Quadrupole Order in the Frustrated Pyrochlore ${\mathrm{Tb}}_{2+x}{\mathrm{Ti}}_{2\ensuremath{-}x}{\mathrm{O}}_{7+y}$

Quadrupole Order in the Frustrated Pyrochlore ${Tb}_{2 + x} {Ti}_{2 - x} O_{7 + y}$

H. Takatsu, S. Onoda, S. Kittaka, A. Kasahara, Y. Kono, T. Sakakibara, Y. Kato, B. Fåk, J. Ollivier, J. W. Lynn, T. Taniguchi, M. Wakita, and H. Kadowaki

Phys. Rev. Lett. 116, 217201 – Published 26 May 2016; Erratum Phys. Rev. Lett. 127, 069902 (2021)

Abstract

A hidden order that emerges in the frustrated pyrochlore ${Tb}_{2 + x} {Ti}_{2 - x} O_{7 + y}$ with $T_{c} = 0.53 K$ is studied using specific heat, magnetization, and neutron scattering experiments on a high-quality single crystal. Semiquantitative analyses based on a pseudospin- $1 / 2$ Hamiltonian for ionic non-Kramers magnetic doublets demonstrate that it is an ordered state of electric quadrupole moments. The elusive spin liquid state of the nominal ${Tb}_{2} {Ti}_{2} O_{7}$ is most likely a $U (1)$ quantum spin-liquid state.

Received 15 June 2015

DOI:https://doi.org/10.1103/PhysRevLett.116.217201

Physics Subject Headings (PhySH)

Frustrated magnetism Quantum spin liquid

Pyrochlores

Condensed Matter, Materials & Applied Physics

Erratum

Erratum: Quadrupole Order in the Frustrated Pyrochlore ${Tb}_{2 + x} {Ti}_{2 - x} O_{7 + y}$ [Phys. Rev. Lett. 116, 217201 (2016)]

H. Takatsu, S. Onoda, S. Kittaka, A. Kasahara, Y. Kono, T. Sakakibara, Y. Kato, B. Fåk, J. Ollivier, J. W. Lynn, T. Taniguchi, M. Wakita, and H. Kadowaki

Phys. Rev. Lett. 127, 069902 (2021)

Authors & Affiliations

H. Takatsu^1,2, S. Onoda^3,4, S. Kittaka⁵, A. Kasahara⁵, Y. Kono⁵, T. Sakakibara⁵, Y. Kato^3,6, B. Fåk⁷, J. Ollivier⁷, J. W. Lynn⁸, T. Taniguchi¹, M. Wakita¹, and H. Kadowaki¹

¹Department of Physics, Tokyo Metropolitan University, Hachioji-shi, Tokyo 192-0397, Japan
²Department of Energy and Hydrocarbon Chemistry, Graduate School of Engineering, Kyoto University, Kyoto 615-8510, Japan
³RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
⁴Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan
⁵Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan
⁶Department of Applied Physics, University of Tokyo, Bunkyo, Tokyo 113-8656, Japan
⁷Institute Laue Langevin, BP 156, F-38042 Grenoble, France
⁸NCNR, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-6102, USA

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Vol. 116, Iss. 21 — 27 May 2016

