# Hierarchy of Spin and Valley Symmetry Breaking in Quantum Hall Single Layer Graphene

## TL;DR

This paper investigates the mechanisms behind symmetry breaking in the Landau levels of single-layer graphene, highlighting the dominant role of electron-phonon coupling combined with Zeeman interaction in splitting the levels.

## Contribution

It identifies the key microscopic mechanism—electron-phonon coupling with Zeeman interaction—that causes full splitting of the $n=0$ Landau levels in graphene.

## Key findings

- Electron-phonon coupling and Zeeman interaction lead to complete Landau level splitting.
- Midgap states exist between Landau levels of opposite valley polarity.
- The results clarify the 'valley-first' versus 'spin-first' symmetry breaking controversy.

## Abstract

We explore several microscopic mechanisms for breaking the $n=0$ fourfold Landau level degeneracy in a single-layer graphene. Valley-scattering random potential, Zeeman interaction, and electron-phonon coupling are considered in the presence of SU(4)-symmetric Coulomb exchange interaction. Among all the mechanisms considered, it is the electron-phonon coupling combined with the Zeeman interaction which leads to the full splitting of the $n=0$ Landau levels. A recent controversy of "valley-first" or "spin-first" breaking of SU(4) symmetry of the $n=0$ graphene Landau level is examined in light of our results. Existence of midgap states between Landau levels of opposite valley polarity are demonstrated.