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

## TL;DR

This paper proposes a model for a thermodynamically stable quantum time-space crystal exhibiting periodic order parameters in real and imaginary times, with unique oscillatory correlation functions, expanding understanding of time-translation symmetry breaking in fermion systems.

## Contribution

It introduces a new model demonstrating a stable quantum time-space crystal with unique oscillatory properties, linked to pseudogap states in superconducting cuprates.

## Key findings

- Existence of a stable quantum time-space crystal with periodic order parameters.
- Correlation functions show non-decaying oscillations despite vanishing average order parameter.
- Predicted inelastic neutron scattering signals at finite energies.

## Abstract

A model demonstrating 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 non-decaying oscillations. An alternative interpretation of the results is based on a 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.