https://doi.org/10.1140/epje/s10189-023-00364-w
Regular Article – Flowing Matter
Active Ising Models of flocking: a field-theoretic approach
1
Dip. di Fisica, Università Sapienza, 00185, Rome, Italy
2
Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, 00185, Rome, Italy
3
DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, CB3 0WA, Cambridge, UK
4
Department of Mathematics, Imperial College London, SW7 2AZ, London, UK
Received:
20
June
2023
Accepted:
12
October
2023
Published online:
26
October
2023
Using an approach based on Doi-Peliti field theory, we study several different Active Ising Models (AIMs), in each of which collective motion (flocking) of self-propelled particles arises from the spontaneous breaking of a discrete symmetry. We test the predictive power of our field theories by deriving the hydrodynamic equations for the different microscopic choices of aligning processes that define our various models. At deterministic level, the resulting equations largely confirm known results, but our approach has the advantage of allowing systematic generalization to include noise terms. Study of the resulting hydrodynamics allows us to confirm that the various AIMs share the same phenomenology of a first-order transition from isotropic to flocked states whenever the self-propulsion speed is nonzero, with an important exception for the case where particles align only pairwise locally. Remarkably, this variant fails entirely to give flocking—an outcome that was foreseen in previous work, but is confirmed here and explained in terms of the scalings of various terms in the hydrodynamic limit. Finally, we discuss our AIMs in the limit of zero self-propulsion where the ordering transition is continuous. In this limit, each model is still out of equilibrium because the dynamical rules continue to break detailed balance, yet it has been argued that an equilibrium universality class (Model C) prevails. We study field-theoretically the connection between our AIMs and Model C, arguing that these particular models (though not AIMs in general) lie outside the Model C class. We link this to the fact that in our AIMs without self-propulsion, detailed balance is not merely still broken, but replaced by a different dynamical symmetry in which the dynamics of the particle density is independent of the spin state.
We dedicate this article to Fyl Pincus, whose enduring contributions to soft matter and biological physics, alongside his enthusiasm, generosity and vision, have enlightened the field for more than half a century.
© The Author(s) 2023
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