# Microscopic Theory of Spin Toroidization in Periodic Crystals

## TL;DR

This paper develops a gauge-invariant semiclassical theory for spin toroidization in periodic crystals, revealing its components, relation to magnetoelectric effects, and demonstrating its bulk nature through a tight-binding model.

## Contribution

It introduces a novel gauge-invariant expression for spin toroidization, linking it to magnetoelectric polarizability and establishing its bulk property in crystals.

## Key findings

- Derived a gauge-invariant formula for spin toroidization.
- Linked spin toroidization to antisymmetric magnetoelectric polarizability.
- Confirmed spin toroidization as a genuine bulk quantity in a model.

## Abstract

Using the semiclassical theory of electron dynamics, we derive a gauge-invariant expression for the spin toroidization in a periodical crystal. We show that the spin toroidization is comprised of two contributions: one is due to the configuration of a classical spin array, while the other comes from the coordinate shift of the electron as spin carrier in response to the inhomogeneous magnetic field. We then establish a direct and elengant relation between our spin toroidization and the antisymmetric magnetoelectric polarizability in insulators. Finally, we demonstrate our spin toroidization in a tight-binding model and show that it is a genuine bulk quantity.