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.