Non-Convergent Perturbation Theory and Misleading Inferences about Parameter Relationships: the Case of Superexchange

TL;DR

This paper critically examines the limitations of perturbation theory in modeling superexchange interactions, revealing that common series expansions can be misleading and do not reliably reflect the true parameter relationships in transition-metal compounds.

Contribution

It demonstrates that no single limiting expansion accurately describes superexchange due to multiple interfering energy scales, challenging traditional perturbative approaches.

Findings

The superexchange $J$ cannot be formally defined in the idealized M-H limit.

Multiple different expansions can produce the same $t^4$ expression, leading to ambiguity.

Fitting experimental data with simple series expansions can result in incorrect parameter inferences.

Abstract

We discuss the well-known three-center cation-anion-cation model for superexchange in insulating transition-metal compounds using limiting expansions for the Anderson-Hubbard model. We find that due to the three interfering energy scales in the model, a limiting expression for the superexchange $J$ for the idealized Mott-Hubbard (M-H) case $t\ll U\ll \Delta$ cannot be formally defined. We further show that no single expansion variable can describe any type of limiting behaviour for superexchange. The well-known $t^4$ expression for M-H insulators, obtained from path-dependent series expansions, is not unique to these systems as it can also be obtained with many other different expansions, in which either the $d-p$ energy difference $\Delta$ or the $d$-electron correlation $U$ can actually be small. At times, and particularly for milder relationships between the parameters, $t\lesssim U\lesssim\Delta,$ the reverse form of the series expansions can yield better agreement with the exact results. This implies that the fitting of experimental data to the simple expressions derived from path-dependent series expansions can lead to qualitatively incorrect relationships between the parameters, fictitiously within the M-H regime.