  This paper introduces a new structural phase field crystal (PFC) type model
that expands the PFC methodology to a wider class of structurally complex
crystal structures than previously possible. Specifically, our new approach
allows for stabilization of graphene, as well as its coexistence with a
disordered phase. It also preserves the ability to model the usual triangular
and square lattices previously reported in 2D PFC studies. Our approach is
guided by the formalism of the classical field theory, wherein the the free
energy functional is expanded to third order in PFC density correlations. It
differs from previous PFC approaches in two main features. First, it utilizes a
hard-sphere repulsion to describe two-point correlations. Second, and more
important, is that it uses a rotationally invariant three-point correlation
function that provides a unified way to control the formation of crystalline
structures that can be described by a specific bond angle, such as graphene,
triangular or square symmetries. Our new approach retains much of the
computational simplicity of previous PFC models and allows for efficient
simulation of nucleation and growth of polycrystalline 2D materials. In
preparation for future applications, this paper details the mathematical
derivation of the model and its equilibrium properties, and uses dynamical
simulations to demonstrate defect structures produced by the model.
