$\mathbb{Z}_2$ Vortex Crystal Candidate in the Triangular $S=1/2$ Quantum Antiferromagnet

TL;DR

This work investigates a spin-orbit coupled triangular-lattice antiferromagnet, (CD$_3$ND$_3$)$_2$NaRuCl$_6$, as a candidate host for a $\mathbb{Z}_2$ vortex crystal. By combining thermodynamic measurements, magnetoelastic probes, and neutron scattering, the authors establish a $j_{\rm eff}=1/2$ ground state, a two-step zero-field transition into a complex multi-$\mathbf{q}$ order, and a rich $H$-$T$ phase diagram with several incommensurate phases. In the high-field regime, inelastic neutron scattering confirms a largely Heisenberg NN triangular-lattice Hamiltonian with $J \approx 1.32$ K, while small bond-dependent anisotropies (potential Kitaev-like terms) are allowed and may stabilize the observed IC states. The results position this material as a prime platform to study the interplay of geometric frustration and spin-orbit coupling and to realize the elusive $\mathbb{Z}_2$ vortex crystal phase.

Abstract

The prospect of merging the paradigms of geometric frustration on a triangular lattice and bond anisotropies in the strong spin-orbit coupling limit holds tremendous promise in the ongoing hunt for exotic quantum materials. Here we identify a new candidate system to realize such physics, the organic quantum antiferromagnet (CD$_3$ND$_3$)$_2$NaRuCl$_6$. We report a combination of thermodynamic, magneto-elastic and neutron scattering experiments on single-crystals to determine the phase diagram in axial magnetic fields $\mathbf{H \parallel c}$ and propose a minimal model Hamiltonian. (CD$_3$ND$_3$)$_2$NaRuCl$_6$ displays an ideal triangular arrangement of Ru$^{3+}$ ions adopting the spin-orbital entangled $j_{\rm eff} = 1/2$ state. It hosts residual magnetic order below $T_{\rm N} = 0.23$ K and a highly unusual $H-T$ phase diagram including three different incommensurate states. Spin-waves in the high-field polarized regime are well described by a Heisenberg-like triangular lattice Hamiltonian with a potential sub-leading bond dependent anisotropy term. We discuss possible candidate magnetic structures in the various observed phases and propose two mechanisms that could explain the field-dependent incommensurability, requiring either a small ferromagnetic Kitaev term or a tiny magneto-elastic $J-J'$ isosceles distortion driven by pseudospin-lattice coupling. We argue that the multi-$\mathbf{q}$ ground state in zero magnetic field is a prime candidate for hosting the $\mathbb{Z}_2$ vortex crystal proposed on the triangular Heisenberg-Kitaev model. (CD$_3$ND$_3$)$_2$NaRuCl$_6$ is the first member in an extended family of quantum triangular lattice magnets, providing a new playground to study the interplay of geometric frustration and spin-orbit effects.

Figures (11)

• Figure 1: Crystal structure of (CD$_3$ND$_3$)$_2$NaRuCl$_6$. (a) Schematic view of the chemical structure, with AA-stacked triangular planes of Ru$^{3+}$ ions. (b) Top-down view of the triangular lattice, emphasizing the three different bond types with parallel-edge RuCl$_6$ octahedra. (c) A typical single-crystal sample of (CD$_3$ND$_3$)$_2$NaRuCl$_6$.
• Figure 2: Single-ion physics and mean-field correlations in (CD$_3$ND$_3$)$_2$NaRuCl$_6$. (a) Schematic of the energy spectrum for a single Ru$^{3+}$ ion. The free-ion multiplet is split by the cubic crystal electric field, the spin-orbit coupling $\lambda$ and a trigonal distortion $\Delta$. An inset visualizes the spatial shape of the pseudospin wavefunctions, both for the ideal cubic case ($\Delta = 0$) and for the parameters $\Delta/\lambda \approx -0.5$ relevant to (CD$_3$ND$_3$)$_2$NaRuCl$_6$. Red/blue color indicates spin up/down of the hole, mixed together in the spin-orbital entangled $j_{\rm eff} = 1/2$ ground state. (b) Magnetic susceptibility for a small probing field $\mu_0 H = 0.1$ T along the principal crystallographic axes. The inset shows a low-temperature Curie-Weiss fit. (c) Magnetization curves at various temperatures (markers), along with Brillouin function expected in absence of two-ion correlations (dashed lines). Red lines in (b,c) represent the calculated magnetic response based on the single-ion + mean-field model discussed in the text.
• Figure 3: Frequency-field diagram of ESR excitations measured at $T = 1.5$ K for both $\mathbf{H \parallel c}$ (green) and $\mathbf{H \parallel a}$ (blue) field orientations. Dashed lines represent linear fits to the function $h\nu = g\mu_{\rm B} \mu_0 H$, while solid red lines show calculations based on the single-ion + mean-field model discussed in the text. ESR spectra at selected frequencies are presented as insets. Characteristic dips in transmittance correspond to the Larmor resonant field at given frequency, while sharp features are caused by the $g = 2.00$ DPPH marker.
• Figure 4: Temperature dependence of specific heat in (CD$_3$ND$_3$)$_2$NaRuCl$_6$. (a) Measured heat capacity $C_p(T)$ in zero magnetic field, exhibiting a two-step transition to magnetic long-range order. Left inset: Zoom-in on the critical region, showing two separate anomalies at $T_{\rm N1}$ and $T_{\rm N2}$ K. Right inset: Arrhenius plot depicting the activated behavior in the ordered state. (b) Entropy change $\Delta S (T)$ obtained through numerical integration of the specific heat.
• Figure 5: Specific heat of (CD$_3$ND$_3$)$_2$NaRuCl$_6$ in axial magnetic fields $\mathbf{H \parallel c}$. (a) Exemplary $C_p(T)$ scans for various fixed fields $H$. (b) Preliminary phase diagram. Color shows $C_p/T$ data, while circles identify sharp anomalies in heat capacity and squares denote the Schottky maximum above $H_{\rm sat}$. (c) Exemplary $C_p(H)$ scan at $T = 0.12$ K. Orange triangles highlight the peak positions in the data. Red arrow marks a broad maximum in $C_p$ discussed in the text.