Attractive Hubbard model as an SO(3) system of competing phases: Supersolid order and its thermal melting

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) nonlinear 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) nonlinear sigma model description. As an application of these ideas, we study the nature of ordering using simulated annealing. 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 overlap of core regions gives rise to coherent long-ranged charge order. We study the effect of thermal fluctuations using classical Monte Carlo simulations. We find a sharp phase transition in the Ising universality class where charge order is lost.