We describe a possible general and simple paradigm in a classical thermal setting for discrete time crystals (DTCs), systems with stable dynamics which is subharmonic to the driving frequency thus breaking discrete time-translational invariance. We c
onsider specifically an Ising model in two dimensions, as a prototypical system with a phase transition into stable phases distinguished by a local order parameter, driven by a thermal dynamics and periodically kicked. We show that for a wide parameter range a stable DTC emerges. The phase transition to the DTC state appears to be in the equilibrium 2D Ising class when dynamics is observed stroboscopically. However, we show that the DTC is a genuine non-equilibrium state. More generally, we speculate that systems with thermal phase transitions to multiple competing phases can give rise to DTCs when appropriately driven.
We study incompressible ground states of bosons in a two-dimensional rotating square optical lattice. The system can be described by the Bose-Hubbard model in an effective uniform magnetic field present due to the lattice rotation. To study ground st
ates of the system, we map it to a frustrated spin model, followed by Schwinger boson mean field theory and projective symmetry group analysis. Using symmetry analysis we identify bosonic fractional quantum Hall states, predicted for bosonic atoms in rotating optical lattices, with possible stable gapped spin liquid states within the Schwinger boson formalism. In particular, we find that previously found fractional quantum Hall states induced by the lattice potential, and with no counterpart in the continuum [G. Moller, and N. R. Cooper, Phys. Rev. Lett. textbf{103}, 105303 (2009)], correspond to $pi$ flux spin liquid states of the frustrated spin model.
