Mean-field thermodynamic quantum time-space crystal: Spontaneous breaking of time-translation symmetry in a macroscopic fermion system

A model demonstrating the existence of a thermodynamically stable quantum time-space crystal has been proposed and studied. This state is characterized by an order parameter periodic in both real and imaginary times. The average of the order parameter over phases of the oscillations vanishes but correlation functions of two or more order parameters show nondecaying oscillations. An alternative interpretation of the results is based on the concept of an operator order parameter introduced for this purpose. The model studied here has been suggested previously, in particular, for describing the pseudogap state in superconducting cuprates. Although many properties of the time-space crystal considered here are close to those of a well known DDW state, static magnetic moments oscillating at (pi, pi) do not exist. Instead, delta peaks at finite energies are predicted in the cross-section of inelastic spin-polarized neutron scattering.