Gapless Spin Excitations in the Field-Induced Quantum Spin Liquid Phase of $\alpha \text{−}{\mathrm{RuCl}}_3$

$\alpha \text{−}{\mathrm{RuCl}}_3$ is a leading candidate material for the observation of physics related to the Kitaev quantum spin liquid (QSL). By combined susceptibility, specific-heat, and nuclear-magnetic-resonance measurements, we demonstrate that $\alpha \text{−}{\mathrm{RuCl}}_3$ undergoes a quantum phase transition to a QSL in a magnetic field of 7.5 T applied in the $ab$ plane. We show further that this high-field QSL phase has gapless spin excitations over a field range up to 16 T. This highly unconventional result, unknown in either Heisenberg or Kitaev magnets, offers insight essential to establishing the physics of $\alpha \text{−}{\mathrm{RuCl}}_3$.

Figure 1

Magnetic phase diagram of $\alpha \text{−}{\mathrm{RuCl}}_3$ with field applied in the $ab$ plane. $T_N$ is determined from magnetization and specific-heat data (Fig. 2). In the QSL phase, the color map represents the exponent, $\alpha$, determined from the power-law form of the NMR spin-lattice relaxation rate, $1{/}^{35}{T}_1\propto T^{\alpha }$ (Fig. 4). $T^{*}$ represents the upper limit of the gapless low-$T$ regime. Inset: schematic representation of zero-field zigzag order in the hexagonal ($ab$) plane.

Figure 2

Magnetic transition with field applied in the $ab$ plane. (a) Magnetization, $M(T)$. Inset: $dM/dT$; vertical lines mark the peaks, which show $T_N$ for fields up to 7.25 T. (b) Low-$T$ specific heat, $C_v$. Data are offset for clarity. The transition for regions with $ABC$ ($AB$) layer stacking is marked by the arrows (vertical lines). (c) ${}^{35}\mathrm{Cl}$ NMR spectra at fields close to 7.5 T, shown at $T=2\mathrm{K}$. Blue arrows mark peaks characteristic of the low-field, ordered phase, green arrows the peaks of the high-field, disordered phase, and red arrows the sharp central peak at which ${{}^{35}K}_n$ and $1{/}^{35}{T}_1$ were measured.

Figure 3

${}^{35}\mathrm{Cl}$ NMR spectra and Knight shift for in-plane fields. ${}^{35}\mathrm{Cl}$ spectra measured at (a) 10.3 and (b) 12 T, shown for $T=2$ and 20 K. $\mathrm{\Delta }f=f-{}^{35}\gamma H$ is the frequency shift, where $f$ is the measured frequency and ${}^{35}\gamma =4.171\mathrm{MHz}/\mathrm{T}$ is the gyromagnetic ratio of ${}^{35}\mathrm{Cl}$. The splitting of the sharp central peak (red and green arrows) is a consequence of the three inequivalent ${\mathrm{Cl}}^{-}$ sites, two of which are not separated at this field angle. The separation of the broader satellite peaks from the center peaks changes little with field. The peak widths are determined by quadrupolar effects [56], discussed in Sec. S5 of the Supplemental Material [55]. (c) NMR Knight shift, ${{}^{35}K}_n(T)$, measured at the central peak and shown for different field values. ${{}^{35}K}_n$ is calculated from $\mathrm{\Delta }f$ after subtracting all $T$-independent quadrupolar and orbital contributions (Sec. S5 [55]). Solid lines are Curie-Weiss fits to the high-$T$ data, of the form ${{}^{35}K}_n=a/(T-\theta )$; we find that $\theta =-45\pm 10\mathrm{K}$ is almost field independent (inset). Dotted lines indicate a “level-off” behavior of ${{}^{35}K}_n$ at low temperatures (marked by the vertical lines).

Figure 4

NMR spin-lattice relaxation rates. $1/{{}^{35}T}_1$ shown for five different applied in-plane fields. At $T>40\mathrm{K}$, $1/{{}^{35}T}_1$ approaches a constant value, $1/{{}^{35}T}_1\simeq 145{\mathrm{s}}^{-1}$, for all fields (dashed line). At low $T$, $1/{{}^{35}T}_1$ follows a power-law $T$ dependence; solid lines are fits to the form $1/{{}^{35}T}_1=b{T}^{\alpha }$ over one decade in $T$. The exponent $\alpha$ (inset) and temperature scale $T^{*}$ (vertical lines) are also shown in Fig. 1.