Population-based 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 SE eigenstates in terms of magnetic sub-level populations. This representation provides a natural description of high-spin fermionic systems, where the underlying $\mathrm{SU}(d)$ symmetry gives rise to extensive degeneracies. We illustrate the usefulness of this framework for deriving effective Hamiltonians for a system weakly coupled to light through spin-orbit interactions, using a second-order Schrieffer--Wolff transformation projected onto the introduced population eigenbasis. These effective models provide a controlled description of the collective spin dynamics and capture the role of population redistribution among different collective spin length sectors induced by interactions. The agreement with the exact Fermi--Hubbard with light coupling dynamics confirms the consistency of the population eigenstates framework as a basis for describing high-spin quantum many-body systems.