Nonperturbative vacuum polarization effects in two-dimensional supercritical Dirac–Coulomb system I. Vacuum charge density

Abstract

Based on the original combination of analytical methods, computer algebra tools and numerical calculations, proposed recently in Refs.[1]-[3], the non-perturbative vacuum polarization effects in the 2+1 D supercritical Dirac-Coulomb system with Z > Zcr,1 are explored. Both the vacuum charge density ρV P ( r) and vacuum energy EV P are considered. The main result of the work is that in the overcritical region EV P turns out to be a rapidly decreasing function ∼ −η ef f Z 3 /R with η ef f > 0 and R being the size of the external Coulomb source. Due to a lot of details of calculation the whole work is divided into two parts I and II. In the present part I we consider the evaluation and behavior of the vacuum density ρV P , which further is used in the part II for evaluation of the vacuum energy, with emphasis on the renormalization, convergence of the partial expansion for ρV P and behavior of the integral induced charge QV P in the overcritical region.

Ferromagnetic Phase in Graphene‐Based Planar Heterostructures Induced by Charged Impurity

“…For the planar case the renormalization of the induced charge density in absence of magnetic field has been considered in detail in Refs. [2,22,23]. In the present case with the dipole-like magnetic field the crucial role is played by the fact that the corresponding vectorpotential behaves for ρ → 0 and ρ → ∞ as A ϕ ∼ ρ and A ϕ ∼ 1/ρ 2 , respectively.…”

“…vac ( ρ) disappears as the first order effect of the perturbation theory, whereas vanishing of the contribution from ̺ (3+) vac,|mj| (ρ) to Q ren vac is verified via direct numerical calculation (in more details this question is considered in Ref. [22] and especially in Ref. [23]).…”

“…After these preliminary steps the renormalization of the vacuum charge density reduces to replacement of the linear term by renormalized first order perturbative density ̺ (1) vac ( ρ), which has been evaluated in Refs. [2,22,23], and takes the form…”

“…The general properties of the renormalized induced charge density coincide with those in absence of magnetic field (see, e.g., Refs. [2,22,23] and citations therein). In particular, the expression (36) provides vanishing of the total induced charge Q ren vac = d 2 ρ ̺ ren vac ( ρ) for Z < Z cr,1 .…”

Super-critical QED-effects via VP-energy

“…In (26) δ tot (l, k) is the total phase shift for the given values of the wavenumber k and orbital momentum l, including the contributions of the scattering states of the problem (23,24) from both continua and both parities.…”