Giant Magnetoelastic-Coupling Driven Spin-Lattice Liquid State in Molybdate Pyrochlores

Andrew Smerald and George Jackeli

Phys. Rev. Lett. 122, 227202 – Published 7 June 2019

Abstract

We propose the idea of a spin-lattice liquid, in which spin and lattice degrees of freedom are strongly coupled and remain disordered and fluctuating down to low temperatures. We show that such a state arises naturally from a microscopic analysis of a class of molybdate pyrochlore compounds, and is driven by a giant magnetoelastic effect. Finally, we argue that this could explain some of the experimental features of $Y_{2} {Mo}_{2} O_{7}$ .

Received 20 December 2018
Revised 29 March 2019

DOI:https://doi.org/10.1103/PhysRevLett.122.227202

Physics Subject Headings (PhySH)

Exchange interaction Glass transition Magnetism Magnetoelastic effect Peierls transition Spin liquid Spin-orbit coupling

Antiferromagnets

Hubbard model Ising model Monte Carlo methods Spin lattice models

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Andrew Smerald^1,* and George Jackeli^1,2,†

¹Max Planck Institut für Festkörperforschung, Heisenbergstraße 1, D-70569 Stuttgart, Germany
²Insitute for Functional Matter and Quantum Technologies, University of Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany

^*andrew.smerald@gmail.com
^†Also at Andronikashvili Institute of Physics, 0177 Tbilisi, Georgia.

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Vol. 122, Iss. 22 — 7 June 2019

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Images

Figure 1
$R_{2} {Mo}_{2} O_{7}$ pyrochlores. (a) Average $F d \bar{3} m$ structure of $R_{2} {Mo}_{2} O_{7}$ , showing the Mo pyrochlore lattice (purple) and surrounding O octahedra (gray) within a unit cell (black cuboid). (b) $J_{eff}^{z^{'}} = \pm 2$ states are represented by arrows pointing in or out of the Mo tetrahedron along the $z^{'}$ local axes. (c) 2-in-2-out lattice displacements (black arrows) create one long (blue), four medium (red), and one short (green) $Mo ─ Mo$ bond on each tetrahedron. (d) $π$ -type $d p d$ hopping path, shown for idealized regular oxygen octahedra. The path shown is $d_{1}^{y z} - (p_{1}^{y} : p_{2}^{x}) - d_{2}^{y z}$ , where Mo sites are labeled 1 and 2 and have associated local coordinates ( $x$ , red; $y$ , green; $z$ , blue). (e) $σ$ -type hopping path allowed by trigonal distortion of the oxygen octahedra and exemplified by $d_{1}^{y z} - (p_{1}^{z} : p_{2}^{z}) - d_{2}^{y z}$ .
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Figure 2
Dependence of the Ising exchange interaction $J_{is}$ on the $Mo ─ O ─ Mo$ bond angle $α$ . (a) $J_{is}$ is plotted for $J_{H} / U = 0.15$ and measured in units of $t^{2} / U$ , where $t = V_{pd π}^{2} / Δ_{dp}$ is the hopping integral along a $Mo ─ O ─ Mo$ $π$ -type bond [34, 47]. For $σ$ bonds we use $V_{pd σ} = - 2.2 V_{pd π}$ [43]. The experimentally determined bond angles for $Y_{2} {Mo}_{2} O_{7}$ are marked [28] and $J_{is}$ passes through zero close to the average bond angle, $α_{av} \approx 127 °$ . (Inset) Dependence of $J_{is}$ on $J_{H} / U$ for $α = 116 °$ (green), $α = 127 °$ (red), and $α = 139 °$ (blue). (b) Experimentally determined bond angles for $Y_{2} {Mo}_{2} O_{7}$ [28].
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Figure 3
Phase diagram of $H_{sl}$ [Eq. (1)] from Monte Carlo simulation. For $g^{2} / (4 K) < J_{is}$ there is no lattice distortion and a crossover (dashed line) to spin-ice behavior occurs at low $T$ . For $g^{2} / (4 K) > J_{is}$ an incoherent lattice distortion occurs ( $δ α > 0$ ) and a spin-lattice liquid (SLL) forms below a first-order transition (red). At lower $T$ a likely second-order transition (blue) picks out loops of length $l_{loop} = 4 m + 4$ , $m = 1, 2, \dots$ [34]. (Inset) Four degenerate 3-1 ground states of a tetrahedron in the SLL state with a fixed lattice distortion (short bond in green and long bond in blue), and with spins pointing in (orange) and out (blue).
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Figure 4
Physical characteristics of $H_{sl}$ [Eq. (1)] from Monte Carlo simulation for $g^{2} / (4 K) = 2 J_{is}$ . Error bars are smaller than the point sizes. (a) Lattice distortion $δ α$ and heat capacity $C / T$ showing two transitions. Results are shown for $L = 3$ (total number of spins $N = 16 L^{3}$ ), for which equilibration is possible across the $lower − T$ transition, and are consistent with simulations with larger $L$ [34]. (b) Fraction of tetrahedra displaying 2-in-2-out (red), 3-1 (blue), or 4-0 (green) spin configurations. (c) Loop-based order parameters showing spin correlations on loops $O_{loop}$ and mismatch between loops of length $l_{loop} = 4 m + 2$ and $l_{loop} = 4 m + 4$ , $O_{mis}$ [34]. (d) Inverse magnetic susceptibility. The fit to $1 / χ \propto T - θ_{cw}$ gives $θ_{cw} = - 0.1 J_{is}$ .
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