Nonexistence of spontaneous symmetry breakdown of time-translation symmetry on general quantum systems: Any macroscopic order parameter moves not!

This review informs that the impossibility of genuine quantum time crystals has been known in the C*-algebraic quantum statistical mechanics since 1970s. The KMS condition implies that spontaneous breakdown of time-translation symmetry is impossible in general quantum systems. In particular, any non-trivial order, such as periodic, quasi-periodic, and chaotic order cannot exist in the time direction of equilibrium states, meaning that genuine quantum time crystals are excluded from the outset.Comment: This is a review paper to be published in Reviews in Mathematical Physic