Anomalous Quantum Hall Effect of 4D Graphene in Background Fields

Abstract

Boriçi-Creutz (\emph{BC}) model describing the dynamics of light quarks in lattice \emph{QCD} has been shown to be intimately linked to the four dimensional extension of 2D graphene refereed below to as four dimensional graphene (\emph{4D-graphene}). Borrowing ideas from the field theory description of the usual \emph{2D} graphene, we study in this paper the anomalous quantum Hall effect (AQHE) of the \emph{BC} fermions in presence of a constant background field strength ${\mathcal{F}}_{\mu \nu }$ with a special focuss on the case ${\mathcal{F}}_{\mu \nu }=\mathcal{B}{ϵ}_{\mu \nu 34}+\mathcal{E}{ϵ}_{12\mu \nu }$ with $\mathcal{B}$ and $\mathcal{E}$ two real\ moduli and $\det {\mathcal{F}}_{\mu \nu }={\mathcal{B}}^2{\times \mathcal{E}}^2$. First, we revisit the anomalous \emph{2D} graphene by using \emph{QFT} method. Then, we consider the AQHE of \emph{BC}fermions for both regular $\det {\mathcal{F}}_{\mu \nu }\ne 0$ and singular $\det F_{\mu \nu }=0$ cases. We show, amongst others, that the exact solutions of the \emph{BC} fermions coupled to constant ${\mathcal{F}}_{\mu \nu }$ have a 5D interpretation; and the filling factor ${\nu }_{BC}$ of the \emph{BC}\ model coupled to constant ${\mathcal{F}}_{\mu \nu }$ is given by $24$ with $N,$ $M$ positive integers. Others features, such as ${\mathcal{F}}_{\mu \nu }^{QCD}\ne 0$ \textrm{and the extension of the obtained results to the lattice fermions like Karsten-Wilzeck (KW) fermions and naive ones}, are also discussed. Key words: Lattice QCD, Boriçi-Creutz fermions, Anomalous Quantum Hall Effect, filling factor, four dimensional graphene, Index theorem, Spectral flow.

Comments: LaTex, 37 pages, 2 figures, To appear in JHEP