Diffusive Nambu-Goldstone modes in quantum time-crystals

Abstract

We study the Nambu-Goldstone (NG) modes associated with spontaneous breaking of the continuous time-translation symmetry. To discuss a quantum time-crystal with the spontaneously-broken continuous time-translation symmetry, we introduce the van der Pol type nonlinear-friction to open quantum systems. By considering small fluctuations around a time-periodic mean-field solution, we show that a gapless collective mode necessarily appears; this is nothing but the NG mode associated with a time crystal. We show that its dispersion relation becomes $\omega =-iC{\mathbf{p}}^2$. We also show that noncommutative breaking of the time-translation and U(1) symmetries results in mixing of the NG modes, and the (typically) propagating NG mode appears, whose dispersion relation becomes $\omega =(\pm C_1-i{C}_2){\mathbf{p}}^2$.

Comments: 5+4 pages, 4+1 figures