Population eigenstates of the SU(d) spin-exchange model for high-spin fermions in optical lattices

Abstract

We investigate the $\mathrm{SU}(d)$ spin exchange model describing ultra-cold fermionic atoms with spin $s\ge 1$ in a one-dimensional optical lattice. The model emerges from the Fermi-Hubbard model in the strongly interacting regime with one atom in each lattice site. The central result of this work is the systematic construction of eigenstates in terms of magnetic sub-level populations, which we call population eigenstates. Exploiting this framework, we derive effective light-induced Hamiltonians via a second-order Schrieffer-Wolff transformation projected onto the population eigenstates. The resulting models reveal a qualitative difference between spin-1/2 and higher-spin systems: whereas spin-1/2 dynamics remains confined to the maximal-spin Dicke manifold, the extensive $\mathrm{SU}(d)$ degeneracies for $s\ge 1$ allow coherent population transfer across sectors of different collective spin length, generating unconventional spin dynamics that cannot be captured by any fixed-spin-manifold description. Agreement with exact Fermi-Hubbard dynamics confirms the framework as a practical foundation for quantum-enhanced correlations and metrological protocols in high-spin fermionic systems.