Codimension-Two Spiral Spin-Liquid in the Effective Honeycomb-Lattice Compound Cs$_3$Fe$_2$Cl$_9$

TL;DR

The work demonstrates a codimension-two spiral spin-liquid on the AB-stacked honeycomb-like Cs$_3$Fe$_2$Cl$_9$ lattice, stabilized by strong intra- and interlayer exchanges and a uniaxial anisotropy $D_z$. By combining elastic and inelastic neutron scattering with classical Monte Carlo modeling of a $J_{1-5}$-$D_z$ Hamiltonian, the authors map an eight-phase magnetic diagram under field, identifying spiral-type and SDW orders and revealing a possible order-by-disorder transition. This study shows how SSLs can arise from intra-sublattice couplings in AB-stacked lattices, providing a new route to spin-liquid physics beyond the conventional honeycomb paradigm and highlighting thermal/quantum ObD possibilities and potential exotic spin textures. The results offer a framework for exploring codimension-two SSLs in related materials and motivate future experiments to probe quantum effects and field-tuned phase transitions in these systems.

Abstract

A codimension-two spiral spin-liquid is a correlated paramagnetic state with one-dimensional ground state degeneracy hosted within a three-dimensional lattice. Here, via neutron scattering experiments and numerical simulations, we establish the existence of a codimension-two spiral spin-liquid in the effective honeycomb-lattice compound Cs$_3$Fe$_2$Cl$_9$, which demonstrates a novel path to spiral spin-liquids by overcoming the long-standing impediment of weak further-neighbor interactions. In the long-range ordered regime, competing spiral and spin density wave orders emerge as a function of applied magnetic field, among which a possible order-by-disorder transition is identified.

Figures (19)

