  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 $\bf H\parallel a$. To exclude
extrinsic artifacts, we carried out several tests. The implications of the
oscillations are discussed.
