Quantum Phases of Time Order in Many-Body Ground States

Understanding phases of matter is of fundamental importance. Prior to the widespread appreciation and acceptance of topological order, the paradigm of spontaneous symmetry breaking, formulated along the Landau-Ginzburg-Wilson (LGW) dogma, is central to understanding phases and transitions between phases associated with order parameters of distinct symmetries. This work identifies ground state phases of quantum many-body system in terms of time order based on twisted vector, which is defined by us as the ground state acted on by a symmetry order parameter operator. Any nontrivial temporal structure in the two-time auto-correlation function of twisted vector implicates time order. The time order phase diagram for spin-1 atomic Bose-Einstein condensate (BEC) and quantum Rabi model are fully worked out. As a special case, the (symmetry protected) time periodic ordered phase implies the presence of continuous time crystal (CTC) and underlines the essence of CTC. We further explore the intriguing phases of time crystalline order and time functional order in non-Hermitian interaction spin models.