Time-crystalline long-range order in squeezed ground state

It is widely believed that ground-state time crystals are not realizable in realistic macroscopic systems. In particular, Watanabe and Oshikawa proved a theorem that implies the absence of the time-dependent long-range order (TDLRO) in the ground states of short-range many-body systems. However, this theorem does not forbid the presence of the ground-state TDLRO for macroscopic quantities. In this work, we investigate a simple bosonic model with a squeezed ground state and point out that the time-dependence of the ground-state TDLRO for the number operator is proportional to the square of the average number in the infinite-squeezing limit, or equivalently, the infinite average-number limit. This result implies the presence of the TDLRO for macroscopic boson number. We also discuss the physical implementations in optical, spin, and tight-binding systems, including the variants. We find an example with the macroscopic TDLRO whose essence is low-lying-state physics and another with $marginal$ TDLRO at the quantum critical point. In addition, we reconsider the definition of the ground-state time crystal in terms of the Floquet picture.