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Möbius-topological auxiliary function for $f$ electrons

Biaoyan Hu

TL;DR

The paper addresses the challenge of reliably determining $f$-electron eigenstates in solids where strong spin–orbit coupling and crystal-field effects create complex, computationally demanding energy landscapes. It introduces a Möbius-topological auxiliary framework built from a modified Legendre function with a sign-corrected half-integer extension, yielding $\psi(r,\theta,\varphi) \propto P_J^M(\cos\theta) e^{iM\varphi}$ whose auxiliary density respects crystal-field symmetry and exhibits inversion symmetry. The radial part is hydrogen-like with an effective nuclear charge $\tilde{Z}$ from Slater’s rules, ensuring normalization via $R(r)=\left(\frac{2\tilde{Z}}{n a_0}\right)^{3/2}R_n^l\left(\frac{2\tilde{Z} r}{n a_0}\right)$ and $l=3$ for $f$-orbitals. This approach enables rapid inference of eigenstate structures without full diagonalization, matching established results (e.g., NdCl$_3$) and providing significant computational simplification. The framework offers potential extensions to other strongly correlated or geometrically frustrated systems, linking symmetry and topology to electronic structure in a practical, predictive manner.

Abstract

$f$-electron systems exhibit a subtle interplay between strong spin--orbit coupling and crystal-field effects, producing complex energy landscapes that are computationally demanding. We introduce auxiliary functions, constructed by extending hydrogen-like wave functions through a modification of the Legendre function. These functions often possess a Möbius-like topology, satisfying $ψ(\varphi) = -ψ(\varphi + 2π)$, while their squared modulus respects inversion symmetry. By aligning $|ψ|^2$ with the symmetry of the crystal field, they allow rapid determination of eigenstate structures without the need for elaborate calculations. The agreement with established results indicates that these functions capture the essential physics while offering considerable computational simplification.

Möbius-topological auxiliary function for $f$ electrons

TL;DR

The paper addresses the challenge of reliably determining -electron eigenstates in solids where strong spin–orbit coupling and crystal-field effects create complex, computationally demanding energy landscapes. It introduces a Möbius-topological auxiliary framework built from a modified Legendre function with a sign-corrected half-integer extension, yielding whose auxiliary density respects crystal-field symmetry and exhibits inversion symmetry. The radial part is hydrogen-like with an effective nuclear charge from Slater’s rules, ensuring normalization via and for -orbitals. This approach enables rapid inference of eigenstate structures without full diagonalization, matching established results (e.g., NdCl) and providing significant computational simplification. The framework offers potential extensions to other strongly correlated or geometrically frustrated systems, linking symmetry and topology to electronic structure in a practical, predictive manner.

Abstract

-electron systems exhibit a subtle interplay between strong spin--orbit coupling and crystal-field effects, producing complex energy landscapes that are computationally demanding. We introduce auxiliary functions, constructed by extending hydrogen-like wave functions through a modification of the Legendre function. These functions often possess a Möbius-like topology, satisfying , while their squared modulus respects inversion symmetry. By aligning with the symmetry of the crystal field, they allow rapid determination of eigenstate structures without the need for elaborate calculations. The agreement with established results indicates that these functions capture the essential physics while offering considerable computational simplification.

Paper Structure

This paper contains 4 sections, 26 equations, 2 figures, 2 tables.

Table of Contents

  1. radial factor of the auxiliary function
  2. About Legendre functions
  3. 4 important properties
  4. Hobson's Legendre function

Figures (2)

  • Figure 1: Comparison of $P_{5/2}^{3/2}(x)$ from Hobson hobson1895onhobson1931theory, Hunter et al.hunter1999Fermionhunter2005properties, and the present work. Hobson's $P_{5/2}^{3/2}(x) = (8x^4 - 12x^2 + 3)/\sqrt{\frac{\pi}{2}\sqrt{(1-x^2)^3}}$ is an even function that diverges as $x \to \pm 1$, with a squared modulus that is non-integrable. Hunter's $P_{5/2}^{3/2}(x) = x (1-x^2)^{3/4}$ is an odd function and integrable. In contrast, as presented here in this work, $P_{5/2}^{3/2}(x) = 8|x|\sqrt{2/\pi}\,(1-x^2)^{3/4}$ is an even function that is integrable.
  • Figure 2: (a) The factor $e^{i\varphi/2}$ is plotted along a circular arc of radius 3, with its vector trajectory forming a Möbius strip. (b) Auxiliary densities distribution of $f$-electron states in Nd$^{3+}$ ($J=9/2$). Kets $|M\rangle$ denote the quantum states. The auxiliary densities exhibit O(2) symmetry, and the surface encloses 50% of the total density.