Two-dimensional localized states in an active phase-field-crystal model

TL;DR

This paper investigates how activity influences localized and periodic crystal states in a two-dimensional active phase-field-crystal model, revealing bifurcation structures, traveling states, and pattern transitions relevant to active matter systems.

Contribution

It provides the first detailed bifurcation analysis of active PFC models in two dimensions, including traveling localized states and pattern transitions due to activity.

Findings

Active states can drift and travel while maintaining structure.

Activity modifies the bifurcation structure, including homoclinic snaking.

Transitions from hexagonal to rhombic and stripe patterns occur with activity.

Abstract

The active phase-field-crystal (active PFC) model provides a simple microscopic mean field description of crystallization in active systems. It combines the PFC model (or conserved Swift-Hohenberg equation) of colloidal crystallization and aspects of the Toner-Tu theory for self-propelled particles. We employ the active PFC model to study the occurrence of localized and periodic active crystals in two spatial dimensions. Due to the activity, crystalline states can undergo a drift instability and start to travel while keeping their spatial structure. Based on linear stability analyses, time simulations and numerical continuation of the fully nonlinear states, we present a detailed analysis of the bifurcation structure of resting and traveling states. We explore, for instance, how the slanted homoclinic snaking of steady localized states found for the passive PFC model is modified by activity. The analysis is carried out for the model in two spatial dimensions. Morphological phase diagrams showing the regions of existence of various solution types are presented merging the results from all the analysis tools employed. We also study how activity influences the crystal structure with transitions from hexagons to rhombic and stripe patterns. This in-depth analysis of a simple PFC model for active crystals and swarm formation provides a clear general understanding of the observed multistability and associated hysteresis effects, and identifies thresholds for qualitative changes in behavior.