Pseudomagnetic field in curved graphene

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