Order-by-disorder near criticality in $XY$ pyrochlore magnets

TL;DR

This paper investigates how thermal fluctuations select a specific ordered state in XY pyrochlore magnets near criticality, revealing an order-by-disorder mechanism that lifts degeneracy and induces weak magnetic moments.

Contribution

It introduces an extended TAP method and cluster-mean-field theory to analyze fluctuation-driven state selection and degeneracy lifting in XY pyrochlore magnets near criticality.

Findings

Fluctuations lift degeneracy, favoring the $ ext{psi}_2$ state over $ ext{psi}_3$.

A weak magnetic moment along the local $ ext{111}$ direction is induced in the $ ext{psi}_2$ state.

Different phase diagrams are obtained from TAP and c-MFT methods, highlighting fluctuation length scale effects.

Abstract

We consider a system of spins on the sites of a three-dimensional pyrochlore lattice of corner-sharing tetrahedra interacting with a predominant effective $xy$ exchange. In particular, we investigate the selection of a long-range ordered state with broken discrete symmetry induced by thermal fluctuations near the critical region. At the standard mean-field theory (s-MFT) level, in a region of the parameter space of this Hamiltonian that we refer to as $\Gamma_5$ region, the ordered state possesses an accidental $U(1)$ degeneracy. In this paper, we show that fluctuations beyond s-MFT lift this degeneracy by selecting one of two states (so-called $\psi_2$ and $\psi_3$) from the degenerate manifold, thus exposing a certain form of order-by-disorder (ObD). We analytically explore this selection at the microscopic level and close to criticality by elaborating upon and using an extension of the so-called TAP method, originally developed by Thouless, Anderson and Palmer to study the effect of fluctuations in spin glasses. We also use a single-tetrahedron cluster-mean-field theory (c-MFT) to explore over what minimal length scale fluctuations can lift the degeneracy. We find the phase diagrams obtained by these two methods to be somewhat different since c-MFT only includes the shortest-range fluctuations. General symmetry arguments used to construct a Ginzburg-Landau theory to lowest order in the order parameters predict that a weak magnetic moment, $m_z$, along the local $\langle 111 \rangle$ (${\hat z}$) direction is generically induced for a system ordering into a $\psi_2$ state, but not so for $\psi_3$ ordering. Both E-TAP and c-MFT calculations confirm this weak fluctuation-induced $m_z$ moment. Using a Ginzburg-Landau theory, we discuss the phenomenology of multiple phase transitions below the paramagnetic phase transition and within the $\Gamma_5$ long-range ordered phase.