Unveiling the nature of three dimensional orbital ordering transitions: the case of $e_g$ and $t_{2g}$ models on the cubic lattice

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 $ν\approx0.66(1)$ is close to the 3D XY value, the exponent $η\approx 0.15(1)$ 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 $Λ\sim ξ^a$, with an unusually small $a\approx1.3$. Finally, for the $t_{2g}$ model we find a {\em first order} transition into a low-temperature lattice-nematic phase without orbital order.

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