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We study spin-1/2 fermions in spin dependent potentials under the \emph{spinmodel approximation}, in which interatomic collisions that change the totaloccupation of single-particle modes are ignored. The spin model approximationmaps the interacting fermion problem to an ensemble of lattice spin models inenergy space, where spin-spin interactions are long-ranged andspin-anisotropic. We show that the spin model approximation is accurate forweak interactions compared to the harmonic oscillator frequency, and capturesthe collective spin dynamics to timescales much longer than would be expectedfrom perturbation theory. We explore corrections to the spin model, and therelative importance of corrections when realistic anharmonic potentialcorrections are taken into account. Additionally, we present numericaltechniques that are useful for analysis of spin models on an energy lattice,including enacting a change of single-particle basis on a many-body state as aneffective time evolution, and fitting of spatially inhomogeneous long-rangeinteractions with exponentials. This latter technique is useful forconstructing matrix product operators for use in DMRG analyses, and may havebroader applicability within the tensor network community.

Simulating fermions in spin-dependent potentials with spin models on an energy lattice