title: [Formula: see text]-symmetric graphene under a magnetic field.
abstract: We propose a [Formula: see text]-symmetrically deformed version of the graphene tight-binding model under a magnetic field. We analyse the structure of the spectra and the eigenvectors of the Hamiltonians around theKandK' points, both in the [Formula: see text]-symmetric and [Formula: see text]-broken regions. In particular, we show that the presence of the deformation parameterVproduces several interesting consequences, including the asymmetry of the zero-energy states of the Hamiltonians and the breakdown of the completeness of the eigenvector sets. We also discuss the biorthogonality of the eigenvectors, which turns out to be different in the [Formula: see text]-symmetric and [Formula: see text]-broken regions.
authors.affiliation: Dipartimento di Energia, Ingegneria dell'Informazione e Modelli Matematici, Facoltà di Ingegneria, Università di Palermo, 90128 Palermo, Italy.
authors.forename: Fabio
authors.identifier: 0000-0002-4454-091X
authors.initials: F
authors.lastname: Bagarello
authors.affiliation: Institute of Industrial Science, University of Tokyo, Komaba 4-6-1, Meguro, Tokyo 153-8505, Japan.
authors.forename: Naomichi
authors.initials: N
authors.lastname: Hatano
keywords: biorthogonal eigenstates; deformed graphene; 𝒫𝒯-symmetric Hamiltonian