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Competition between superconductivity and charge order is a recurring themein contemporary condensed matter physics. This is quintessentially captured inthe attractive Hubbard model, a simple theoretical model where the competitioncan be directly tuned. In previous studies by the current authors, it has beensuggested that the Hubbard model maps to an $SO(3)$ non-linear sigma model,where the phase competition becomes manifest. In this article, we rigorouslydemonstrate this mapping and use it to study thermal disordering of asupersolid. Starting with the attractive Hubbard model in the presence of anorbital field, we take the limit of strong coupling where a pseudospindescription emerges. The in-plane pseudospin components representsuperconducting pairing while the out-of-plane component encodes charge densitywave order. We obtain an effective spin-$1/2$ Hamiltonian with ferromagneticin-plane couplings and antiferromagnetic z-z couplings. In addition, theorbital field gives rise to a textured Dzyaloshinskii-Moriya interaction thathas the same periodicity as the magnetic unit cell. In order to examine thenature 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 modeldescription. As an application of these ideas, we study the nature of orderingusing simulated annealing and classical Monte Carlo simulations. The groundstate represents a supersolid with coexisting superconductivity and chargeorder. It can be viewed as a `meron crystal', a regular arrangement ofsuperconducting vortices with charge-ordered cores. The coherent overlap ofcore regions gives rise to coherent long-ranged charge order. As thetemperature is raised, this charge order is lost via a sharp phase transitionin the Ising universality class.

The attractive Hubbard model as an $SO(3)$ system of competing phases: supersolid order and its thermal melting