Bipartiteness transition in a two-dimensional antiferromagnetic tetratic

We explore the effect of strong Ising antiferromagnetic correlations on dislocation-mediated melting in two dimensions. In the solid phase, fundamental dislocations disrupt the bipartiteness of the underlying square lattice; as a result, pairs of dislocations are linearly confined by string-like antiferromagnetic domain walls. It has previously been argued that an antiferromagnetic tetratic phase can arise when double dislocations proliferate while single dislocations remain bound. By identifying an emergent Ising gauge field and constructing an effective field theory, we demonstrate that the antiferromagnetic and paramagnetic tetratic phases are smoothly connected to each other. Despite the absence of a thermodynamic phase transition, we argue that a novel "bipartiteness" transition separates the antiferromagnetic and paramagnetic tetratic regimes: given an arrangement of particles and dislocations sampled from the Gibbs ensemble, we ask whether it is possible to unambiguously define antiferromagnetic correlations between the particles by constructing a nearly bipartite lattice. In the antiferromagnetic tetratic regime, dislocations can always be paired in a well-defined homology class, allowing for long-range antiferromagnetic correlations on the resulting lattice. In contrast, the paramagnetic tetratic does not admit a fixed bipartite structure, resulting in short-range antiferromagnetic correlations.