  Competition between superconductivity and charge order is a recurring theme
in contemporary condensed matter physics. This is quintessentially captured in
the attractive Hubbard model, a simple theoretical model where the competition
can be directly tuned. In previous studies by the current authors, it has been
suggested that the Hubbard model maps to an $SO(3)$ non-linear sigma model,
where the phase competition becomes manifest. In this article, we rigorously
demonstrate this mapping and use it to study thermal disordering of a
supersolid. Starting with the attractive Hubbard model in the presence of an
orbital field, we take the limit of strong coupling where a pseudospin
description emerges. The in-plane pseudospin components represent
superconducting pairing while the out-of-plane component encodes charge density
wave order. We obtain an effective spin-$1/2$ Hamiltonian with ferromagnetic
in-plane couplings and antiferromagnetic z-z couplings. In addition, the
orbital field gives rise to a textured Dzyaloshinskii-Moriya interaction that
has the same periodicity as the magnetic unit cell. In order to examine the
nature of ordering in this spin model, we consider it in the classical limit.
We assume slowly varying fields, leading to the $SO(3)$ non-linear sigma model
description. As an application of these ideas, we study the nature of ordering
using simulated annealing and classical Monte Carlo simulations. The ground
state represents a supersolid with coexisting superconductivity and charge
order. It can be viewed as a `meron crystal', a regular arrangement of
superconducting vortices with charge-ordered cores. The coherent overlap of
core regions gives rise to coherent long-ranged charge order. As the
temperature is raised, this charge order is lost via a sharp phase transition
in the Ising universality class.
