Incommensurate Charge Density Wave as Quantum Space-Time Crystal

A quantum time crystal (QTC) is a novel quantum mechanical ground state that was recently proposed by Wilczek. Although many QTC models have been proposed, it is not clear yet if such states are possible. We propose the idea that an incommensurate charge density wave (ICDW) forms a quantum space crystal (QSC), which is an extension of a QTC in the spatial domain. Consequently, a rotating ring-shaped ICDW naturally forms a quantum space-time crystal (QSTC), which combines the two concepts of QTC and QSC. The breaking of space-time translation symmetry is understood using time-dependent density matrices. Furthermore, we show that this model can be observed in real systems such as TaS$_3$ ring crystals at finite temperature. Our results suggest that QTC/QSTC can exist.