Reentrance in a Hamiltonian flocking model

TL;DR

This work investigates whether reentrant phase separation, common in active matter, can arise in a conservative, Hamiltonian flocking model. It introduces a Hamiltonian flocking model with spin–velocity coupling $K$, repulsion $U_R$, and spin–spin coupling, and derives overdamped Langevin dynamics that couple translation and rotation through a state-dependent mobility matrix, providing an effective drive without self-propulsion. The key finding is a non-monotonic reentrance: increasing $K$ first promotes clustering and then suppresses it via kinetic frustration, effectively reducing dynamics to 1D sliders at large $K$; a scaling argument $\,\Pi(K) \propto \frac{K^3}{(\gamma_t \gamma_r + K^2)^2}$ predicts a peak at $K_{\mathrm{peak}} = \sqrt{3\gamma_t \gamma_r}$, in agreement with simulations. This work bridges reentrant physics across equilibrium and active matter within a Hamiltonian, thermodynamically consistent setting and suggests experimental realizations in spin-coupled granular or Quincke-roller systems.

Abstract

The clustering of self-motile and repulsive particles, so-called motility-induced phase separation (MIPS), is one of the clearest signatures of active physics. Typically, increasing the amplitude of self-motility increases the degree of clustering, however for high enough self-motility the homogeneous phase is reentered. Here, we report that such reentrance naturally emerges in a Hamiltonian (conservative) model known to recapitulate properties of (active) bird flocks, and exhibits clustering behaviour reminiscent of MIPS. We numerically demonstrate the reentrance of the homogeneous phase and identify the underlying mechanism as a competition between the amplitude of a spin-velocity coupled drive and mobility-limited kinetic frustration. Specifically, we reveal that strong spin-velocity coupling suppresses transverse diffusion, thereby leading the system into an arrest that closes the window for phase separation. Overall, our work offers a Hamiltonian, conservative, bridge between reentrant physics across equilibrium and non-equilibrium materials.

Figures (5)

• Figure 1: Hamiltonian flocking model (HFM) and qualitative observation of reentrance.(a) Pictorial description of the essential model ingredients defined in \ref{['eq:Hamilton']}. Colors indicate orientation. (b) Steady-state snapshots at fixed $N=1000$, $\eta=0.32$, $T=0.6$, and $J=1$, for varying spin-velocity coupling strengths $K$ (top-left of snapshots). Particles are colored by their orientation $\cos\theta_i$. Upon monotonically increasing $K$ the system exhibits reentrance: from a homogeneous phase (pink square) to clustered (green square) back to the homogeneous phase.
• Figure 2: Numerical characterization of reentrant phase behavior. (a) Representative snapshots of the slab simulations. From left to right: Initial configuration ($t=0$); steady states at $K=0, 1.5, 3, \text{ and } 30$. System size: $N=5000$ and $L_x = 3 L_y$. (b) Difference between high and low densities $\Delta\rho\sigma^2$ vs. scaled coupling $K/\sqrt{J}$ for varying $\sqrt{\gamma_t\gamma_r}$. The order parameter attains its maximum around $K = \sqrt{3\gamma_t\gamma_r}$. (c) Phase coexistence diagram in the $K/\sqrt{J}$-$\rho$ plane for different $N$. Circles denote dilute phase, squares the dense phase, and gray crosses denote the single-phase. (d) Order parameter $\Delta\rho \sigma^2$ as a function of the scaled ferromagnetic coupling $J/T$ at fixed $K=3/2$. The order parameter increases monotonically with $J/T$ and eventually saturates, confirming that stronger alignment interactions enhance phase separation. (e) Phase diagram in the $K/\sqrt{J}$-$\rho$ plane for different global packing fractions $\eta \in \{0.24, 0.32, 0.34\}$. Solid lines with symbols represent the binodal coexistence densities, while gray crosses indicate homogeneous states. Unless otherwise indicated parameters used: $N=1000$, $T=0.6$, $\gamma_t=\gamma_r=1$, $\eta=0.32$, and $J=1$.
• Figure 3: Dynamical and structural properties across reentrance. This figure is organized into four columns corresponding to different physical quantities. (a) Particle speed analysis. (i) Heatmap of the particle speed distribution $P(v)$ vs scaled coupling $K/\sqrt{J}$. (ii) PDFs of particle speed $v$ for selected $K/\sqrt{J}$. (b) Local magnetization analysis. (i) Heatmap of the local magnetization distribution $P(m_{\text{local}})$ vs scaled coupling $K/\sqrt{J}$. (ii) PDFs of local magnetization for selected $K/\sqrt{J}$. (c) Local signed magnetization analysis. (i) Heatmap of the local signed magnetization distribution $P(m_{x})$ vs scaled coupling $K/\sqrt{J}$. (ii) PDFs of local signed magnetization for selected $K/\sqrt{J}$. (d) Macroscopic order parameters. (i) The mean center-of-mass velocity magnitude $\langle |\mathbf{v}_{\text{CoM}}| \rangle$ as a function of $K/\sqrt{J}$. (ii) The time-averaged global magnetization $\langle M \rangle$ as a function of $K/\sqrt{J}$, averaging over 10 simulation runs. All data are for $N=5000$, $T=0.6, \eta=0.32, J=1$, and $\gamma_t=\gamma_r=1$.
• Figure 4: Underlying reentrance mechanisms.(a) Verification of kinetic frustration: Normalized mobility of longitudinal ($v_\parallel$, blue circles), transverse ($v_\perp$, red squares), and angular ($\dot{\theta}$, green triangles) degrees of freedom as a function of the scaled coupling $K/\sqrt{J}$. The simulations are performed with $N=1000$ particles at $J=1.0$ and $\gamma_t=\gamma_r=1$. The dashed black line represents the theoretical prediction derived from the Langevin dynamics (see Eq. \ref{['eq:variances']}). At large coupling, the transverse and rotational variances decay as $K^{-2}$ (slope $-2$), while longitudinal motion remains unaffected, signifying a transition from isotropic diffusion to highly anisotropic diffusion. (b) The rotational autocorrelation function $C_s(t) = \langle \mathbf{S}_i(t) \cdot \mathbf{S}_i(0) \rangle$ vs. time $t$ for increasing scaled coupling strength $K/\sqrt{J}$. The inset presents the same data colored by the order parameter $\Delta\rho\sigma^2$. (c) Scaling ansatz of reentrance: Comparison of the density contrast $\Delta\rho\sigma^2$ (symbols) with the scaling ansatz $\Pi(K) \propto K^3/(\gamma_t\gamma_r + K^2)^2$ (solid curves, normalized to match the peak height of the simulation data) across different damping regimes. The vertical dotted lines indicate the theoretical peak position $K_{\mathrm{peak}} = \sqrt{3\gamma_t\gamma_r}$.
• Figure 5: Steady-state snapshots across different aspect ratios. Results are shown for $L_x/L_y = 1, 2,$ and $3$ with fixed parameters $N=5000$, $K=30$ , $T=0.6$ , $\eta=0.32$, $J=1$, and $\gamma_t=\gamma_r=1$. Increasing $L_x/L_y$ stabilizes a nearly flat spin domain wall, leading to long-lived bipolar spin domains at large $K$ while the density remains essentially homogeneous.