Essentially Nonperturbative Vacuum Polarization Effects in a Two-Dimensional Dirac–Coulomb System with Z > Zcr: Vacuum Charge Density

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

“…[22] and especially in Ref. [23]). The change in Q ren vac can occur only for Z > Z cr,1 due to discrete levels diving into the lower continuum, wherein each dived level ψ n,mj ( ρ) shifts the total induced charge by (−|e|), while the charge density ̺ ren vac ( ρ) undergoes the following change [20,24] ∆̺ ren vac ( ρ) = −|e| ψ ǫn=−1,mj ( ρ) † ψ ǫn=−1,mj ( ρ) .…”

“…This result justifies also the WK contours deformation to imaginary axis, considered above, because in the purely Coulomb planar systems the latter is reliably verified in Refs. [2,22,23]. In turn, the properties of the limit |m j | → ∞ follow directly from the systems (11)(12), since in this case the magnetic field again turns out to be a perturbation in the Coulomb background.…”

“…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

“…(41) At the same time, the discrete spectrum with the same conditions on l (that means l → ∞ or at least l subject to condition (27)) corresponds with a high precision to well-known solution of the Coulomb-Schroedinger problem for a point-like source with the same Z. In this limit the main contribution to the total sum of discrete levels comes from the vicinity of condensation point ǫ n,l → 1, where both parities reveal the same properties and so can be freely treated at the same footing.…”