Observation of a finite-energy phase transition in a one-dimensional quantum simulator

Equilibrium phase transitions in many-body systems have been predicted and observed in two and three spatial dimensions but have long been thought not to exist in one-dimensional systems. It was suggested that a phase transition in one dimension can occur in the presence of long-range interactions. However, an experimental realization has so far not been achieved due to the requirement to both realize interactions over sufficiently long distances and to prepare equilibrium states. Here we demonstrate a finite-energy phase transition in one dimension by implementing a long-range interacting model in a trapped-ion quantum simulator. We show that finite-energy states can be generated by time-evolving initial product states and letting them thermalize under the dynamics of a many-body Hamiltonian. By preparing initial states with different energies, we study the finite-energy phase diagram of a long-range interacting quantum system. We observe a ferromagnetic equilibrium phase transition as well as a crossover from a low-energy polarized paramagnet to a high-energy unpolarized paramagnet, in agreement with numerical simulations. Our work presents a scheme for preparing finite-energy states in quantum simulation platforms, enabling access to phases at finite energy density.