Classical $\mathbb{Z}_2$ spin liquid on the generalized four-color Kitaev model

Abstract: While $U$(1) spin liquids have been extensively studied in both quantum and classical regimes, exact classical $\mathbb{Z}_2$ spin liquids arising from models with nearest-neighbor, bilinear spin interactions are still rare. In this Letter, we explore the four-color Kitaev model as a minimal model for stabilizing classical $\mathbb{Z}_2$ spin liquids across a broad family of tricoordinated lattices. By formulating a $\mathbb{Z}_2$ lattice gauge theory, we identify this spin liquid as being described by an emergent Gauss's law with effective charge-2 condensation, and deconfined fractionalized bond-charge excitations. We complement our findings with Monte Carlo simulations, revealing a crossover from a high-temperature paramagnet to a low-temperature liquid phase characterized by residual entropy, classical $\mathbb{Z}_2$ flux order, and diffuse spin structure factors.