Thermodynamic Quantum Time-Space Crystal: Spontaneous Breaking of Time-Translation Invariance

Abstract

We suggest and study a model of interacting fermions demonstrating existence of a thermodynamically stable quantum time-space crystal. An order parameter characterizing this state is periodic in both real and imaginary times. The imaginary-time dependence is used for calculation of the free energy of the system, while the real-time dependence leads to non-decaying oscillation of correlation functions of two- and more times. Single-time physical quantities do not depend on time. The oscillation of correlation functions at different time scan in principle be observed in scattering experiments.