The critical transition of Coulomb impurities in gapped graphene
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Springer
Received: May Revised: July Accepted: July Published: August
11, 13, 31, 27,
2020 2020 2020 2020
M. Asorey and A. Santagata Centro de Astropart´ıculas y F´ısica de Altas Energ´ıas, Departamento de F´ısica Te´ orica, Universidad de Zaragoza, C/ Pedro Cerbuna 12, 50009 Zaragoza, Spain
E-mail: [email protected], [email protected] 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. Keywords: Boundary Quantum Field Theory, Integrable Field Theories, Renormalization Group, Topological States of Matter ArXiv ePrint: 2005.04175
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP08(2020)144
JHEP08(2020)144
The critical transition of Coulomb impurities in gapped graphene
Contents 1 Introduction
1
2 Charged impurities in graphene 2.1 2D Dirac Hamiltonian in a Coulomb background 2.2 Boundary conditions for different regimes
3 4 7 9 11 11 12 12 12 12 13
4 Spectral flows of bound states 4.1 Spectral flow and boundary conditions 4.2 Spectral flow and impurity charges
13 13 17
5 Conclusions
21
1
Introduction
The stability of non-relativistic hydrogenoid atoms is one of the essential features that contributed to consolidate the quantum theory. However in relativistic quantum mechanics there is a critical value of the central point-like charge Ze from where on atom stability is lost [1–4]. This is one of the surprising consequences of relativistic invariance in QED. The phenomenon can be understood in a heuristic way as a falling to the center catastrophe. The critical value in QED is reached when the spectrum of bound states of Dirac equation becomes complex which occurs for Z > 137. In fact what happens in QED is that when one of the bound states reaches
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