  Two-dimensional polar liquid crystals have been discovered recently in
monolayers of anisotropic molecules. Here, we provide a systematic theoretical
description of liquid-crystalline phases for polar particles in two spatial
dimensions. Starting from microscopic density functional theory, we derive a
phase-field-crystal expression for the free-energy density which involves three
local order-parameter fields, namely the translational density, the
polarization, and the nematic order parameter. Various coupling terms between
the order-parameter fields are obtained which are in line with macroscopic
considerations. Since the coupling constants are brought into connection with
the molecular correlations, we establish a bridge from microscopic to
macroscopic modeling. Our theory provides a starting point for further
numerical calculations of the stability of polar liquid-crystalline phases and
is also relevant for modeling of microswimmers which are intrinsically polar.
