Dissipative time crystals originating from parity-time symmetry*

Crystals are many-body systems where continuous space-translation symmetry is spontaneously broken. In analogy with this, dynamical many-body states in closed systems which spontaneously break continuous time-translation symmetry, namely, time crystals, have been proposed by Wilczek in 2012. After that, this concept has been expanded to non-equilibrium systems. In particular, non-trivial steady states characterized by persistent periodic oscillation induced by coupling with an external environment are called dissipative time crystals. Among them, those that emerge only in the thermodynamic limit are here called boundary time crystals (BTCs). BTCs have been recently actively investigated but their origin has not been clarified. 

In this presentation, I show that a class of boundary time crystals appears when the PT symmetry is realized in collective spin systems with GKSL dynamics. First, I show that the most basic model for BTCs with interaction due to a transverse magnetic field and excitation decay (hereafter referred to as the 1 spin BTC model) satisfies a proposed definition of Liouvillian PT symmetry if the parity transformation is appropriately chosen. Second, I prove that the PT symmetry breaking of a stationary state occurs at the BTC phase transition point in the large spin limit. Lastly, I perform a perturbative analysis of a class of the 1 spin models, including the 1 spin BTC model under weak dissipation. Consequently, I show that the BTCs appear in the first-order correction due to the balanced total gain and loss. These results strongly imply that BTCs are time crystals originating from PT symmetry.