Four-coloring model on the square lattice: A critical ground state

We study critical properties of the four-coloring model, which is given by the equal-weighted ensemble of all possible edge colorings of the square lattice with four different colors. We map the four-coloring model onto an interface model for which we propose an effective Gaussian field theory, which allows us to calculate correlation functions of operators in the coloring model. The critical exponents are given by the stiffness of the interface, which we calculate exactly using recent results on the statistical topography of rough interfaces. Our numerical exponents, from Monte Carlo simulations of the four-coloring model, are in excellent agreement with the analytical calculations. These results support the conjecture that the scaling limit of the four-coloring model is given by the Su(4${)}_{\mathit{k}=1\mathrm{}}$ Wess-Zumino-Witten model. Moreover, we show that our effective field theory is the free-field representation of the Su(4${)}_{\mathit{k}=1\mathrm{}}$ Wess-Zumino-Witten model. Finally, we discuss connections to loop models, and some predictions of finite temperature properties of a particular Potts model for which the four-coloring model is the ground state.