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A model demonstrating existence of a thermodynamically stable quantumtime-space crystal has been proposed and studied. This state is characterizedby an order parameter periodic in both real and imaginary times. The average ofthe order parameter over phases of the oscillations vanishes but correlationfunctions of two or more order parameters show non-decaying oscillations. Analternative interpretation of the results is based on a concept of an operatororder parameter introduced for this purpose. The model studied here has beensuggested previously, in particular, for describing the pseudogap state insuperconducting cuprates. Although many properties of the time-space crystalconsidered here are close to those of a well known DDW state, static magneticmoments oscillating at $ ( \pi ,\pi) $ do not exist. Instead, $\delta$-peaks atfinite energies are predicted in the cross-section of inelastic spin-polarizedneutron scattering.

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