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Anomalous Quantum Hall Effect of 4D Graphene in Background Fields

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 $\mathcal{F}_{μν}$ with a special focuss on the case $\mathcal{F}_{μν}=\mathcal{B}ε_{μν34}+\mathcal{E}ε_{12μν}$ with $\mathcal{B}$ and $ \mathcal{E}$ two real\ moduli and $\det \mathcal{F}_{μν}=\mathcal{B}^{2}\mathcal{\times E}^{2}$. 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 $\det \mathcal{F}_{μν}\neq 0$ and singular $ \det F_{μν}=0$ cases. We show, amongst others, that the exact solutions of the \emph{BC} fermions coupled to constant $\mathcal{F}_{μν}$ have a 5D interpretation; and the filling factor $ν_{BC}$ of the \emph{BC}\ model coupled to constant $\mathcal{F}_{μν}$ is given by $24% \frac{(2N+1) (2M+1)}{2}$ with $N,$ $M$ positive integers. Others features, such as $\mathcal{F}_{μν}^{QCD}\neq 0$ \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, Boriçi-Creutz fermions, Anomalous Quantum Hall Effect, filling factor, four dimensional graphene, Index theorem, Spectral flow.