Fate of partial order on the trillium and distorted windmill lattices

TL;DR

This paper studies how classical Heisenberg spins on trillium and distorted windmill lattices transition from degenerate partial order to magnetic order, revealing different mechanisms and consistent spin correlations above transition temperatures.

Contribution

It demonstrates the first order transition to magnetic order and distinguishes order by disorder from unique ground states in these lattices.

Findings

First order transition to magnetic order for both lattices

Order by disorder drives the windmill lattice ordering

Static structure factors show agreement between theory and simulations above transition

Abstract

The classical Heisenberg model on the trillium and distorted windmill lattices exhibits a degenerate ground state within large-$N$ theory, where the degenerate wavevectors form a surface and line, in 3-dimensional space, respectively. We name such states partially ordered to represent the existence of long-range order along the direction normal to these degenerate manifolds. We investigate the effects of thermal fluctuations using Monte Carlo (MC) methods, and find a first order transition to a magnetically ordered state for both cases. We further show that the ordering on the distorted windmill lattice is due to order by disorder, while the ground state of the trillium lattice is unique. Despite these different routes to the realization of low temperature ordered phases, the static structure factors obtained by large-$N$ theory and MC simulations for each lattice show quantitative agreement in the cooperative paramagnetic regime at finite temperatures. This suggests that a remnant of the characteristic angle-dependent spin correlations of partial order remains above the transition temperatures for both lattices. The possible relevance of these results to $\beta$-Mn, CeIrSi, and MnSi is discussed.