Pseudo-magnetic field in curved graphene

The general covariance of the Dirac equation is exploited in order to explore the curvature effects appearing in the electronic properties of graphene. Two physical situations are then considered: the weak curvature regime, with $\left|R\right|<1/L^2$, and the strong curvature regime, with $1/L^2\ll \left|R\right|<1/d^2$, where $R$ is the scalar curvature, $L$ is a typical size of a sample of graphene and $d$ is a typical size of a local domain where the curvature is pronounced. In the first scenario, we found that the curvature transforms the conical nature of the dispersion relation due to a shift in the momentum space of the Dirac cone. In the second scenario, the curvature in the local domain affects the charge carriers in such a manner that bound states emerge; these states are declared to be pseudo-Landau states because of the analogy with the well known Landau problem; here the curvature emulates the role of the magnetic field. Seeking more tangible curvature effects we calculate e.g. the electronic internal energy and heat capacity of graphene in the small curvature regime and give an expresssion for the ground state energy in the strong curvature regime.