Order-by-disorder in magnets with frustrated spin interactions -- classical and large-$S$ limits via the spin functional integral

TL;DR

This paper explores how fluctuations lift degeneracies in frustrated magnetic systems, using spin functional integration to analyze order-by-disorder effects and pseudo-Goldstone gaps in classical and large-S limits.

Contribution

It introduces a method to compute order-by-disorder effects and pseudo-Goldstone gaps using spin functional integration, extending previous predictions to new models and interaction types.

Findings

Order-by-disorder induces gradient-dependent anisotropic interactions.

Fluctuations generate an effective potential determining the ground state.

Explicit calculations of pseudo-Goldstone gaps for specific compass models.

Abstract

We investigate spin systems with extensive degeneracies in the classical ground states due to anisotropic frustrated spin interactions, where the degeneracy is not protected by symmetry. Using spin functional integration, we study the lifting of the degeneracies by fluctuations called order-by-disorder (ObD), and the associated gap in the spin-wave spectrum. It is shown that ObD corresponds to gradient-dependent anisotropic interactions of the pseudo-Goldstone modes, which vanish for a classical uniform spin configuration. Fluctuations generate a gradient-independent effective potential which determines the ground state and the pseudo-Goldstone gap. Furthermore, we recover previous predictions for the pseudo-Goldstone gap in type-I and II ObD with two-spin interactions in the large spin-$S$ limit or the classical small temperature limit, by computing the gap explicitly for the type-II cubic compass model and the type-I square compass model. We show that these two limits correspond to the one-loop approximation for the effective potential. We also discuss other types of order by disorder due to $m$-spin interactions where $m>2$.