Realizing exactly solvable SU (N) magnets with thermal atoms

We show that n thermal fermionic alkaline-earth atoms in a at-bottom trap allow one to robustly implement a spin model displaying two symmetries: the Sn symmetry that permutes atoms occupying di erent vibrational levels of the trap and the SU(N) symmetry associated with N nuclear spin states. The symmetries makes the model exactly solvable, which, in turn, enables the analytic study of dynamical processes such as spin di usion in this SU(N) system. We also show how to use this system to generate entangled states that allow for Heisenberg-limited metrology. This highly symmetric spin model should be experimentally realizable even when the vibrational levels are occupied according to a high-temperature thermal or an arbitrary non-thermal distribution.