# The flea on the Magnetic Elephant

## TL;DR

The paper shows how a tiny symmetry violation in a magnetic system can drastically change where particles are localized.

## Contribution

The paper demonstrates a magnetic counterpart to the 'flea on the elephant' effect in quantum systems.

## Key findings

- Exponentially small symmetry violation can localize eigenfunctions dominantly in one magnetic well.
- The effect is similar to the 'flea on the elephant' phenomenon in Schrödinger operators.
- Tunneling between symmetric magnetic wells leads to equal localization until symmetry is broken.

## Abstract

We investigate a two-dimensional magnetic Laplacian with two radially symmetric magnetic wells. Its spectral properties are determined by the tunneling between them. If the tunneling is weak and the wells are mirror symmetric, the two lowest eigenfunctions are localized in both wells being distributed roughly equally. In this note, we show that an exponentially small symmetry violation can in this situation have a dramatic effect, making each of the eigenfunctions localized dominantly in one well only. This is reminiscent of the ‘flea on the elephant’ effect for Schrödinger operators; our result shows that it has a purely magnetic counterpart.