Space and time play in many respects a similar role in modern physics. At the same time, there are many known materials that have stable crystalline structures characterized by a periodic dependence of physical parameters on coordinates but not on time. Are thermodynamic states with a periodic time dependence forbidden by fundamental laws of nature?

Several years ago, Frank Wilczek proposed that quantum time crystals might exist using a rather simple model with a superconducting ring and this work has attracted great attention.   It has been shown somewhat later that the phase proposed by Wilczek was not an equilibrium state  and even a no-go theorem rejecting this possibility has been published. Nevertheless, the proposal was very stimulating and it has been suggested that some non-equilibrium states demonstrate a periodic time dependence. This has recently been confirmed experimentally  and study of systems demonstrating periodic time dependence out of equilibrium is a hot topic now.    

   Here it is shown solving explicitly a model for interacting electrons that the  thermodynamically stable quantum space-time crystal state with a periodic dependence on both space coordinates and time is nevertheless possible. The model under study is a spin-fermion model with overlapping “hot spots” introduced by us recently  for description of underdoped superconducting cuprates. It has been demonstrated previously that, within this model, one could obtain d-wave superconductivity, charge-density waves with the d-wave formfactor, Pomeranchuk deformation of the Fermi surface, and the so -called  d-density wave (DDW) state characterized by loop currents oscillating with the double period of the lattice.

Surprisingly, in addition to all these states, one more state exists within this model and this observation is discussed in the present talk. The order parameter of this new state describes the loop currents with the same periodic space dependence as in the DDW state. However, it oscillates also both in real t and imaginary τ times (period of the oscillation in the imaginary time is commensurate with the inverse temperature 1/T). Generally, the order parameter can be well described by an elliptic function leading to this double-periodic dependence.

  Explicit calculations show that, in a certain region of parameters of the model, such a quantum space-time crystal has the lowest free energy and therefore is stable. Although the average order parameter  equals zero, the correlation function of two parameters   has a long range order in both space and time, thus demonstrating a possibility of the existence of the thermodynamic space-time crystal. This new state looks a good candidate for describing the still mysterious "pseudogap state" in superconducting cuprates but the phenomenon is clearly more general and other interesting applications are anticipated