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Supercriticality of novel type induced by electric dipole in gapped graphene

We reveal a new type of supercritical behavior in gapped graphene with two oppositely charged impurities by studying the two-dimensional Dirac equation for quasiparticles with the Coulomb potential regularized at small distances accounting the lattice effects. By utilizing the variational Galerkin--Kantorovich method, we show that for supercritical electric dipole the wave function of the electron bound state changes its localization from the negatively charged impurity to the positively charged one as the distance between the impurities changes. Such a migration of the wave function corresponds to the electron and hole spontaneously created from the vacuum in bound states screening the positively and negatively charged impurities of the supercritical electric dipole, respectively. We generalize our results to a particle-hole asymmetric case, where the charges of impurities differ in signs and absolute values and demonstrate that the necessary energetic condition for the supercriticality of novel type to occur is that the energy levels of single positively and negatively charged impurities traverse together the energy distance separating the upper and lower continua. The robustness of the supercriticality of novel type is confirmed by the study of an exactly solvable 1D problem of the Dirac equation with the square well and barrier potential modeling an electric dipole potential.