Microscopic Description of Electric and Magnetic Toroidal Multipoles in Hybrid Orbitals

TL;DR

This paper develops a formalism for electric and magnetic toroidal multipoles in hybrid orbitals, revealing their role as fundamental order parameters and their couplings with other degrees of freedom in electronic systems.

Contribution

It introduces a quantum-mechanical operator formalism for electric and magnetic toroidal multipoles and explores their significance as primary order parameters in hybridized orbital systems.

Findings

Electric and magnetic toroidal multipoles can be primary order parameters.

Emergent cross-couplings like magneto-electric effects are demonstrated.

Higher-order toroidal multipoles beyond dipole and monopole are significant.

Abstract

We present a general formalism of multipole descriptions under the space-time inversion group. We elucidate that two types of atomic toroidal multipoles, i.e., electric and magnetic, are fundamental pieces to express electronic order parameters in addition to ordinary electric and magnetic multipoles. By deriving quantum-mechanical operators for both toroidal multipoles, we show that electric (magnetic) toroidal multipole higher than dipole (monopole) can become a primary order parameter in a hybridized-orbital system. We also demonstrate emergent cross-correlated couplings between electric, magnetic, and elastic degrees of freedom, such as magneto-electric and magneto(electro)-elastic couplings, under toroidal multipole orders.