Order from structural disorder in the $XY$ pyrochlore antiferromagnet ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$

The effect of structural disorder is investigated for an $XY$ pyrochlore antiferromagnet with continuous degeneracy of classical ground states. Two types of disorder—vacancies and weakly fluctuating exchange bonds—lift degeneracy, selecting the same subset of classical ground states. Analytic and numerical results demonstrate that such an “order by structural disorder” mechanism competes with the effect of thermal and quantum fluctuations. Our theory predicts that a small amount of nonmagnetic impurities in ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ will stabilize the coplanar ${\psi }_3$ $\left(m_{x^2-y^2}\right)$ magnetic structure as opposed to the ${\psi }_2$ $\left(m_{3z^2-r^2}\right)$ state found in pure material.

Figure 1

Ordered magnetic structures of the $XY$ pyrochlore antiferromagnet: (a) the coplanar Palmer-Chalker state $m_{xy}$, (b) the noncoplanar state $m_{3z^2-r^2}$, and (c) the coplanar state $m_{x^2-y^2}$. (d) Schematic phase diagram of the model (1) showing the lowest-energy magnetic structures depending on the ratio $J_{\perp }^a/J_{\perp }$. The circle marks the value of $J_{\perp }^a/J_{\perp }$ in ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$.

Figure 2

Zero-temperature results for the clock order parameter $m_6$ obtained for an $XY$ pyrochlore antiferromagnet with $n_{\mathrm{imp}}=5\%$ of nonmagnetic impurities for several cluster sizes $L$ and different values of the anisotropic exchange $J_{\perp }^a$. Positive and negative values of $m_6$ correspond to $m_{3z^2-r^2}$ and $m_{x^2-y^2}$ magnetic states, respectively.

Figure 3

Monte Carlo results for the antiferromagnetic $m$ (upper panel) and the clock $m_6$ (lower panel) order parameters for the $XY$ pyrochlore antiferromagnet (1) with $J_{\perp }^a=0.5$ and $5\%$ of vacancies. The dotted vertical line indicates the transition temperature. Dashed lines show the behavior of $m$ and $m_6$ for a pure system with $L=16$.

Figure 4

Temperature dependence of the magnetic order parameters for different impurity concentrations. Monte Carlo results are for $J_{\perp }^a=0.3$ and $L=16$.

Figure 5

Temperature dependence of the magnetic order parameters for fixed impurity concentration $n_{\mathrm{imp}}=5\%$ and different values of $J_{\perp }^a$. Monte Carlo results are for $L=16$ clusters.

Figure 6

Inset: fit of the spin-wave energy correction $\Delta E_{\mathrm{g}.\mathrm{s}.}$ (data points) to $cos6\phi$ angular dependence (full lines). Main panel: comparison of the fitted amplitude $C_6$ of the cosine function (circles) to the analytic expression (solid line) from the leading term in the harmonic expansion (B5).