Nonexistence of spontaneous symmetry breakdown of time-translation symmetry: a review

Abstract

The time invariance of equilibrium states has been established in the C * -algebraic framework assuming an infinite-volume Heisenberg time evolution as C * -dynamics. This fundamental property of equilibrium states 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 appear in the time direction of equilibrium states for general quantum models, meaning that genuine quantum time crystals are strictly forbidden. We compare our statement on the impossibility of quantum time crystals with the result by Watanabe-Oshikawa which has been considered as a milestone in the study on quantum time crystals. Our no-go statement based on the Kubo-Martin-Schwinger condition for C * -dynamics is not only mathematically rigorous but also more general than the main result of Watanabe-Oshikawa and its improvement by Watanabe-Oshikawa-Koma based on temporal correlation functions under cut-off quantum time evolutions.