Theory of multiple-phase competition in pyrochlore magnets with anisotropic exchange with application toYb2Ti2O7, Er2Ti2O7, andEr2Sn2O7

Abstract

The family of magnetic rare-earth pyrochlore oxides ${\mathrm{R}}_{2}{\mathrm{M}}_{2}{\mathrm{O}}_{7}$ plays host to a diverse array of exotic phenomena, driven by the interplay between geometrical frustration and spin-orbit interaction, which leads to anisotropy in both magnetic moments and their interactions. In this article we establish a general, symmetry-based theory of pyrochlore magnets with anisotropic exchange interactions. Starting from a very general model of nearest-neighbor exchange between Kramers ions, we find four distinct classical ordered states, all with $\mathbf{q}=0$, competing with a variety of spin liquids and unconventional forms of magnetic order. The finite-temperature phase diagram of this model is determined by Monte Carlo simulation, supported by classical spin-wave calculations. We pay particular attention to the region of parameter space relevant to the widely studied materials ${\mathrm{Er}}_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}, {\mathrm{Yb}}_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$, and ${\mathrm{Er}}_{2}{\mathrm{Sn}}_{2}{\mathrm{O}}_{7}$. We find that many of the most interesting properties of these materials can be traced back to the ``accidental'' degeneracies where phases with different symmetries meet. These include the ordered ground-state selection by fluctuations in ${\mathrm{Er}}_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$, the dimensional reduction observed in ${\mathrm{Yb}}_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$, and the lack of reported magnetic order in ${\mathrm{Er}}_{2}{\mathrm{Sn}}_{2}{\mathrm{O}}_{7}$. We also discuss the application of this theory to other pyrochlore oxides.