Time crystals: Can diamagnetic currents drive a charge density wave into rotation?

It has been recently argued that an inhomogeneous system could rotate spontaneously in its ground state —hence a “time crystal” which is periodic in time. In this letter we present a very simple example: a superfluid ring threaded by a magnetic field which develops a charge density wave (CDW). A simple calculation shows that diamagnetic currents cannot drive rotation of the CDW, with a clear picture of the cancellation mechanism.