# Classes of metastable Thermodynamic Quantum Time Crystals

## TL;DR

This paper classifies metastable thermodynamic quantum time crystals in fermionic systems, showing that true equilibrium states are time-independent, and proves a no-go theorem for certain long-range interaction cases.

## Contribution

It introduces a classification of metastable quantum time crystals and analytically proves a no-go theorem for long-range interactions in fermionic systems.

## Key findings

- Time-independent solutions have lower free energy than time crystals.
- Quantum time crystals are metastable in certain fermionic systems.
- A no-go theorem is established for long-range interaction cases.

## Abstract

We found that thermodynamic quantum time crystals in fermi systems, defined as quantum orders oscillating periodically in the imaginary Matsubara time with zero mean, are metastable for two general classes of solutions. Mean-field time independent solutions proved to have lower free energy manifesting true thermodynamic equilibrium with either single or multiple (competing) charge, spin and superconducting symmetry breaking orders. The 'no-go' theorem is proven analytically for a case of long-range interactions between fermions in momentum space in electron-hole and Cooper channels.