The many faces of order-by-disorder in rare-earth pyrochlore magnets

Order-by-disorder (ObD) is a concept of central importance in the field of frustrated magnetism. Saddled with large accidental degeneracies, a subset of states, those that support the largest quantum and/or thermal fluctuations, may be selected to form true long-range order. More formally, one often begins describing a system in terms of some order parameter $m$ with the low-energy description framed in terms of an effective action $\Gamma(m)$. In each ObD scenario, one starts from an artificial limit where there is an accidental degeneracy; that is the effective action at this point, $\Gamma_0(m)$, has an accidental symmetry. One may then view ObD phenomena as cases where the corrections to $\Gamma_0(m)$ arise through some form of fluctuation corrections, may they be thermal, quantum or virtual, towards an enlarged higher energy Hilbert space. In the rare-earth pyrochlore oxides, of formula $R_2$$M_2$O$_7$, the trivalent magnetic rare-earth ions $R^{3+}$ (e.g $R=$Gd, Er, Yb; $M=$Ti, Sn is non-magnetic) reside on a three-dimensional pyrochlore lattice of corner-sharing tetrahedra. This architecture is prone to a high degree of magnetic frustration, with the $R_2M_2$O$_7$ pyrochlore materials having been found over the past twenty years to display a gamut of exotic phenomena. In this talk, I will discuss three such phenomena: (i) the intermediate partially-ordered multiple-$k$ state between $0.7$K and $1$K in the Gd$_2$Ti$_2$O$_7$ Heisenberg antiferromagnet~\footnote{$\,$PRL \textbf{114}, 130601 (2015)}, (ii) the ordered $\psi_2$ state selection in the $XY$ Er$_2$Ti$_2$O$_7$ antiferromagnet~\footnote{$\,$arXiv:1510.04292} and (iii) the puzzling high sample sensitivity of the Yb$_2$Ti$_2$O$_7$ ``quantum spin ice'' candidate~\footnote{$\,$arXiv:1505.05499}. I will argue that in all three cases, some form of fluctuation corrections to their simplest $\Gamma_0(m)$ description play a significant role in the state selection and experimentally observed behaviors.