0.04,-0.04)$, (b, e) $(0.2,0.323031,0.05,-0.05,0.04,-0.04)$, (c, f) $(0.2,0.4,0.05,-0.05,0.04,-0.04)$. Note that at $(0.2,0.279083,0.05,-0.05,0.04,-0.04)$ a Cahn-Hilliard instability occurs, which is not explicitly shown.