Anomalous Quantum Hall Effect of 4D Graphene in Background Fields

Bori\c{c}i-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 $\mathcal{F}_{\mu \nu}=\mathcal{B}\epsilon_{\mu \nu 34}+\mathcal{E}\epsilon_{12\mu \nu}$ with $\mathcal{B}$ and $\mathcal{E}$ two real\ moduli and $\det \mathcal{F}_{\mu \nu}=\mathcal{B}^{2}\mathcal{\times E}^{2}$. 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 $\det \mathcal{F}_{\mu \nu}\neq 0$ 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 $\mathcal{F}_{\mu \nu}^{QCD}\neq 0$ \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, Bori\c{c}i-Creutz fermions, Anomalous Quantum Hall Effect, filling factor, four dimensional graphene, Index theorem, Spectral flow.