Reentrance in a Hamiltonian flocking model

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.

💡 Research Summary

This paper investigates whether the re‑entrant transition—where a system moves from a homogeneous phase to a phase‑separated state and then back to homogeneity as a single control parameter is increased—requires non‑conservative energy injection, as is typical in active matter, or can also arise in a fully Hamiltonian setting. To answer this, the authors study a two‑dimensional Hamiltonian flocking model (HFM) that incorporates three ingredients: (i) a short‑range purely repulsive Weeks‑Chandler‑Andersen (WCA) potential, (ii) a ferromagnetic spin‑spin interaction of finite range, and (iii) a spin‑velocity coupling term of strength K that appears directly in the Hamiltonian as a linear contribution to each particle’s momentum (p_i = m ṙ_i + K S_i). This coupling is mathematically analogous to the minimal coupling of a charged particle to a magnetic vector potential, with the spin acting as an internal gauge field.

The dynamics are rendered overdamped by coupling the system to a thermal bath via Langevin equations. Translational and rotational frictions (γ_t, γ_r) and Gaussian white noises satisfy the fluctuation‑dissipation theorem. The resulting mobility matrix M_i(θ_i) depends on K; as K grows, the determinant Δ = γ_t γ_r + K² increases, reducing both translational and rotational mobilities. Consequently, strong spin‑velocity coupling suppresses transverse diffusion, a phenomenon the authors term “kinetic frustration”.

Numerical simulations are performed in periodic square boxes and in elongated slab geometries (aspect ratio L_x = 3 L_y) to facilitate the observation of coexisting liquid‑gas interfaces. Typical parameters are N = 250–5000 particles, packing fraction η ≈ 0.3, temperature T = 0.6, spin magnitude S = 1, ferromagnetic strength J = 1, and γ_t = γ_r = 1. The spin‑velocity coupling K is varied from 0 to 30. Each run consists of 4 × 10⁷ integration steps with a time step Δt = 10⁻⁴; the first 30 % of the trajectory is discarded as equilibration.

The central observable is the density contrast Δρ = ρ_high − ρ_low obtained from local density histograms. As K is increased, Δρ first rises from zero (homogeneous gas) to a pronounced maximum (clear liquid‑gas coexistence) and then falls back to zero, establishing a non‑monotonic, re‑entrant dependence Δρ(K). The peak occurs at K_peak ≈ √(3 γ_t γ_r), a scaling that the authors rationalize by balancing the spin‑velocity drive against the effective friction. Varying the product γ_t γ_r shifts the location of the peak but does not destroy the re‑entrance, confirming its robustness to changes in the bath viscosity.

System‑size analysis (N = 250, 1000, 5000) shows that the qualitative shape of Δρ(K) is unchanged, while the magnitude of Δρ grows with N, consistent with finite‑size scaling observed in active Brownian particle MIPS studies. Changing the global packing fraction η within 0.24–0.34 merely shifts the overall densities of the coexisting phases without eliminating the re‑entrance.

The ferromagnetic coupling J also plays a crucial role. By fixing K = 1.5 and varying J/T, the authors find a critical value (J/T)_* ≈ 1.2 below which Δρ ≈ 0 because thermal noise destroys spin alignment. Above this threshold, Δρ rises sharply and saturates, indicating that spin alignment is a prerequisite for the effective attraction generated by the spin‑velocity term.

Microscopic dynamics reveal that the particle speed distribution P(v) evolves from a Maxwell‑Boltzmann form at K = 0 to a sharply peaked low‑speed distribution at large K, reflecting the kinetic frustration mentioned above. The slowdown is accompanied by a dramatic change in local order: the magnitude of the coarse‑grained magnetization m_local becomes bimodal in the phase‑separated regime, while the signed magnetization m_x exhibits a trimodal distribution (central peak near zero representing the vapor phase and two symmetric peaks ±m_x corresponding to oppositely polarized domains inside the dense cluster). At very large K, density becomes homogeneous again, but m_x retains a bimodal shape, indicating the persistence of long‑lived polar domains separated by a nearly flat domain wall in the slab geometry. In square boxes, such domain walls are unstable and the system relaxes to a disordered homogeneous state.

The authors interpret these findings as evidence that re‑entrance does not require explicit non‑conservative energy input; rather, it emerges from a competition between an internally generated drive (the spin‑velocity coupling) and the mobility‑limiting kinetic frustration that this drive itself creates. When K is moderate, the drive is strong enough to align spins and generate an effective attraction, leading to clustering. When K becomes too large, the same coupling immobilizes particles, suppressing the necessary rearrangements for phase separation and restoring homogeneity. This mechanism mirrors the re‑entrance observed in active matter where very high Péclet numbers suppress MIPS, but here it is realized in a strictly Hamiltonian, energy‑conserving framework.

In summary, the paper provides a clear numerical demonstration that a Hamiltonian flocking model with spin‑velocity coupling exhibits re‑entrant phase behavior analogous to that seen in active systems. The work bridges equilibrium and non‑equilibrium physics by showing that the essential ingredient is the balance between drive and kinetic frustration, not the presence of non‑conservative forces. This insight opens avenues for designing conservative systems that mimic active matter phenomenology and for re‑examining the universality of re‑entrant transitions across a broad class of materials.