Spiral Spin Liquid on a Honeycomb Lattice

Shang Gao, Michael A. McGuire, Yaohua Liu, Douglas L. Abernathy, Clarina dela Cruz, Matthias Frontzek, Matthew B. Stone, and Andrew D. Christianson

Phys. Rev. Lett. 128, 227201 – Published 1 June 2022

Abstract

Spiral spin liquids are correlated paramagnetic states with degenerate propagation vectors forming a continuous ring or surface in reciprocal space. On the honeycomb lattice, spiral spin liquids present a novel route to realize emergent fracton excitations, quantum spin liquids, and topological spin textures, yet experimental realizations remain elusive. Here, using neutron scattering, we show that a spiral spin liquid is realized in the van der Waals honeycomb magnet ${FeCl}_{3}$ . A continuous ring of scattering is directly observed, which indicates the emergence of an approximate U(1) symmetry in momentum space. Our work demonstrates that spiral spin liquids can be achieved in two-dimensional systems and provides a promising platform to study the fracton physics in spiral spin liquids.

Received 23 November 2021
Revised 18 January 2022
Accepted 25 March 2022

DOI:https://doi.org/10.1103/PhysRevLett.128.227201

Physics Subject Headings (PhySH)

Frustrated magnetism Spin liquid

Honeycomb lattice

Inelastic neutron scattering Quasi elastic neutron scattering

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Shang Gao^1,2,*, Michael A. McGuire ², Yaohua Liu¹, Douglas L. Abernathy ¹, Clarina dela Cruz ¹, Matthias Frontzek¹, Matthew B. Stone ¹, and Andrew D. Christianson²

¹Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
²Materials Science & Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

^*sgao.physics@gmail.com

Low-energy structure of spiral spin liquids

Han Yan (闫寒) and Johannes Reuther

Phys. Rev. Research 4, 023175 (2022)

