We calculate exactly the vacuum polarization charge density in the field of a subcritical Coulomb impurity, Z|e|/r, in graphene. Our analysis is based on the exact electron Green's function, obtained by using the operator method, and leads to results that are exact in the parameter Zα, where α is the "fine structure constant" of graphene. Taking into account also electron-electron interactions in the Hartree approximation, we solve the problem self-consistently in the subcritical regime, where the impurity has an effective charge Z eff , determined by the localized induced charge. We find that an impurity with bare charge Z = 1 remains subcritical, Z eff α < 1/2, for any α, while impurities with Z = 2, 3 and higher can become supercritical at certain values of α.PACS numbers: 81.05. Uw, 73.43.Cd It has been known for a long time that the single electron dynamics in a monolayer of graphite (graphene) is described by a massless two-component Dirac equation [1,2,3]. A surge of interest in the problem was caused by the recent successful fabrication of graphene [4] and measurements of transport properties [5,6,7,8,9,10], including an unconventional form of the quantum Hall effect. Due to the Coulomb interaction between electrons, graphene represents a peculiar two-dimensional (2D) version of massless Quantum Electrodynamics (QED) [3]. It appears to be much simpler than conventional QED because the interaction is described by the instantaneous 1/r Coulomb's law. On the other hand the Fermi velocity v F ≈ 10 6 m/s ≈ c/300 (c is the velocity of light), and therefore the "fine structure constant" α = e 2 / v F ∼ 1, leading to a strong-coupling version of QED. Below we set = v F = 1. Screening of a charged nucleus due to vacuum polarization is an effect of fundamental importance in QED. This problem was investigated in detail both in the subcritical and supercritical regimes [11,12,13,14]. The problem of charged impurity screening in graphene, which also can be treated in terms of vacuum polarization, has recently received a lot of attention [15,16,17,18,19,20,21,22,23], due to the importance of the problem for transport properties involving charged impurities, as well as for our general understanding of the theory of graphene.To leading order in the weak coupling expansion, Zα ≪ 1, the induced charge is negative and localized at the impurity position, ρ ind = −|e| π 2 (Zα)δ(r), which leads to screening of the impurity potential [20,21,22,24]. We denote by Z|e| the impurity charge, and e = −|e| is the effective electron charge; from now on we refer to Z as the impurity charge with the understanding that it is measured in units of |e|. In graphene, the strong-coupling problem Zα ∼ 1 was recently addressed [20], and it was found that the supercritical regime occurs for Zα > 1/2, where a 1/r 2 tail appears in the induced charge density, while in the subcritical regime Zα < 1/2, the induced charge is always localized at the impurity site. Analytical results were also supplemented by numerical lattice calculations [21], lea...

The self-energy and the vertex radiative corrections to the effect of parity nonconservation in heavy atoms are calculated analytically in orders Zalpha2 and Z2alpha3ln((lambda(C)/r(0)), where lambda(C) and r(0) are the Compton wavelength and the nuclear radius, respectively. The sum of the radiative corrections is -0.85% for Cs and -1.41% for Tl. Using these results, we have performed analysis of the experimental data on atomic parity nonconservation. The values obtained for the nuclear weak charge, Q(W)=-72.90(28)exp(35)theor for Cs, and Q(W)=-116.7(1.2)exp(3.4)(theor) for Tl, agree with predictions of the standard model. As an application of our approach, we have also calculated analytically the dependence of the Lamb shift on the finite size of the nucleus.

The effect of vacuum polarization in the field of an infinitesimally thin solenoid at distances much larger than the radius of solenoid is investigated. The induced charge density and induced current are calculated. Though the induced charge density turned out to be zero, the induced current is a finite periodical function of the magnetic flux ⌽. The expression for this function is found exactly in a value of the flux. The induced current is equal to zero at the integer values of ⌽ / ⌽ 0 as well as at half-integer values of this ratio, where ⌽ 0 =2បc / e is the elementary magnetic flux. The latter is a consequence of the Furry theorem and periodicity of the induced current with respect to magnetic flux. As an example we consider the graphene in the field of solenoid perpendicular to the plane of a sample. The Aharonov-Bohm effect, 1 scattering of a charged particle off an infinitesimally thin solenoid, which is absent in classical electrodynamics, has been investigated in numerous papers, see review.2 Both nonrelativistic 1 and relativistic 3-6 equations have been considered. Similar effects having topological origin have been studied in quantum field theory in Refs 7 and 8. Intensive investigation of the topological effects in condensed-matter systems has been performed recently both experimentally and theoretically in Refs. 9-12. New possibilities to study topological effects in quantum electrodynamics ͑QED͒ have appeared after recent successful fabrication of a monolayer graphite ͑graphene͒, see Ref.13 and recent review.14 The single-electron dynamics in graphene is described by a massless two-component Dirac equation [15][16][17][18] so that graphene represents a peculiar twodimensional ͑2D͒ version of massless QED. This version is essentially simpler than conventional QED because effects of retardation are absent in graphene. However, the "fine structure constant" ␣ = e 2 / បv F ϳ 1, since the Fermi velocity v F Ϸ 10 6 m / s Ϸ c / 300 ͑where c is the velocity of light͒, and therefore we have a strong-coupling version of QED. Below we set ប = c =1.Existence of induced charge density in the electric field of heavy nucleus due to vacuum polarization is one of the most important effects of QED. This problem was investigated in detail in many papers, see, e.g., Refs. 19-22. Charged impurity screening in graphene can also be treated in terms of vacuum polarization. [23][24][25][26][27][28][29][30][31][32][33] In Ref. 32, the induced charge density in graphene has been investigated analytically using convenient integral representation for the Green's function of the two-dimensional Dirac equation of electron in a Coulomb field. Calculation of the induced charge has been performed exactly in the charge of impurity. In Ref. 32, the Green's function has been obtained following the method based on the operator technique suggested in Ref. 34. In the present Brief Report, we use similar integral representation for the Green's function to derive the induced current in the field of infinitesimally thin solenoid. Calculati...

Traditional wave kinetics describes the slow evolution of systems with many degrees of freedom to equilibrium via numerous weak non-linear interactions and fails for very important class of dissipative (active) optical systems with cyclic gain and losses, such as lasers with non-linear intracavity dynamics. Here we introduce a conceptually new class of cyclic wave systems, characterized by non-uniform double-scale dynamics with strong periodic changes of the energy spectrum and slow evolution from cycle to cycle to a statistically steady state. Taking a practically important example-random fibre laser-we show that a model describing such a system is close to integrable non-linear Schrödinger equation and needs a new formalism of wave kinetics, developed here. We derive a non-linear kinetic theory of the laser spectrum, generalizing the seminal linear model of Schawlow and Townes. Experimental results agree with our theory. The work has implications for describing kinetics of cyclical systems beyond photonics.