Thermodynamics of vison crystals in an anisotropic quantum spin liquid

Using unbiased Monte Carlo simulations and variational analysis, we present the ground state and finite-temperature phase diagrams of an exactly solvable spin-orbital model with Kitaev-type interactions on a square lattice. We show that an array of gapped and gapless vison crystals, characterized by the periodic arrangement of $Z_2$ flux excitations, can be stabilized as a function of external magnetic field and exchange anisotropy. In particular, we discover a variety of "quarter phases" wherein different 16-site periodic patterns emerge, with only a quarter of the fluxes adopting zero-flux configurations. In contrast, the rest remain in $\pi$-flux configurations. Vison crystals break translational symmetry and undergo finite-temperature phase transitions. We investigate the finite-temperature properties of these phases and report the corresponding critical and crossover temperatures. Our results reveal an array of novel phases in exactly solvable extensions of the Kitaev model, wherein local and topological orders can coexist.