In the class of materials called spin liquids, a magnetically ordered state cannot be attained even at milliKelvin temperatures because of conflicting constraints on each spin (for e.g. from geometric or exchange frustration). The resulting quantum spin-liquid (QSL) state is currently of intense interest because it exhibits novel excitations as well as wave-function entanglement. The layered insulator $alpha$-RuCl$_3$ orders as a zigzag antiferromagnet below $sim$7 K in zero magnetic field. The zigzag order is destroyed when a magnetic field $bf H$ is applied parallel to the zigzag axis a. Within the field interval (7.3, 11) Tesla, there is growing evidence that a QSL state exists. Here we report the observation of oscillations in its thermal conductivity below 4 K. The oscillation amplitude is very large within the interval (7.3, 11) T and strongly suppressed on either side. Paradoxically, the oscillations are periodic in 1/emph{H}, analogous to quantum oscillations in metals, even though $alpha$-RuCl$_3$ is an excellent insulator with a gap of 1.9 eV. By tilting $bf H$ out of the plane, we find that the oscillation period is determined by the in-plane component $H_a$. As the temperature is raised above 0.5 K, the oscillation amplitude decreases exponentially. The decrease anticorrelates with the emergence above $sim$2 K of an anomalous planar thermal Hall conductivity measured with $bf Hparallel a$. To exclude extrinsic artifacts, we carried out several tests. The implications of the oscillations are discussed.
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