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

TL;DR

This study uses large-scale Monte Carlo simulations to explore phase transitions in 3D orbital models, revealing a continuous transition with unique critical behavior in the $e_g$ model and a first-order transition to a nematic phase in the $t_{2g}$ model.

Contribution

It provides the first detailed finite-temperature analysis of 3D $e_g$ and $t_{2g}$ orbital models, highlighting their distinct phase transition characteristics.

Findings

The $e_g$ model exhibits a continuous phase transition with unconventional critical exponents.

A U(1) symmetry emerges at the critical point and persists below it.

The $t_{2g}$ model undergoes a first-order transition to a nematic phase.

Abstract

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 $\nu\approx0.66(1)$ 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 $a\approx1.3$. Finally, for the $t_{2g}$ model we find a {\em first order} transition into a low-temperature lattice-nematic phase without orbital order.