Probing $\alpha -{\mathrm{RuCl}}_3$ Beyond Magnetic Order: Effects of Temperature and Magnetic Field

Recent studies have brought $\alpha \text{−}{\mathrm{RuCl}}_3$ to the forefront of experimental searches for materials realizing Kitaev spin-liquid physics. This material exhibits strongly anisotropic exchange interactions afforded by the spin-orbit coupling of the $4d$ Ru centers. We investigate the dynamical response at finite temperature and magnetic field for a realistic model of the magnetic interactions in $\alpha \text{−}{\mathrm{RuCl}}_3$. These regimes are thought to host unconventional paramagnetic states that emerge from the suppression of magnetic order. Using exact diagonalization calculations of the quantum model complemented by semiclassical analysis, we find a very rich evolution of the spin dynamics as the applied field suppresses the zigzag order and stabilizes a quantum paramagnetic state that is adiabatically connected to the fully polarized state at high fields. At finite temperature, we observe large redistributions of spectral weight that can be attributed to the anisotropic frustration of the model. These results are compared to recent experiments and provide a road map for further studies of these regimes.

Figure 1

(a) Schematic phase diagram of the model Hamiltonian (1) for $\alpha \text{−}{\mathrm{RuCl}}_3$ at finite $T$ and $B$. $T_N$ is the Néel temperature, and $\mathrm{\Theta }$ is the Curie-Weiss constant. The variable blue color shading indicates a crossover to the high-field regime. (b) 24-site cluster employed in ED calculations showing the orientation of the cubic $x$, $y$, $z$ axes, and $C2/m$ unit cell. The crystallographic axes correspond to $a=[11\overline{2}]$, $b=[1\overline{1}0]$, and $c^{*}=[111]$ in cubic coordinates. The numbers label sites defining the ${\mathbb{Z}}_2$ flux operator ${\widehat{W}}_p$. Nearest neighbor $X$, $Y$, and $Z$ bonds are red, green, and blue, respectively.

Figure 2

Evolution of the $T=0$ static correlations under magnetic field computed via ED. (a) Magnetization $M(B)$. Experimental data at $T$ = 2 K from [23]. (b) Static structure factor for $\mathbf{k}=\mathrm{\Gamma }$, $M$, and $Y$. (c) Ground state fidelity susceptibility ${\chi }_F$ and second derivative of the ground state energy. The peak in both indicates a single phase transition at $B_c\sim 6\mathrm{T}$. (d) ${\mathbb{Z}}_2$ flux density compared to known limits: the Kitaev spin liquid (KSL) has $⟨\widehat{n}⟩=0$ at $B=0$, while classical collinear ordered states have $⟨\widehat{n}⟩\approx 0.5$. The present model has $⟨\widehat{n}⟩\gtrsim 0.5$ at all fields (blue line).

Figure 3

(a) Brillouin zone definition. (b) Summary of low-energy contributions to ${\mathcal{S}}^{\mu \nu }$ from different zigzag domains at $B=0$. (c)–(g) $T=0$ inelastic neutron scattering intensity $\mathcal{I}(\mathbf{k},\omega )\propto f(k{)}^2\sum _{\mu \nu }({\delta }_{\mu ,\nu }-{\widehat{k}}_{\mu }{\widehat{k}}_{\nu }){\mathcal{S}}^{\mu \nu }(\mathbf{k},\omega )$ under applied field, computed with ED; $f(k)$ is the magnetic form factor for ${\mathrm{Ru}}^{3+}$ [58]. (c) $B=0$, (d),(e) $\mathbf{B}||b$, (f),(g) $\mathbf{B}||a$. (h),(i) Summary of low-energy contributions to ${\mathcal{S}}^{\mu \nu }$ for $B>B_c$. (j)–(m) Polarized electron spin resonance absorption $\omega {\chi }^{\prime \prime }(\omega )\propto \omega {\mathcal{S}}^{\mu \mu }(0,\omega )$, with $\mu ||{\mathbf{h}}_{\omega }$, at the level of (j),(k) LSWT and (l),(m) ED. LSWT results include only the domain $\mathbf{Q}=\mathrm{Y}$ for $B<B_c$. ED results combine data from various 20- and 24-site clusters as in [28]. Spectra were Gaussian broadened by a width of 0.5 meV and integrated over $k_{c^{*}}$ consistent with [17]. The color scale of each figure is independent.

Figure 4

Neutron scattering intensity for $T>0$, as a function of $\mathbf{k}$ (a)–(c) and $T$ for (e) $\mathbf{k}=\mathrm{\Gamma }$ and (f) $\mathbf{k}=\mathrm{M}$, combining the results of multiple clusters. (d) Kitaev flux density $⟨\widehat{n}⟩$ and normalized real space static correlations computed for the cluster in Fig. 1 for nearest neighbors $⟨\gamma \gamma ⟩\equiv ⟨S_1^{\gamma }{S}_2^{\gamma }{⟩}_T$, $⟨\alpha \beta ⟩\equiv ⟨S_1^{\alpha }{S}_2^{\beta }{⟩}_T$, second nearest neighbors $⟨2nn⟩\equiv ⟨{\mathbf{S}}_1·{\mathbf{S}}_3{⟩}_T$, and third nearest neighbors $⟨3nn⟩\equiv ⟨{\mathbf{S}}_1·{\mathbf{S}}_4{⟩}_T$. Except for $⟨\widehat{n}⟩$, values are normalized by their $T=0$ value. Site labels refer to Fig. 1. The color scale of each figure is independent.