We perform large scale finite-temperature Monte Carlo simulations of the classical $e_g$ and $t_{2g}$ orbital models on the simple cubic lattice in three dimensions. The $e_g$ model displays a continuous phase transition to an orbitally ordered phase. While the correlation length exponent $\nu \approx 0.66(1)$ is close to the 3D $XY$ value, the exponent $\eta \approx 0.15(1)$ differs substantially from $O(N)$ values. At $T_c$ a $U(1)$ symmetry emerges, which persists for $T<T_c$ below a crossover length scaling as $\Lambda \sim {\xi }^a$, with an unusually small $a\approx 1.3$. Finally, for the $t_{2g}$ model we find a first order transition into a low-temperature lattice-nematic phase without orbital order.

• Received 19 January 2011

DOI:https://doi.org/10.1103/PhysRevLett.106.197201