Magnetic Excitations and Continuum of a Possibly Field-Induced Quantum Spin Liquid in $\alpha \text{−}{\mathrm{RuCl}}_3$

We report on terahertz spectroscopy of quantum spin dynamics in $\alpha \text{−}{\mathrm{RuCl}}_3$, a system proximate to the Kitaev honeycomb model, as a function of temperature and magnetic field. We follow the evolution of an extended magnetic continuum below the structural phase transition at $T_{s2}=62\mathrm{K}$. With the onset of a long-range magnetic order at $T_N=6.5\mathrm{K}$, spectral weight is transferred to a well-defined magnetic excitation at $\hslash {\omega }_1=2.48\mathrm{meV}$, which is accompanied by a higher-energy band at $\hslash {\omega }_2=6.48\mathrm{meV}$. Both excitations soften in a magnetic field, signaling a quantum phase transition close to $B_c=7\mathrm{T}$, where a broad continuum dominates the dynamical response. Above $B_c$, the long-range order is suppressed, and on top of the continuum, emergent magnetic excitations evolve. These excitations follow clear selection rules and exhibit distinct field dependencies, characterizing the dynamical properties of a possibly field-induced quantum spin liquid.

Figure 1

Absorption coefficient $\alpha$ as a function of temperature $T$ for the photon energies (a) $\hslash {\omega }_1=2.48\mathrm{meV}$ and (b) $\hslash {\omega }_2=6.38\mathrm{meV}$, respectively, measured with the terahertz wave vector perpendicular to the crystallographic $ab$ plane of $\alpha \text{−}{\mathrm{RuCl}}_3$. Clear hysteresis is revealed at the structural phase transitions between the low-temperature rhombohedral and the high-temperature monoclinic phases. The solid lines are guides for the eyes. The structural phase boundaries [14] are indicated by the dashed lines at $T_{s2}=62\mathrm{K}$ and $T_{s1}=166\mathrm{K}$, and the magnetic phase transition by the dotted line at $T_N=6.5\mathrm{K}$.

Figure 2

(a) Ratio of transmission as obtained at 4.5 K ($T<T_N$) and 7 K ($T>T_N$) as a function of photon energy in $\alpha \text{−}{\mathrm{RuCl}}_3$. Two dips appear below $T_N=6.5\mathrm{K}$ at $\hslash {\omega }_1=2.48\mathrm{meV}$ (mode $M_1$) and $\hslash {\omega }_2=6.38\mathrm{meV}$ (band $M_2$), as marked by the arrow and asterisk, respectively. (b) Evolution of the absorption-coefficient spectra with decreasing temperature below $T_{s2}$, where the 60 K spectrum is taken as a reference. A broad continuum extending over the whole spectral range evolves with decreasing temperature down to $T_N$. Below $T_N$, the spectral weight transfers to lower frequency with a sharp peak developing at $\hslash {\omega }_1$. The oscillations are due to multiple interference at the sample surfaces.

Figure 3

Terahertz absorption spectra of $\alpha \text{−}{\mathrm{RuCl}}_3$ measured at 2.4 K below and above the critical field $B_c=7\mathrm{T}$ for (a) the terahertz magnetic field parallel (${\mathbf{h}}^{\omega }\parallel \mathbf{B}$) and (b) perpendicular to the external magnetic field (${\mathbf{h}}^{\omega }\perp \mathbf{B}$), respectively, with the spectra of 10 K taken as references. The spectra are shifted vertically by a constant for clarity. The arrows indicate the low-energy magnetic excitations with sharp absorption lines, while the asterisks mark the high-energy excitations that are rather broad. At the critical field, a broad continuum is observed that extends over the whole spectral range. Above $B_c$, various excitations emerge on the top of the continuum. (c),(d) Contour plot of the absorption coefficient as a function of field and photon energy for ${\mathbf{h}}^{\omega }\parallel \mathbf{B}$ and ${\mathbf{h}}^{\omega }\perp \mathbf{B}$. The symbols correspond to the peak positions of the low-energy modes and the maximum positions of the high-energy bands in (a) and (b). For the broad bands, the linewidths are indicated by the error bars. Above $B_c$, for ${\mathbf{h}}^{\omega }\parallel \mathbf{B}$ the lower-lying mode $L_{1,\parallel }$ follows a linear field dependence with an apparent $g$ value of $g^{*}=11.1(1)$ assuming a magnetic-dipole excitation [solid line in (c)], while for ${\mathbf{h}}^{\omega }\perp \mathbf{B}$ the field dependence of the higher-lying mode $L_{2,\perp }$ is described by the linear Zeeman term with $g^{*}=10.2(2)$ [solid line in (d)]. The vertical dashed line marks the critical field $B_c=7\mathrm{T}$.