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In the class of materials called spin liquids, a magnetically ordered statecannot be attained even at milliKelvin temperatures because of conflictingconstraints on each spin (for e.g. from geometric or exchange frustration). Theresulting quantum spin-liquid (QSL) state is currently of intense interestbecause it exhibits novel excitations as well as wave-function entanglement.The layered insulator $\alpha$-RuCl$_3$ orders as a zigzag antiferromagnetbelow $\sim$7 K in zero magnetic field. The zigzag order is destroyed when amagnetic field $\bf H$ is applied parallel to the zigzag axis a. Within thefield interval (7.3, 11) Tesla, there is growing evidence that a QSL stateexists. Here we report the observation of oscillations in its thermalconductivity below 4 K. The oscillation amplitude is very large within theinterval (7.3, 11) T and strongly suppressed on either side. Paradoxically, theoscillations are periodic in 1/\emph{H}, analogous to quantum oscillations inmetals, even though $\alpha$-RuCl$_3$ is an excellent insulator with a gap of1.9 eV. By tilting $\bf H$ out of the plane, we find that the oscillationperiod is determined by the in-plane component $H_a$. As the temperature israised above 0.5 K, the oscillation amplitude decreases exponentially. Thedecrease anticorrelates with the emergence above $\sim$2 K of an anomalousplanar thermal Hall conductivity measured with $\bf H\parallel a$. To excludeextrinsic artifacts, we carried out several tests. The implications of theoscillations are discussed.

Oscillations of the thermal conductivity observed in the spin-liquid state of $\alpha$-RuCl$_3$