True nature of long-range order in a plaquette orbital model

TL;DR

This paper rigorously proves the existence of orientational long-range order in a classical plaquette orbital model at low temperatures, clarifying the nature of order and symmetry effects observed in numerical studies.

Contribution

It provides a first-principles proof of OLRO in the classical model and discusses implications for quantum systems and numerical simulation methods.

Findings

OLRO exists at low temperatures in the classical model

Magnetic order is prevented by symmetries

Numerical Neél order is an artifact of OLRO and energy choices

Abstract

We analyze the classical version of a plaquette orbital model that was recently introduced and studied numerically by S. Wenzel and W. Janke. In this model, edges of the square lattice are partitioned into $x$ and $z$-types that alternate along both coordinate directions and thus arrange into a checkerboard pattern of $x$ and $z$-plaquettes; classical O(2)-spins are then coupled ferromagnetically via their first components over the $x$-edges and via their second components over the $z$-edges. We prove from first principles that, at sufficiently low temperatures, the model exhibits orientational long-range order (OLRO) in one of the two principal lattice directions. Magnetic order is precluded by the underlying symmetries. A similar set of results is inferred also for quantum systems with large spin although the $\frac12$ instance currently seems beyond the reach of rigorous methods. We point out that the Ne\'el order in the plaquette energy distribution observed in numerical simulations is an artefact of the OLRO and a judicious choice of the plaquette energies. In particular, this order seems to disappear when the plaquette energies are adjusted to vanish at the ground-state level. We also discuss the specific role of the underlying symmetries in Wenzel and Janke's simulations and propose an enhanced method of numerical sampling that could in principle significantly increase the speed of convergence.