Why magnetic monopole becomes dyon in topological insulators

TL;DR

This paper provides a microscopic explanation for the Witten effect in topological insulators, showing how a magnetic monopole becomes a dyon due to localized zero modes and domain-wall effects, confirmed through analytical and numerical methods.

Contribution

It introduces a detailed microscopic model incorporating Wilson terms and gauge smearing to explain the monopole-dyon transition in topological insulators.

Findings

Wilson term induces a positive mass shift creating a domain-wall.

Domain-wall localizes chiral zero modes ensuring their stability.

Fractional electric charge explained via Atiyah-Singer index theorem.

Abstract

The Witten effect predicts that a magnetic monopole acquires a fractional electric charge inside topological insulators. In this work, we give a microscopic description of this phenomenon, as well as an analogous two-dimensional system with a vortex. We solve the Dirac equation of electron field both analytically in continuum and numerically on a lattice, by adding the Wilson term and smearing the gauge field within a finite range to regularize the short-distance behavior of the system. Our results reveal that the Wilson term induces a strong positive mass shift, creating a domain-wall around the monopole/vortex. This small, yet finite-sized domain-wall localizes the chiral zero modes and ensures their stability through the Atiyah-Singer index theorem, whose cobordism invariance is crucial in explaining why the electric charge is fractional.