The study of dynamic phase transitions has attracted considerable research effort over the past decade. One of my current interests is searching for exotic scenarios beyond the framework of equilibrium phase transitions. Here we establish the duality between the many-body dynamics and the static Hamiltonian ground state for both classical and quantum systems. We construct a frustration-free Hamiltonian in which the ground-state phase transition is a strict duality to the chaotic transition of the dynamic system. With this duality, we show that the corresponding ground-state phase transition is characterized by a bulk-to-surface response. This is called the “peratic” meaning defined by the response to the boundary. In classical systems, we show how time-like dimensions appear in the static ground state. For quantum systems, the ground state is a superposition of geometric lines on a two-dimensional array, encoding the dynamic His-Floquet evolutionary history of one-dimensional disordered spin chains. Predicting the peratic phase transition has direct implications for quantum simulation platforms such as Rydberg atoms, superconducting qubits, and anisotropic spin glass materials. This discovery will shed light on the unification of dynamic phase transitions and equilibrium systems.



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