# Atomic-Scale Magnetic Toroidal Dipole under Odd-Parity Hybridization

## TL;DR

This paper investigates atomic-scale magnetic toroidal dipoles in solids with odd-parity hybridization, revealing their types, stabilization mechanisms, and effects on magneto-electric responses using a two-orbital model.

## Contribution

It introduces a microscopic two-orbital model capturing atomic-scale MTDs with odd-parity hybridization and analyzes their stabilization and magneto-electric effects.

## Key findings

- Two types of MTDs: in-plane and out-of-plane.
- Coulomb interactions stabilize different MTD orientations.
- Odd-parity hybridization modulates magneto-electric effects.

## Abstract

Magnetic toroidal dipole (MTD) is one of a fundamental constituent to induce magneto-electric effects in the absence of both spatial inversion and time-reversal symmetries. We report on a microscopic investigation of the atomic-scale MTD in solids by taking into account the orbital degree of freedom with a different parity. We construct an effective two-orbital $d$-$f$ tight-binding model on a polar tetragonal system for describing the atomic-scale MTD, which are obtained by incorporating the atomic spin-orbit coupling and odd-parity hybridization. The effective model exhibits two types of the MTDs: in-plane $x, y$ components activated through spontaneous ferromagnetic ordering or external magnetic field and an out-of-plane $z$ component by a spontaneous odd-parity hybridization without spin moments. We show that the intra-orbital (inter-orbital) Coulomb interaction in multi-orbital systems plays an important role in stabilizing the in-plane (out-of-plane) MTD orderings. We also examine the magneto-electric effect under each MTD ordering by calculating a linear response tensor. We show that the odd-parity hybridization enhances the magneto-electric effect for the in-plane MTDs, while it suppresses that for the out-of-plane MTD.