Flocking with discrete symmetry: the 2d Active Ising Model

TL;DR

This paper investigates the active Ising model on a 2D lattice, revealing it exhibits a liquid-gas phase transition with a critical point in the Ising universality class, and develops a continuum theory to describe its behavior.

Contribution

It introduces a minimal flocking model with discrete symmetry, analyzes its phase transition as a liquid-gas type, and constructs a continuum theory matching microscopic results.

Findings

The transition to collective motion is a liquid-gas phase transition.

Critical point at zero velocity belongs to the Ising universality class.

The continuum theory predicts phase diagram shapes and coexistence properties.

Abstract

We study in detail the active Ising model, a stochastic lattice gas where collective motion emerges from the spontaneous breaking of a discrete symmetry. On a 2d lattice, active particles undergo a diffusion biased in one of two possible directions (left and right) and align ferromagnetically their direction of motion, hence yielding a minimal flocking model with discrete rotational symmetry. We show that the transition to collective motion amounts in this model to a bona fide liquid-gas phase transition in the canonical ensemble. The phase diagram in the density/velocity parameter plane has a critical point at zero velocity which belongs to the Ising universality class. In the density/temperature "canonical" ensemble, the usual critical point of the equilibrium liquid-gas transition is sent to infinite density because the different symmetries between liquid and gas phases preclude a supercritical region. We build a continuum theory which reproduces qualitatively the behavior of the microscopic model. In particular we predict analytically the shapes of the phase diagrams in the vicinity of the critical points, the binodal and spinodal densities at coexistence, and the speeds and shapes of the phase-separated profiles.