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Figure 1
(a) Phase diagram of the effective Hamiltonian Eq. (1) determined from CMC simulations. The shading (color) represents $T_{c}$ . Two quadrupole LRO phases, the planar antiferropseudospin (PAF) and planar ferropseudospin (PF) phases, exist in the vicinity of the SI phase [11]. Classical SI is replaced by a $U (1)$ QSL in quantum theory [9]. The region enclosed by the dotted line represents an acceptable parameter region for the experimental data on ${Tb}_{2.005} {Ti}_{1.995} O_{7 + y}$ . The cross mark indicates the typical values $(δ, q) = (0, 0.85)$ . (b) Schematic view of the deformation of the $f$ -electron charge density due to the PAF order on the pyrochlore lattice.
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Figure 2
(a) Temperature dependence of magnetic susceptibility for an applied field along [111]. The solid line is $χ_{corr}$ (see text for details). (b) Comparison between $[χ_{\exp} - χ_{corr}]^{- 1} = χ_{gnd}^{- 1}$ and QMC calculation. Note, $1 emu = 10^{- 3} A m^{2}$ .
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Figure 3
(a) Neutron inelastic magnetic spectra of the polycrystalline ${Tb}_{2.005} {Ti}_{1.995} O_{7 + y}$ sample taken at $T = 0.1 K$ . (b) Calculated $S (Q, E)$ for the PAF phase with $J_{n n, eff} > 0$ using $J_{n n, eff} = 1.48 K$ ( $J_{n n} = 1.00 K$ ) and $(δ, q) = (0, 0.85)$ . $J_{n n, eff}$ is determined by the analysis of $χ_{gnd}$ . (c) Calculated $S (Q, E)$ for the PAF phase with $J_{n n, eff} < 0$ using $J_{n n, eff} = - 1.77 K$ ( $J_{n n} = - 2.25 K$ ) and $(δ, q) = (- 0.5, - 1.0)$ . $J_{n n, eff}$ is determined by the analysis of $χ_{gnd}$ . (d) $T$ dependence of intensities of the single-crystal neutron Bragg scattering at (002) and ( $\frac{1}{2} \frac{1}{2} \frac{3}{2}$ ).
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Figure 4
(a) Temperature dependence of the observed specific heat $C_{P} (T, H)$ for $H ∥ [111]$ . (b) Temperature-field map of $C_{P} (T, H)$ for $H ∥ [111]$ . Filled circles in the map are peak positions of $C_{P} (T, H)$ . (c),(d) CMC results of specific heat $C (T, H)$ for (c) $(δ, q) = (0, 0.85)$ with $J_{n n} = 1.00 K$ and for (d) $(δ, q) = (- 0.54, 0.5)$ with $J_{n n} = 0.92 K$ . The values of $J_{n n}$ have been determined from the comparison of $χ_{gnd}$ with the QMC results for each case. Solid, dashed, and dotted lines in (b), (c), and (d) are guides to the eyes. Labels in maps (b) and (c) indicate assigned states from the analysis with Eq. (1) and CMC simulations [40]; i.e., (I) a paramagnetic paraquadrupole state, (II) the 3D PAF state, and (III) the 2D PAF state.
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Figure 5
Field dependence of the magnetization $M$ under the [111] magnetic field. (a) Experimental data on ${Tb}_{2.005} {Ti}_{1.995} O_{7 + y}$ above and below $T_{c} = 0.53 K$ . (b) CMC results obtained with the same parameters as for Figs. 3 and 4.
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Physical Review Letters

Quadrupole Order in the Frustrated Pyrochlore Tb2+xTi2−xO7+y

H. Takatsu, S. Onoda, S. Kittaka, A. Kasahara, Y. Kono, T. Sakakibara, Y. Kato, B. Fåk, J. Ollivier, J. W. Lynn, T. Taniguchi, M. Wakita, and H. Kadowaki

Phys. Rev. Lett. 116, 217201 – Published 26 May 2016; Erratum Phys. Rev. Lett. 127, 069902 (2021)

Abstract

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Erratum

Erratum: Quadrupole Order in the Frustrated Pyrochlore Tb2+xTi2−xO7+y [Phys. Rev. Lett. 116, 217201 (2016)]

H. Takatsu, S. Onoda, S. Kittaka, A. Kasahara, Y. Kono, T. Sakakibara, Y. Kato, B. Fåk, J. Ollivier, J. W. Lynn, T. Taniguchi, M. Wakita, and H. Kadowaki

Phys. Rev. Lett. 127, 069902 (2021)

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Quadrupole Order in the Frustrated Pyrochlore ${Tb}_{2 + x} {Ti}_{2 - x} O_{7 + y}$

Erratum: Quadrupole Order in the Frustrated Pyrochlore ${Tb}_{2 + x} {Ti}_{2 - x} O_{7 + y}$ [Phys. Rev. Lett. 116, 217201 (2016)]