Ground state phase diagram of generic $XY$ pyrochlore magnets with quantum fluctuations

Ground state phase diagram of generic $X Y$ pyrochlore magnets with quantum fluctuations

Anson W. C. Wong, Zhihao Hao, and Michel J. P. Gingras

Phys. Rev. B 88, 144402 – Published 3 October 2013

Abstract

Motivated by recent experimental and theoretical progress on the Er $_{2}$ Ti $_{2}$ O $_{7}$ pyrochlore $X Y$ antiferromagnet, we study the general problem of quantum order-by-disorder in pyrochlore $X Y$ systems. We consider the nearest-neighbor pseudo-spin-1/2 Hamiltonian for such a system characterized by anisotropic spin-spin couplings $J_{e} \equiv$ ${J_{\pm}, J_{\pm \pm}, J_{z \pm}, J_{z z}}$ 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 $Γ_{5}$ irreducible representation, the $ψ_{2}$ and $ψ_{3}$ (basis) states, are rather accurately determined by a cubic equation in $(J_{\pm} J_{\pm \pm}) / J_{z \pm}^{2}$ . Depending on the value of $J_{z z}$ , there can be one or three phase boundaries that separate alternating regions of $ψ_{2}$ and $ψ_{3}$ states. In particular, we find for sufficiently small $J_{z z} / J_{\pm}$ a narrow $ψ_{2}$ sliver sandwiched between two $ψ_{3}$ regions in the $J_{\pm \pm} / J_{\pm}$ vs $J_{z \pm} / J_{\pm}$ phase diagram. From our results, one would be able to predict which state ( $ψ_{2}$ or $ψ_{3}$ ) 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 $X Y$ pyrochlore systems to minute changes in their spin Hamiltonian. Finally, using the experimentally determined $J_{e} \equiv$ ${J_{\pm}, J_{\pm \pm}, J_{z \pm}, J_{z z}}$ 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 $ψ_{2}$ ground state induced via a quantum order-by-disorder mechanism. As an avenue of research in $X Y$ 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 $X Y$ -like, could be interesting to investigate.

Received 3 May 2013

DOI:https://doi.org/10.1103/PhysRevB.88.144402

Authors & Affiliations

Anson W. C. Wong^1,2, Zhihao Hao², and Michel J. P. Gingras^2,3,4

¹Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, V6T 1Z1, Canada
²Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
³Perimeter Institute for Theoretical Physics, 31 Caroline North, Waterloo, Ontario, N2L 2Y5, Canada
⁴Canadian Institute for Advanced Research, 180 Dundas Street West, Suite 1400, Toronto, Ontario, M5G 1Z8, Canada

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Vol. 88, Iss. 14 — 1 October 2013

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Images

Figure 1
Spin configurations of $ψ_{2}$ (left) and $ψ_{3}$ (right) states. See main text for description of the spin orientation for each state.Reuse & Permissions
Figure 2
$δ E (j_{\pm \pm}, 0)$ (top) is an odd function of $j_{\pm \pm}$ and $δ E (0, j_{z \pm})$ (bottom) is an even function of $j_{z \pm}$ . The insets show, in the form of log-log plots, the fit (solid blue line) of $| δ E (j_{\pm \pm}, 0) |$ to $j_{\pm \pm}^{3}$ and $δ E (0, j_{z \pm})$ to $j_{z \pm}^{6}$ for $j_{\pm \pm} > 0$ and $j_{z \pm} > 0$ , respectively. The fits (blue solid lines) were performed over the range $[0.01, 0.1]$ for both $j_{\pm \pm}$ and $j_{z \pm}$ . The deviation of the data (symbols) from the fit reflects the slight departure of $δ E (ϕ)$ from the strict $\cos (6 ϕ)$ form and thus the cubic polynomial in $x$ in Eq. (4).Reuse & Permissions
Figure 3
Phase boundaries separating the $ψ_{2}$ and $ψ_{3}$ states for different values of $j_{z z} \equiv J_{z z} / J_{\pm}$ . The solid red lines are phase boundaries between the $Γ_{5}$ manifold and the Palmer-Chalker (PC, or $ψ_{4}$ state of the $Γ_{7}$ irrep[13]), or the splayed ferromagnetic (SF) state, or, within $Γ_{5}$ , between the $ψ_{2}$ and $ψ_{3}$ states. Within the $Γ_{5}$ manifold, $ψ_{2}$ is selected within the blue shaded regions. The various symbols correspond to values of $J_{e} \equiv {J_{\pm}$ , $J_{z \pm}$ , $J_{\pm \pm}$ and $J_{z z}}$ for pyrochlore systems previously studied.[12, 17, 18, 21] The $X Y$ antiferromagnet model of Ref. 16 is at $j_{z z} = j_{z \pm} = 0$ and $j_{\pm \pm} = 2$ , that is, at the $ψ_{2}$ (PC) boundary where quantum fluctuations select $ψ_{2}$ .[16, 17] The red circle in $ψ_{2}$ ( $J_{z z} / J_{\pm} = - 0.5$ panel) is for Er $_{2}$ Ti $_{2}$ O $_{7}$ .[18] Despite uncertainties in its $J_{e}$ couplings, we find that Er $_{2}$ Ti $_{2}$ O $_{7}$ remains deeply in the $ψ_{2}$ region and does not cross in either $ψ_{3}$ or $ψ_{4}$ (PC). The yellow squares in $ψ_{2}$ ( $J_{z z} / J_{\pm} = 0$ panel) are for the model of Ref. 17 with anisotropic coupling $j_{a} \equiv J_{a} / J$ in their notation. With $J_{\pm \pm} / J_{\pm} = (4 - j_{a}) / (2 + j_{a})$ , their model approaches the $ψ_{2}$ (PC) boundary as $j_{a} \to 0^{+}$ . The green triangle ( $J_{z z} / J_{\pm} = - 0.5$ , $J_{z z} / J_{\pm} = 0$ , $J_{z z} / J_{\pm} = 0.5$ , $J_{z z} / J_{\pm} = 2.0$ , $J_{z z} / J_{\pm} = 3.0$ panels) on the $ψ_{3}$ /SF boundary is for the Heisenberg pyrochlore antiferromagnet with indirect Dzyaloshinskii-Moriya interactions[15] of varying strength studied in Ref. 21 which, coincidentally, resides on the $Γ_{5}$ /SF boundary at the classical level (Appendix ). The orange diamond corresponds to Yb $_{2}$ Ti $_{2}$ O $_{7}$ .[12] Because of the uncertainty in $J_{\pm}$ , $J_{z \pm}$ , $J_{\pm \pm}$ , and $J_{z z}$ , Yb $_{2}$ Ti $_{2}$ O $_{7}$ “inhabits” the $J_{z z} / J_{\pm} = 2.0$ and $J_{z z} / J_{\pm} = 3.0$ panels. The green box delineates the ranges $J_{\pm \pm} / J_{\pm} \in [0.6, 1.5]$ and $J_{z \pm} / J_{\pm} \in [- 3.8, - 2.2]$ , which extends well to the left of the vertical $J_{\pm \pm} / J_{\pm}$ axis, hence the left pointing arrows. This figure illustrates that the Hamiltonian of Yb $_{2}$ Ti $_{2}$ O $_{7}$ sits in the same SF phase recently reported for Yb $_{2}$ Sn $_{2}$ O $_{7}$ ,[24] far from the $Γ_{5}$ manifold ( $ψ_{2}$ and $ψ_{3}$ ) as well as the $ψ_{4}$ (PC) state.Reuse & Permissions

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