Hierarchical time crystals

Spontaneous symmetry breaking is one of the central organizing principles in physics. Time crystals have emerged as an exotic phase of matter, spontaneously breaking the time translational symmetry, and are mainly categorized as discrete or continuous. While these distinct types of time crystals have been extensively explored as standalone systems, intriguing effects can arise from their mutual interaction. Here, we demonstrate that a time-independent coupled system of discrete and continuous time crystals induces a simultaneous twofold temporal symmetry breaking, resulting in a hierarchical time crystal phase. Interestingly, one of the subsystems breaks a discrete temporal symmetry that is absent from the time-independent Liouvillian generating the dynamics, but instead emerges dynamically, giving rise to a convoluted non-equilibrium phase. We demonstrate that hierarchical time crystals are robust, emerging for fundamentally different coupling schemes and persisting across wide ranges of system parameters.