Supersymmetry and Unconventional Quantum Hall Effect in Graphene

TL;DR

This paper provides a unified theoretical framework for the quantum Hall effect in graphene using supersymmetric quantum mechanics, revealing the role of SUSY in zero-energy states and extending the Dirac Hamiltonian to include new indices affecting the Hall conductivity.

Contribution

It introduces a supersymmetric approach to describe quantum Hall states in graphene and extends the Dirac Hamiltonian with new indices to account for dispersion and Berry phase effects.

Findings

Zero-energy state arises from SUSY symmetry with equal Zeeman and Landau level splitting.

Nonzero energy states form supermultiplets with spin SU(2) symmetry.

Quantized Hall conductivity depends on new indices and their relation to dispersion and Berry phase.

Abstract

We present a unified description of the quantum Hall effect in graphene on the basis of the 8-component Dirac Hamiltonian and the supersymmetric (SUSY) quantum mechanics. It is remarkable that the zero-energy state emerges because the Zeeman splitting is exactly as large as the Landau level separation, as implies that the SUSY is a good symmetry. For nonzero energy states, the up-spin state and the down-spin state form a supermultiplet possessing the spin SU(2) symmetry. We extend the Dirac Hamiltonian to include two indices $j_{\uparrow}$ and $j_{\downarrow}$, characterized by the dispersion relation $E(p) \propto p^{j_{\uparrow}+j_{\downarrow}}$ and the Berry phase $\pi (j_{\uparrow}-j_{\downarrow})$. The quantized Hall conductivity is shown to be $\sigma_{xy}=\pm (2n+j_{\uparrow}+j_{\downarrow}) 2e^{2}/h$.