  One of the most striking many-body phenomena in nature is the sudden change
of macroscopic properties as the temperature or energy reaches a critical
value. Such equilibrium transitions have been predicted and observed in two and
three spatial dimensions, but have long been thought not to exist in
one-dimensional (1D) systems. Fifty years ago, Dyson and Thouless pointed out
that a phase transition in 1D can occur in the presence of long-range
interactions, but an experimental realization has so far not been achieved due
to the requirement to both prepare equilibrium states and realize sufficiently
long-range interactions. Here we report on the first experimental demonstration
of a finite-energy phase transition in 1D. We use the simple observation that
finite-energy states can be prepared by time-evolving product initial states
and letting them thermalize under the dynamics of a many-body Hamiltonian. By
preparing initial states with different energies in a 1D trapped-ion quantum
simulator, 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 a system of up to $23$ spins, in excellent agreement
with numerical simulations. Our work demonstrates the ability of quantum
simulators to realize and study previously inaccessible phases at finite energy
density.
