Effect of spatial inhomogeneity on the mapping between strongly interacting fermions and weakly interacting spins

TL;DR

This paper investigates how spatial inhomogeneities like boundaries and impurities affect the mapping between strongly interacting fermions and weakly interacting spins, combining analytical and numerical methods across various models.

Contribution

It provides new insights into the robustness of the fermion-spin mapping in inhomogeneous systems, including boundaries, impurities, and interfaces, using both analytical and numerical approaches.

Findings

Mapping persists despite impurities when using harmonic mean for interactions.

Analytical results show local mapping in inhomogeneous systems.

Numerical results confirm the analytical predictions in interfaces and superlattices.

Abstract

A combined analytical and numerical study is performed of the mapping between strongly interacting fermions and weakly interacting spins, in the framework of the Hubbard, t-J and Heisenberg models. While for spatially homogeneous models in the thermodynamic limit the mapping is thoroughly understood, we here focus on aspects that become relevant in spatially inhomogeneous situations, such as the effect of boundaries, impurities, superlattices and interfaces. We consider parameter regimes that are relevant for traditional applications of these models, such as electrons in cuprates and manganites, and for more recent applications to atoms in optical lattices. The rate of the mapping as a function of the interaction strength is determined from the Bethe-Ansatz for infinite systems and from numerical diagonalization for finite systems. We show analytically that if translational symmetry is broken through the presence of impurities, the mapping persists and is, in a certain sense, as local as possible, provided the spin-spin interaction between two sites of the Heisenberg model is calculated from the harmonic mean of the onsite Coulomb interaction on adjacent sites of the Hubbard model. Numerical calculations corroborate these findings also in interfaces and superlattices, where analytical calculations are more complicated.