Effective Crystalline Electric Field Potential in a j-j Coupling Scheme

TL;DR

This paper develops an effective model based on a $j$-$j$ coupling scheme to accurately describe local $f$-electron states, including crystalline electric field effects, bridging the gap between weak and intermediate coupling regimes for realistic materials.

Contribution

It introduces a systematic method to incorporate CEF effects into the $j$-$j$ coupling scheme up to first order in $1/\lambda$, improving the description of $f$-electron systems in intermediate coupling.

Findings

CEF energy levels are accurately reproduced in the intermediate coupling regime.

The model effectively bridges $LS$ and $j$-$j$ coupling schemes for real materials.

Application to filled skutterudites demonstrates the model's practical relevance.

Abstract

We propose an effective model on the basis of a $j$-$j$ coupling scheme to describe local $f$-electron states for realistic values of Coulomb interaction $U$ and spin-orbit coupling $\lambda$, for future development of microscopic theory of magnetism and superconductivity in $f^n$-electron systems, where $n$ is the number of local $f$ electrons. The effective model is systematically constructed by including the effect of a crystalline electric field (CEF) potential in the perturbation expansion in terms of $1/\lambda$. In this paper, we collect all the terms up to the first order of $1/\lambda$. Solving the effective model, we show the results of the CEF states for each case of $n$=2$\sim$5 with $O_{\rm h}$ symmetry in comparison with those of the Stevens Hamiltonian for the weak CEF. In particular, we carefully discuss the CEF energy levels in an intermediate coupling region with $\lambda/U$ in the order of 0.1 corresponding to actual $f$-electron materials between the $LS$ and $j$-$j$ coupling schemes. Note that the relevant energy scale of $U$ is the Hund's rule interaction. It is found that the CEF energy levels in the intermediate coupling region can be quantitatively reproduced by our modified $j$-$j$ coupling scheme, when we correctly take into account the corrections in the order of $1/\lambda$ in addition to the CEF terms and Coulomb interactions which remain in the limit of $\lambda$=$\infty$. As an application of the modified $j$-$j$ coupling scheme, we discuss the CEF energy levels of filled skutterudites with $T_{\rm h}$ symmetry.