Emergent Continuous Time Crystal in Dissipative Quantum Spin System without Driving

Time crystal, a nonequilibrium phenomenon extending spontaneous symmetry breaking into the temporal dimension, holds fundamental significance in understanding quantum many-body physics. In this work, we explore the nonequilibrium phase diagram of a two-dimensional dissipative Heisenberg spin system in the absence of explicit driving. We numerically identify the emergence of novel nonstationary oscillatory states by analyzing the spin dynamics. These states are categorized as limit cycle and chaos based on the Lyapunov exponent. Remarkably, the observed limit cycle behavior represents a continuous time crystal (CTC), spontaneously breaking the continuous time translation symmetry of the system. We further confirm those oscillatory behaviors by studying the stability against local perturbations applied to the system. Finally, we investigate the robustness of the emergent CTC by introducing isotropic Gaussian-type white noise into the interactions. This study provides many insights into the intricate interplay between dissipation-induced decay processes and interaction-induced spin precession, deepening our understanding of dissipative quantum many-body systems.