Abstract
The effect of supercritical charge impurities in graphene is very similar to the supercritical atomic collapses in QED for Z > 137, but with a much lower critical charge. In this sense graphene can be considered as a natural testing ground for the analysis of quantum field theory vacuum instabilities. We analyze the quantum transition from subcritical to supercritical charge regimes in gapped graphene in a common framework that preserves unitarity for any value of charge impurities. In the supercritical regime it is possible to introduce boundary conditions which control the singular behavior at the impurity. We show that for subcritical charges there are also non-trivial boundary conditions which are similar to those that appear in QED for nuclei in the intermediate regime 118 < Z < 137. We analyze the behavior of the energy levels associated to the different boundary conditions. In particular, we point out the existence of new bound states in the subcritical regime which include a negative energy bound state in the attractive Coulomb regime. A remarkable property is the continuity of the energy spectral flow under variation of the impurity charge even when jumping across the critical charge transition. We also remark that the energy levels of hydrogenoid bound states at critical values of charge impurities act as focal points of the spectral flow.
Highlights
Quarks, which reaches very fast supercritical values in the infrared
The results show a continuous behavior of the corresponding energy levels, the spectral flow is very peculiar: energy levels of hydrogenoid spectrum in the critical regime are focal points of the spectra of subcritical and supercritical regimes
The peculiar behavior of the supercritical regime is reflected by the increasing number of energy levels inside the energy gap, but the continuity of the spectral flow is always preserved along the transitions between the different spectral regimes
Summary
Graphene is a two dimensional layer of carbon atoms arranged on a honeycomb lattice of hexagons. For some values of the impurity charge not all the boundary conditions β0j give rise to normalizable asymptotic zero modes Such boundary conditions do not lead by the procedure described above to a well defined selfadjoint Dirac Hamiltonian. As we shall see later on, it is always possible to find an alternative boundary condition β0j for the same value of impurity charge whose zero mode is normalizable and leads to a well defined selfadjoint Dirac Hamiltonian.. As we shall see later on, it is always possible to find an alternative boundary condition β0j for the same value of impurity charge whose zero mode is normalizable and leads to a well defined selfadjoint Dirac Hamiltonian.3 By this method we have replaced the convergent flow of UV cut-off boundary conditions just by the choice of a simple asymptotic boundary condition (2.14). This phenomenon which is present in QCD provides in that case the physical argument for the opening a mass gap and an anomalous breaking of conformal symmetry [12,13,14,15]
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