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The active phase-field-crystal (active PFC) model provides a simplemicroscopic mean field description of crystallization in active systems. Itcombines the PFC model (or conserved Swift-Hohenberg equation) of colloidalcrystallization and aspects of the Toner-Tu theory for self-propelledparticles. We employ the active PFC model to study the occurrence of localizedand periodic active crystals in two spatial dimensions. Due to the activity,crystalline states can undergo a drift instability and start to travel whilekeeping their spatial structure. Based on linear stability analyses, timesimulations and numerical continuation of the fully nonlinear states, wepresent a detailed analysis of the bifurcation structure of resting andtraveling states. We explore, for instance, how the slanted homoclinic snakingof steady localized states found for the passive PFC model is modified byactivity. The analysis is carried out for the model in two spatial dimensions.Morphological phase diagrams showing the regions of existence of varioussolution types are presented merging the results from all the analysis toolsemployed. We also study how activity influences the crystal structure withtransitions from hexagons to rhombic and stripe patterns. This in-depthanalysis of a simple PFC model for active crystals and swarm formation providesa clear general understanding of the observed multistability and associatedhysteresis effects, and identifies thresholds for qualitative changes inbehavior.

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