  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.
