Magnetic Field Induced Quantum Spin Liquid in the Two Coupled Trillium Lattices of ${\mathrm{K}}_{2}{\mathrm{Ni}}_{2}({\mathrm{SO}}_{4}{)}_{3}$

Magnetic Field Induced Quantum Spin Liquid in the Two Coupled Trillium Lattices of $K_{2} {Ni}_{2} ({SO}_{4})_{3}$

Ivica Živković, Virgile Favre, Catalina Salazar Mejia, Harald O. Jeschke, Arnaud Magrez, Bhupen Dabholkar, Vincent Noculak, Rafael S. Freitas, Minki Jeong, Nagabhushan G. Hegde, Luc Testa, Peter Babkevich, Yixi Su, Pascal Manuel, Hubertus Luetkens, Christopher Baines, Peter J. Baker, Jochen Wosnitza, Oksana Zaharko, Yasir Iqbal, Johannes Reuther, and Henrik M. Rønnow

Phys. Rev. Lett. 127, 157204 – Published 6 October 2021

Abstract

Quantum spin liquids are exotic states of matter that form when strongly frustrated magnetic interactions induce a highly entangled quantum paramagnet far below the energy scale of the magnetic interactions. Three-dimensional cases are especially challenging due to the significant reduction of the influence of quantum fluctuations. Here, we report the magnetic characterization of $K_{2} {Ni}_{2} ({SO}_{4})_{3}$ forming a three-dimensional network of ${Ni}^{2 +}$ spins. Using density functional theory calculations, we show that this network consists of two interconnected spin-1 trillium lattices. In the absence of a magnetic field, magnetization, specific heat, neutron scattering, and muon spin relaxation experiments demonstrate a highly correlated and dynamic state, coexisting with a peculiar, very small static component exhibiting a strongly renormalized moment. A magnetic field $B ≳ 4 T$ diminishes the ordered component and drives the system into a pure quantum spin liquid state. This shows that a system of interconnected $S = 1$ trillium lattices exhibits a significantly elevated level of geometrical frustration.

Received 3 May 2021
Revised 4 August 2021
Accepted 8 September 2021

DOI:https://doi.org/10.1103/PhysRevLett.127.157204

Physics Subject Headings (PhySH)

Magnetism Quantum spin liquid

Magnetization measurements Muon spin resonance Neutron diffraction Specific heat measurements Time-of-flight neutron spectroscopy

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Ivica Živković ^1,*, Virgile Favre ¹, Catalina Salazar Mejia ², Harald O. Jeschke ³, Arnaud Magrez⁴, Bhupen Dabholkar ⁵, Vincent Noculak^6,7, Rafael S. Freitas ⁸, Minki Jeong ⁹, Nagabhushan G. Hegde¹, Luc Testa ¹, Peter Babkevich¹, Yixi Su ¹⁰, Pascal Manuel¹¹, Hubertus Luetkens¹², Christopher Baines¹², Peter J. Baker ¹¹, Jochen Wosnitza^2,13, Oksana Zaharko¹⁴, Yasir Iqbal ⁵, Johannes Reuther^6,7, and Henrik M. Rønnow¹

¹Laboratory for Quantum Magnetism, Institute of Physics, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
²Hochfeld-Magnetlabor Dresden (HLD-EMFL) and Würzburg-Dresden Cluster of Excellence ct.qmat, Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany
³Research Institute for Interdisciplinary Science, Okayama University, Okayama 700-8530, Japan
⁴Crystal Growth Facility, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
⁵Department of Physics and Quantum Centers in Diamond and Emerging Materials (QuCenDiEM) Group, Indian Institute of Technology Madras, Chennai 600036, India
⁶Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany
⁷Helmholtz-Zentrum für Materialien und Energie, Hahn-Meitner-Platz 1, 14109 Berlin, Germany
⁸7 Instituto de Física, Universidade de São Paulo, 05508-090 São Paulo, Brazil
⁹School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom
¹⁰Jülich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), Forschungszentrum Jülich, Lichtenbergstrasse 1, D-85747 Garching, Germany
¹¹ISIS Pulsed Neutron and Muon Source, STFC Rutherford Appleton Laboratory, Harwell Science and Innovation Campus, Didcot, Oxfordshire OX11 0QX, United Kingdom
¹²Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen, Switzerland
¹³Institut für Festkörper- und Materialphysik, TU Dresden, 01062 Dresden, Germany
¹⁴Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institut, CH-5253 Villigen, Switzerland

^*Corresponding author. ivica.zivkovic@epfl.ch

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Vol. 127, Iss. 15 — 8 October 2021

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Figure 1
(a) Unit cell of $K_{2} {Ni}_{2} ({SO}_{4})_{3}$ . A $Ni − O − S − O − Ni$ super-superexchange path contributing to a trillium coupling is marked by dashed lines. (b) Exchange network between nickel sites. A ten-site loop formed by the strongest exchange $J_{4}$ is marked by arrows. (c) Exchange couplings determined by DFT energy mapping. The vertical line indicates the $U$ value where the calculated Curie-Weiss temperature matches the experimental value.
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Figure 2
(a) $T$ dependence of $χ_{dc}$ along the three orthogonal directions (left axis) for $B = 0.1 T$ , and the inverse susceptibility $1 / χ_{dc}$ (right axis) with $B | | [111]$ . The solid gray line represents the Curie-Weiss law. (b) Magnetic field dependence of magnetization, together with the Brillouin function for $S = 1$ and $g = 2.34$ (dashed lines) and Monte Carlo simulations (solid lines). (c) Zero field temperature dependence of the total specific heat of $K_{2} {Ni}_{2} ({SO}_{4})_{3}$ , together with the total specific heat of $K_{2} {Mg}_{2} ({SO}_{4})_{3}$ representing the phonon contribution. (d) $T$ dependence of magnetic entropy. An inset shows the magnetic field dependence of the exponent $n$ in the power law $C_{p} = A T^{n}$ . (e) $T$ dependence of the magnetic specific heat for several magnetic field values. (f) $T - B$ phase diagram of $K_{2} {Ni}_{2} ({SO}_{4})_{3}$ .
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Figure 3
(a) Pure magnetic scattering pattern derived from the spin-polarized neutron diffraction data at 0.5 K (blue diamonds) and from pseudofermion functional renormalization group simulation (solid line). (b) Neutron powder diffraction at 0.1 K and 10 K. (c) Combined data from polarized neutrons [blue diamonds from panel (a)] and powder diffraction [blue (10 K) and red points (0.1 K) from panel (b)]. Blue shading indicates an upper limit of the contribution from the ordered component, yellow shading represents the contribution from the fluctuating component, (d) Time-of-flight data at 0.5 K. (e) $E$ dependence of the integrated intensity for two $Q$ ranges.
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Figure 4
(a) ZF- $μ SR$ spectra at selected temperatures. The solid lines represent fitted curves following the model in Eq. (2). (b) A comparison of the applicability of two models at 0.02 K (shifted down for clarity) and 3 K. (c) Temperature dependence of the $μ SR$ rates extracted from two models. Circles and triangles represent relaxation rates from the model in Eq. (2), diamonds from the model in Eq. (1). The inset displays the temperature dependence of the exponent $β_{2}$ from Eq. (2) and $β$ from Eq. (1).
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Magnetic Field Induced Quantum Spin Liquid in the Two Coupled Trillium Lattices of K2Ni2(SO4)3

Phys. Rev. Lett. 127, 157204 – Published 6 October 2021

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Magnetic Field Induced Quantum Spin Liquid in the Two Coupled Trillium Lattices of $K_{2} {Ni}_{2} ({SO}_{4})_{3}$