Ground state phase diagram of generic<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>X</mml:mi><mml:mi>Y</mml:mi></mml:mrow></mml:math>pyrochlore magnets with quantum fluctuations

Motivated by recent experimental and theoretical progress on the Er${}_{2}$Ti${}_{2}$O${}_{7}$ pyrochlore $XY$ antiferromagnet, we study the general problem of quantum order-by-disorder in pyrochlore $XY$ systems. We consider the nearest-neighbor pseudo-spin-1/2 Hamiltonian for such a system characterized by anisotropic spin-spin couplings ${J}_{e}\ensuremath{\equiv}$${{J}_{\ifmmode\pm\else\textpm\fi{}},{J}_{\ifmmode\pm\else\textpm\fi{}\ifmmode\pm\else\textpm\fi{}},{J}_{z\ifmmode\pm\else\textpm\fi{}},{J}_{zz}}$ and construct zero-temperature phase diagrams. Combining symmetry arguments and spin-wave calculations, we show that the ground state phase boundaries between the two candidate ground states of the ${\ensuremath{\Gamma}}_{5}$ irreducible representation, the ${\ensuremath{\psi}}_{2}$ and ${\ensuremath{\psi}}_{3}$ (basis) states, are rather accurately determined by a cubic equation in $({J}_{\ifmmode\pm\else\textpm\fi{}}{J}_{\ifmmode\pm\else\textpm\fi{}\ifmmode\pm\else\textpm\fi{}})/{J}_{z\ifmmode\pm\else\textpm\fi{}}^{2}$. Depending on the value of ${J}_{zz}$, there can be one or three phase boundaries that separate alternating regions of ${\ensuremath{\psi}}_{2}$ and ${\ensuremath{\psi}}_{3}$ states. In particular, we find for sufficiently small ${J}_{zz}/{J}_{\ifmmode\pm\else\textpm\fi{}}$ a narrow ${\ensuremath{\psi}}_{2}$ sliver sandwiched between two ${\ensuremath{\psi}}_{3}$ regions in the ${J}_{\ifmmode\pm\else\textpm\fi{}\ifmmode\pm\else\textpm\fi{}}/{J}_{\ifmmode\pm\else\textpm\fi{}}$ vs ${J}_{z\ifmmode\pm\else\textpm\fi{}}/{J}_{\ifmmode\pm\else\textpm\fi{}}$ phase diagram. From our results, one would be able to predict which state (${\ensuremath{\psi}}_{2}$ or ${\ensuremath{\psi}}_{3}$) may be realized in a real material given its set of ${J}_{e}$ couplings. Our results further illustrate the very large potential sensitivity of the ground state of $XY$ pyrochlore systems to minute changes in their spin Hamiltonian. Finally, using the experimentally determined ${J}_{e}\ensuremath{\equiv}$${{J}_{\ifmmode\pm\else\textpm\fi{}},{J}_{\ifmmode\pm\else\textpm\fi{}\ifmmode\pm\else\textpm\fi{}},{J}_{z\ifmmode\pm\else\textpm\fi{}},{J}_{zz}}$ and $g$-tensor values for Er${}_{2}$Ti${}_{2}$O${}_{7}$, we show that the heretofore neglected long-range $1/{r}^{3}$ magnetostatic dipole-dipole interactions do not change the conclusion that Er${}_{2}$Ti${}_{2}$O${}_{7}$ has a ${\ensuremath{\psi}}_{2}$ ground state induced via a quantum order-by-disorder mechanism. As an avenue of research in $XY$ pyrochlore materials distinct from the rare-earth pyrochlore oxides, we propose that the Cd${}_{2}$Dy${}_{2}$Se${}_{4}$ chalcogenide spinel, in which the Dy${}^{3+}$ ions form a pyrochlore lattice and may be $XY$-like, could be interesting to investigate.

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