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Issue

Vol. 128, Iss. 22 — 3 June 2022

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Figure 1
(a) The ${Fe}^{3 +}$ ions ( $S = 5 / 2$ ) in ${FeCl}_{3}$ form honeycomb lattices with $A B C$ -type stacking along the $c$ axis. Red solid arrows indicate the nearest-, second-, and third-neighbor couplings $J_{1}$ , $J_{2}$ , and $J_{3}$ , respectively. Yellow dot-dashed arrow indicates the interlayer couplings $J_{c 1}$ . Blue dashed arrows indicate the second-layer couplings $J_{s 1}$ , $J_{s 2}$ , and $J_{s 3}$ . (b) A SSL state is realized on the honeycomb lattice at $| J_{2} / J_{1} | > 1 / 6$ (blue shaded in the bottom panel) with propagation vectors forming a continuous ring in reciprocal space, which we refer to as the spiral ring. The black curve in the top panel shows the position of a representative propagation vector ( $q$ , $q$ , 0) over the spiral ring as a function of $| J_{2} / J_{1} |$ . Inset shows the complete spiral rings at $| J_{2} / J_{1} | = 0.25$ , 0.5, and 0.7 as indicated by circular markers over the black curve. The red filled marker indicates the location of ${FeCl}_{3}$ with an effective ratio of $| J_{2} / J_{1} | \sim 0.25$ as determined from the $J_{1} − J_{2} − J_{c}$ minimal model (see text).
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Figure 2
(a)–(c) Temperature evolution of the quasi-elastic spin correlations in the $l = - 1.5$ plane measured on CORELLI at $T = 5$ , 10, and 20 K. The white hexagon outlines the first nuclear Brillouin zone. Data are integrated in the range of $l = [- 1.6, - 1.4]$ . Measurements at 50 K have been subtracted as the background. In (b) and (c), the double rings spanning the whole panel are background scattering from the sample environment, which is less evident in (a) due to the different intensity scale. (d) Calculated diffuse neutron scattering pattern in the $l = - 1.5$ plane using the minimal $J_{1} − J_{2} − J_{c 1}$ model at $T = 10 K$ . The coupling strengths are $J_{1} = - 0.3$ , $J_{2} = 0.075$ , and $J_{c 1} = 0.15 meV$ . Variations in coupling strengths do not qualitatively affect the diffuse pattern as long as $J_{2} / J_{1} = - 0.25$ with $J_{1}$ and $J_{c 1}$ being ferromagnetic and antiferromagnetic, respectively.
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Figure 3
(a)–(f) Diffuse neutron scattering intensity measured at 10 K together with the simulated results. Data in the $l = - 1.5$ , $- 1.7$ , and $- 1.3$ planes are shown in the left half of panels (a)–(c), respectively. Data in the $k^{'} = 0$ , $k^{'} = - 0.15$ , and $h = 0.25$ planes are shown in the upper half of panels (d)–(f), respectively. In panels (a)–(c), the white hexagon outlines the first nuclear Brillouin zone. Dashed arrows in panel (a) indicate the directions of the vertical slices in panels (d)–(f). (g),(h) Exchange paths for the interlayer interactions (g) $J_{c 2}$ (red solid line) and (h) $J_{c 3}$ (red solid line). In panel (h), the $J_{c 3}^{'}$ interactions (green solid line) are symmetrically inequivalent with the $J_{c 3}$ interactions in spite of their equal distances. (i) Comparison of the experimental and calculated diffuse scattering intensities using the $J_{123} − J_{c 123}$ model. The fitted coupling strengths are $J_{1} = - 0.249 (4)$ , $J_{2} = 0.089 (1)$ , $J_{3} = 0.026 (1)$ , $J_{c 1} = 0.019 (9)$ , $J_{c 2} = 0.042 (2)$ , $J_{c 3} = 0.030 (2) meV$ . Uncertainties are estimated from 50 independent runs. The goodness-of-fit factor is $χ^{2} = 1.77$ . (j) Temperature dependence of the reduced magnetic susceptibility $χ T$ (black circle) together with the calculated values based upon the SCGA method described in the text (red line). Data are measured on a powder sample in a 1 T magnetic field [34]. Error bars representing the standard deviations are smaller than the symbols.
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Figure 4
(a) Experimental INS spectra $S (Q, ω)$ measured on SEQUOIA with an incident neutron energy of $E_{i} = 8 meV$ at $T = 4 K$ along the high symmetry directions. Data are symmetrized according to the $R \bar{3}$ symmetry. The positions of the high symmetry points in the $l = 2$ and $l = - 1.5$ planes are shown in the inset, with the first nuclear Brillouin zone being outlined by the black hexagon. Intensity is plotted in a log scale. (b) Calculated INS spectra using the linear spin wave theory for a $J_{123} − J_{c 123} − J_{s 123}$ model with fitted coupling strengths of $J_{1} = - 0.28 (3)$ , $J_{2} = 0.095 (9)$ , $J_{3} = 0.008 (5)$ , $J_{c 1} = 0.05 (3)$ , $J_{c 2} = 0.024 (5)$ , $J_{c 3} = 0.026 (6)$ , $J_{s 1} = 0.01 (1)$ , $J_{s 2} = - 0.007 (3)$ , and $J_{s 3} = - 0.003 (4) meV$ . Uncertainties are estimated from 20 independent runs. A weak easy plane single-ion anisotropy of $1 μ eV$ is included to stabilize the helical ground state. The calculated spectra are convoluted by a Gaussian function with a fitted full-width at half-maximum (FWHM) of 0.35 meV. Intensity is plotted in a log scale as that for the experimental data. (c–f) Comparison between the experimental and calculated constant-energy slices at $E = 0.5$ (c), 0.7 (d), 0.9 (e), and 1.5 (f) meV. Data are integrated in an energy range of $\pm 0.1 meV$ . Intensity in panel (f) is multiplied by 8 times for better visibility. The same linear intensity scale as shown in panel (c) is utilized for panels (c-f).
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