Dynamical phase transitions in a collective Heisenberg model simulated with harmonically trapped ultracold fermions

The collective Heisenberg spin model is a textbook model for magnetism and superconductivity. It describes localized spins interacting via long-range exchange interactions. We discuss a quantum simulation of this model using tens of thousands of potassium atoms trapped in a three-dimensional harmonic oscillator, without an optical lattice. Using a Feshbach resonance to tune the interactions between spin-up and spin-down potassium atoms to be weak, mode-changing collisions can be suppressed during the time of the experiment. Atoms remain in their initial single-particle eigenmodes, which form a lattice in mode space. We study spin dynamics initiated with a $\pi/2$ pulse, and observe competition between interactions and an inhomogenous effective magnetic field (due to a real-space field curvature, which maps onto a mode-space field gradient). Varying both the strength of interactions and of inhomogeneity, we observe a dynamical phase transition between ferromagnetic dynamics and ungapped paramagnetic dynamics. We find excellent agreement with calculations based on a mean-field treatment of a collective Heisenberg model, with all-to-all couplings.