Classical and quantum time crystals in a levitated nanoparticle without drive

A time crystal is defined as a phase of matter spontaneously exhibiting a periodicity in time. Previous studies focused on discrete quantum time crystals under periodic drive. Here, we propose a time crystal model based on a levitated charged nanoparticle in a static magnetic field without drive. Both the classical time crystal in thermal equilibrium and the quantum time crystal in the ground state can emerge in the spin rotational mode, under a strong magnetic field or large charge-to-mass ratio limit. Additionally, the time polycrystal is defined and naturally appears in this model. Our model paves a way for realizing time crystals in thermal equilibrium.