Spontaneously broken discrete time translation symmetry in driven quantum Brownian motion

Spontaneously broken discrete time translation symmetry (DTTS) in Floquet systems, often be dubbed as discrete time crystal or Floquet time crystal, draws a lot of attention in recent years. While most experimental realizations and theoretical studies of the DTTS breaking involve many-body localization to avoid heating, the effect of dissipation is an obvious alternative to prevent heating. In this work, we consider the quintessential model of quantum dissipation — the Caldeira-Leggett model and take into account the presence of a periodically driven external potential. Employ a perturbative approach within the framework of Feynman-Vernon influence functional enables us to demonstrate analytically that DTTS breaking would occur in such a simple model. The interplay between the driving frequency, the strength of dissipation, and the temperature are studied in detail. A crossover between the quantum and classical limit is examined as well. Owing to the ubiquitousness of the ordinary Caldeira-Leggett model, our findings pave a new avenue toward exploring the novel phenomena of DTTS breaking in broad systems.