Classes of metastable thermodynamic quantum time crystals

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.

Figure 1

Schematic layout of QOP variation in Matsubara time. Chord (1), short dashes, corresponds to $M_1(\tau +1/T)=M_1\left(\tau \right)$ and $M_2\left(\tau \right)=\text{const}$. Chord (2), long dashes, corresponds to $M_1\left(\tau \right)+i{M}_2\left(\tau \right)\equiv M\left(\tau \right)e^{i\varphi }$, with $\varphi =\text{const}$.

Figure 2

QOP function $M_1\left(\tau \right)$ as a function of Matsubara time $\tau$. Curves are marked according to different parameter sets: (1) $T=0.51$; $k=0.99999$; $n=1$; (2) $T=0.8$; $k=0.999$; $n=1$; (3) $T=0.51,k=0.84,n=4$. Temperature $T$ is measured in arbitrary units.

Figure 3

Density of Floquet states ${\tilde{\alpha }}_q$ (solid lines) for the same parameter sets (1)–(3) as indicated in Fig. 2. Dashed line is COP density of states (23) with dispersion $\pm \sqrt{{P_1^{c2}+{\epsilon }_q^2}^{\phantom{\int }}}$ under condition (41) of equal COP and QOP mean square amplitudes defined in (22). Dotted line is a guide for the eyes.