  The study of dynamical phase transitions has been attracting considerable
research efforts in the last decade. One theme of present interest is to search
for exotic scenarios beyond the framework of equilibrium phase transitions.
Here, we establish a duality between many-body dynamics and static Hamiltonian
ground states for both classical and quantum systems. We construct frustration
free Hamiltonians whose ground state phase transitions have rigorous duality to
chaotic transitions in dynamical systems. By this duality, we show the
corresponding ground state phase transitions are characterized by
bulk-to-surface response, which are then dubbed "peratic" meaning defined by
response to the boundary. For the classical system, we show how the time-like
dimension emerges in the static ground states. For the quantum system, the
ground state is a superposition of geometrical lines on a two dimensional
array, which encode the dynamical Floquet evolution history of one dimensional
disordered spin chains. Our prediction of peratic phase transition has direct
consequences in quantum simulation platforms such as Rydberg atoms and
superconducting qubits, as well as anisotropic spin glass materials. The
discovery would shed light on the unification of dynamical phase transitions
with equilibrium systems.