• Figure 1: (a) AB-stacked triangular bilayers formed by the Fe$^{3+}$ ions in Cs$_3$Fe$_2$Cl$_9$. Atoms belonging to the neighboring bilayers are shown in red and blue, respectively. The $J_1$, $J_2$, and $J_3$ bonds are shown by thick black lines, thin black lines, and thin gray lines, respectively. The $J_4$ and $J_5$ bonds are indicated by red curved lines. Arrows indicate the spin directions of the collinear ground state with $\bm{q} = (\frac{1}{2}, 0, 0)$. (b) The AB-stacked triangular bilayers viewed along the $c$ axis. (c) Refinement result of the powder neutron diffraction data measured on POWGEN at $T$ = 1.6 K. Data points are shown as red circles. The calculated pattern is shown as the black solid line. The vertical bars indicate the positions of the structural (upper) and magnetic (lower) Bragg peaks for Cs$_3$Fe$_2$Cl$_9$. The blue line at the bottom shows the difference of measured and calculated intensities. The goodness-of-fit parameters are $R_\textrm{p}=18.6\%$ and $R_\textrm{wp}=10.2\%$.
• Figure 2: (a,c,e,g) Inelastic neutron scattering spectra $S(q,\omega)$ for Cs$_3$Fe$_2$Cl$_9$ powders measured on CNCS with $E_i=3.32$ meV at $T=2$ (a) and 10 K (c), and with $E_i=1.0$ meV at $T=2$ (e) and 10 K (g). (b,f) Simulated spectra for the fitted $J_{1\textrm{-}5}$-$D_z$ model using linear spin wave theory (LSWT). The spectra in (b) and (f) are convolved with the instrumental energy resolution for $E_i = 3.32$ and 1.0 meV, respectively. (d,h) Simulated spectra for the fitted $J_{1\textrm{-}5}$-$D_z$ model using the Landau-Langevin-Gilbert (LLG) method calculated at $T = 10$ K.
• Figure 3: (a) Left half shows the diffuse neutron scattering pattern in the ($h$, $k$, 0) plane for Cs$_3$Fe$_2$Cl$_9$ measured at $T = 6$ K on CORELLI. Data are symmetrized according to the $6/mmm$ Laue class. The right panel is the calculated diffuse neutron scattering pattern for the fitted $J_{1\textrm{-}5}$-$D_z$ model using the SCGA method assuming a reduced $T = 5$ K to compensate for the underestimated critical correlations. Both the experimental and calculated data were integrated over $l=[-0.1,0.1]$ reciprocal lattice units, r.l.u. The solid-line hexagon indicates the boundary of the first Brillouin zone. Triangular-shaped lobes around the $K$-$(\frac{1}{3}, \frac{1}{3}, 0)$ points are the spiral surface for a honeycomb-lattice model with a frustration ratio of $J_2^\textrm{h}/J_1^\textrm{h} = 0.72$. (b) Similar experimental (left half) and calculated (right half) diffuse scattering patterns in the ($h$, $k$, 1) plane with an integration range of $l=[0.9,1.1]$ r.l.u. (c) Experimental (left half) and calculated (right half) diffuse scattering patterns with an integration range of $l=[-0.1,0.1]+[0.9,1.1]$ r.l.u. (d) Experimental (left half) and calculated (right half) diffuse scattering patterns in the $(h,h,l)$ plane with an integration width of 0.1 r.l.u. along $(h,-h,0)$. (e) Scattering intensity along $(h,h,0)$ integrated from the $(h,k,0)+(h,k,1)$ pattern in panel (c) with an integration width of 0.06 r.l.u. along $(h,-h,0)$ as outlined by the dashed-line rectangle in the left half of panel (c). (f) $l$-dependence of the scattering intensity integrated in the area of $[0.125, 0.5]$ and $[-0.125, 0.125]$ r.l.u. along the $(h,h,0)$ and $(h,-h,0)$ directions, respectively. This area is outlined in the right half of panel (c) by a dashed-line rectangle. In panels (e) and (f), the red solid line shows the calculated results for the fitted $J_{1\textrm{-}5}$-$D_z$ model using the SCGA method.
• Figure 4: (a) $H$-$T$ phase diagram for Cs$_3$Fe$_2$Cl$_9$ reproduced from Ref. ishii_field_2021 with magnetic field applied along the $c$ axis. Red squares are phase transitions revealed in heat capacity measurements ishii_field_2021. Blue points correspond to data described in the current work. Empty blue circles are the experimental conditions for measurements in phases III and V supp. (b) $H$-$T$ phase diagram for the $J_{1\textrm{-}5}$-$D_z$ model obtained from classical Monte Carlo simulations. Pseudocolor corresponds to the calculated heat capacity $C_p$. Cross (Dot) marks are phase boundaries determined from $C_p(T)$ [$M(T)$]. (c) Diffraction intensity along the $(h+1,h-1,0)$ line as indicated by the dashed-line rectangle in panel (g) in phases III-VI. The integration width perpendicular to the line is 0.05 r.l.u. (d) Diffraction intensity along the $(-2h,4h,0)$ line as indicated by the dashed-line rectangle in panel (h) in phases III-VI. The integration width perpendicular to the line is 0.05 r.l.u. (e-h) Diffraction pattern in the ($h$, $k$, 0) plane collected on WAND$^2$ in (e) phase I with $T = 1.5$ K and $H = 0$ T, (f) phase II with $T = 1.5$ K and $H = 3.2$ T, (g) phase IV with $T = 1.5$ K and $H = 5.5$ T, and (h) phase VI with $T = 5.05$ K and $H = 5.5$ T. In each panel, reflections belonging to the characteristic propagation vectors are indicated by black arrows. In panel (g), the 2$\bm{q}$ reflection is indicated by the white circle. The shape of spiral surface is shown in dashed line. (i-l) Magnetic structures for phase (i) I, (j) III and IV, (k) V, and (l) VI. The $xy$ components of the ordered spins are indicated by black arrows. The $z$ components are encoded by colors. In each panel, the bottom left (top right) part depicts the spin configuration for two sublattices (one sublattice).
• Figure S1: Refinement result of the powder neutron diffraction data measured on POWGEN at $T$ = 15 K. Data points are shown as red circles. The calculated pattern is shown as the black solid line. The vertical bars indicate the positions of the structural Bragg peaks for Cs$_3$Fe$_2$Cl$_9$. The blue line at the bottom shows the difference between measured and calculated intensities.