Dirac fermions in graphene under curved spacetime: a study of Beltrami geometry and $${{\mathcal {CPT}}}$$ symmetry

Abstract We analyze the dynamics of Dirac fermions in graphene set within Beltrami spacetime, characterized by a negative curvature. This research investigates how both curvature and external electromagnetic fields influence electron motion, particularly in terms of Landau level quantization and the quantum Hall effect. By utilizing Darboux transformations, we construct novel solvable potentials along with their associated solutions. Furthermore, we assess the role of $${\mathcal {CPT}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>CPT</mml:mi> </mml:math> symmetry in shaping the Hamiltonian and derive the metric operator $$\eta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>η</mml:mi> </mml:math> within the framework of pseudo-Hermitian quantum mechanics. This study sheds light on the intricate relationship between curvature, symmetry, and quantum behavior in graphene-based systems. We have tested our results using numerical calculations.