Spherical-tensor description of the Jahn--Teller--Hubbard molecule and local electron--phonon entanglement

TL;DR

This paper uses a spherical-tensor formalism to analyze electron-phonon interactions and entanglement in a Jahn--Teller-Hubbard molecule, revealing the nature of multipole moments and their coupling properties.

Contribution

It introduces a novel application of spherical-tensor formalism to study electron-phonon entanglement and quadrupole operators in a molecular system with complex multiplet structure.

Findings

Both electronic quadrupole and lattice displacement moments vanish in the ground state.

Composite quadrupole operators involving electrons and phonons are introduced and shown not to couple to standard quadrupole operators.

Ground state entanglement involves superpositions of multi-phonon states with specific angular momenta.

Abstract

We investigate the localized-electron character of the Mott-insulating phase in A$_3$C$_{60}$ using a single-site multiorbital electron model coupled to anisotropic molecular vibrations (Jahn--Teller phonons). We apply the spherical-tensor formalism, a framework originally developed in nuclear physics, to analyze the electron--phonon-coupled ground-state multiplet. Focusing on multipole moments, we find that both the conventional electronic quadrupole moment and the lattice displacement associated with the molecular vibrations vanish, even though the degenerate ground-state multiplet implies the presence of quadrupolar degrees of freedom. By analyzing these degrees of freedom within the spherical-tensor framework, we introduce composite (two-body) quadrupole operators involving both electrons and phonons and study their parameter dependence numerically. Furthermore, using quasispin selection rules, we demonstrate that the composite quadrupole does not couple to either the conventional quadrupole or lattice-displacement operators, thereby distinguishing it fundamentally from standard quadrupolar degrees of freedom. In addition, we investigate the nature of the electron--phonon entanglement and characterize it from the viewpoint of angular momentum. Analysis of the entanglement spectrum reveals that the ground state consists of superpositions of multi-phonon states with angular momenta $L_{\rm ph}=2$ and $L_{\rm ph}=3$, formed through coupling to three-electron states with $L=1$ and $L=2$.