Monte Carlo study of the critical properties of the three-dimensional 120-degree model

We report on large scale finite-temperature Monte Carlo simulations of the classical ${120}^{\circ }$ or $e_g$ orbital-only model on the simple cubic lattice in three dimensions with a focus towards its critical properties. This model displays a continuous phase transition to an orbitally ordered phase. While the correlation length exponent $\nu \approx 0.665$ is close to the 3D XY value, the exponent $\eta \approx 0.15$ differs substantially from O(N) values. We also introduce a discrete variant of the $e_g$ model, called $e_g$-clock model, which is found to display the same set of exponents. Further, an emergent U(1) symmetry is found at the critical point $T_c$, which persists for $T<T_c$ below a crossover length scaling as $\mathrm{\Lambda }\sim {\xi }^a$, with an unusually small $a\approx 1.3$.