# Spin-1/2 Landau levels in the symmetric gauge from the zero energy modes

## TL;DR

This paper introduces a new analytical method to derive Landau level eigenstates and eigenvalues in the symmetric gauge using zero energy modes, applicable to graphene systems, avoiding numerical solutions.

## Contribution

A novel mechanism to construct Landau level eigenstates directly from zero modes of the Dirac-Weyl equation, simplifying analysis in symmetric gauge systems.

## Key findings

- Eigenstates for negative magnetic fields derived from positive chirality zero modes.
- Eigenstates for positive magnetic fields derived from negative chirality zero modes.
- Method applicable to bilayer graphene Hamiltonian analysis.

## Abstract

Starting from the zero modes of the Dirac-Weyl equation for Landau levels in the symmetric gauge, we propose a novel mechanism to construct the eigenvalues and its eigenfunctions. We show that the problem may be addressed without numerical calculation and only solving the Dirac-Weyl equation for the zero modes. Specifically, the eigenstates associated to the negative magnetic field configurations may be constructed from the zero mode with positive chirality. In addition, we obtain that the eigenstates associated to the positive magnetic field configurations may be constructed from the zero mode with negative chirality. Finally, we show that our mechanism may be used to obtain the eigenvalues and eigenfunctions of the Hamiltonian corresponding to bilayer graphene system.