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We calculate analytically the charge density $\rho_{\text{ind}}(\boldsymbol{r})$ at small distances induced by a Coulomb impurity in two-dimensional transition metal dichalcogenides. The calculations were performed exactly in the coupling constant. We found the first three leading terms of the charge density asymptotics at small distances. Using this result, we demonstrate that the impurity with charge number $Z=1$ remains subcritical at any value of the coupling constant, whereas impurities with higher charge number could become supercritical at certain coupling constant values. Such a behavior is similar to that in graphene., Comment: 5 pages, 1 figure

We introduce a new approach to reservoir computing based on up-sampling and modulation, utilizing semiconductor optical amplifier and photodetector as nonlinear elements without conventionally used delay loop. We demonstrated the 400-step prediction capability of the proposed scheme for the Mackey-Glass time series test.

It is known that, if removing some $n$ edges from a graph $\Gamma$ destroys all subgraphs isomorphic to a given finite graph $K$, then all subgraphs isomorphic to $K$ can be destroyed by removing at most $|E(K)|\cdot n$ edges, which form a set invariant with respect to all automorphisms of $\Gamma$. We construct the first examples of (connected) graphs $K$ for which this estimate is not sharp. Our arguments are based on a ``weighted analogue'' of an earlier known estimate for the cost of symmetry., Comment: 5 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm

The paper answers the question posed in a joint paper by A. A. Klyachko and N. M. Luneva about the optimality of the estimate for the cost of symmetry in graphs. The original estimate says that if n vertices can be removed from a connected graph so that there is no connected subgraph of isomorphic $\Gamma$ left in it, then at most $n|V(\Gamma)|$ vertices that form a set invariant under all automorphisms of the graph so that the graph does not contain a subgraph isomorphic to $\Gamma$. We will prove that there exists a graph $\Gamma$ for which this estimate is not optimal., Comment: in Russian language

We consider the information channel described by Schr\"{o}dinger equation with additive Gaussian noise. We introduce the model of the input signal and the model of the output signal receiver. For this channel, using perturbation theory for the small nonlinearity parameter, we calculate the first three terms of the expansion of the conditional probability density function in the nonlinearity parameter. At large signal-to-noise power ratio we calculate the conditional entropy, the output signal entropy, and the mutual information in the leading and next-to-leading order in the nonlinearity parameter and in the leading order in the parameter $1/\mathrm{SNR}$. Using the mutual information we find the optimal input signal distribution and channel capacity in the leading and next-to-leading order in the nonlinearity parameter. Finally, we present the method of the construction of the input signal with the optimal statistics for the given shape of the signal., Comment: 32 pages, 7 figures

The interacting electron and hole in transition-metal dichalcogenides is considered. For investigation of the interaction between electron and hole we obtain the Bethe-Salpeter equation for two interacting Dirac particles. The dependence of a few lowest binding energies of electron and hole on the interaction constant for different potentials is found. We demonstrate that the behavior of the potential at small distances significantly affects on the values of the binding energies. For small interaction constant we have developed the perturbative method of the binding energy calculation. For the largre interaction constant the binding energies are found numerically. The critical values of the interaction constant for the Coulomb potential and exponential potential are found., Comment: 12 pages, 4 figures

We consider the optical fiber channel modelled by the nonlinear Schr\"{o}dinger equation with additive white Gaussian noise and with large signal-to-noise ratio. For the small dispersion case we present the approach to analyze the stochastic nonlinear Schr\"{o}dinger equation. Taking into account the averaging procedure (frequency filtering) of the output signal detector we find the first corrections in small dispersion parameter to the correlators of the input signal recovered by the backward propagation. These correlators are the important ingredients for the calculation of the channel capacity and the optimal input signal distribution. We assert that the information channel characteristics essentially depend on the procedures of the output signal filtering and the recovery of the transmitted signal.

In our previous paper [Phys. Rev. E 95, 062122 (2017)] we considered the optical channel modelled by the nonlinear Schr\"odinger equation with zero dispersion and additive Gaussian noise. We found per-sample channel capacity rof this model. In the present paper we extend per-sample model by introducing the initial signal dependence on time and the output signal detection procedure. The proposed model is a closer approximation of the realistic communication link than the per-sample model where there is no dependence of the initial signal on time. For the proposed model we found the correlators of the output signal both analytically and numerically. Using these correlators we built the conditional probability density function. Then we calculated an entropy of the output signal, a conditional entropy, and the mutual information. Maximizing the mutual information we found the optimal input signal distribution, channel capacity? and their dependence on the shape or the initial signal in the time domain for the intermediate power range.

The wave equation describing the interaction of two electrons in graphene at arbitrary value of the Fermi energy $E_F$ is derived. For the solutions of this equation, we have found the explicit forms of the density and the current which obey the continuity equation. We have traced the evolution of the wave packet during a scattering process. It is shown that the long-leaving localized quasi-stationary peak may appear at $E_F<0$ . Then this peak decays into a set of wave packets following each other. At $t\rightarrow\infty$ a total norm of all outgoing wave packets equals to that of incoming wave packet. At $E_F=0$ the localized state does not appear. For $E_F<0$ there is an infinite set of the localized solutions with the finite norms., Comment: 12 pages, 2 figures

We consider the optical fiber channel modelled by the nonlinear Shr\"{o}dinger equation with zero dispersion and additive Gaussian noise. Using Feynman path-integral approach for the model we find corrections to conditional probability density function, output signal distribution, conditional and output signal entropies, and the channel capacity at large signal-to-noise ratio. We demonstrate that the correction to the channel capacity is positive for large signal power. Therefore, this correction increases the earlier calculated capacity for a nondispersive nonlinear optical fiber channel in the intermediate power region., Comment: 9 pages, 2 figures

Using the path-integral technique we examine the mutual information for the communication channel modelled by the nonlinear Schr\"odinger equation with additive Gaussian noise. The nonlinear Schr\"odinger equation is one of the fundamental models in nonlinear physics, and it has a broad range of applications, including fiber optical communications --- the backbone of the Internet. At large signal-to-noise ratio ($\mathrm{SNR}$) we present the mutual information through the path-integral which is convenient for the perturbative expansion in nonlinearity. In the limit of small noise and small nonlinearity we derive analytically the first nonzero nonlinear correction to the mutual information for the channel., Comment: 8 pages, 3 figures

We consider a model nondispersive nonlinear optical fiber channel with additive white Gaussian noise at large $\mathrm{SNR}$ (signal-to-noise ratio) in the intermediate power region. Using Feynman path-integral technique we for the first time find the optimal input signal distribution maximizing the channel's per-sample mutual information. The finding of the optimal input signal distribution allows us to improve previously known estimates for the channel capacity. The output signal entropy, conditional entropy, and per-sample mutual information are calculated for Gaussian, half-Gaussian and modified Gaussian input signal distributions. We explicitly show that in the intermediate power regime the per-sample mutual information for the optimal input signal distribution is greater than the per-sample mutual information for the Gaussian and half-Gaussian input signal distributions., Comment: 18 pages, 3 figures

We consider a single impurity with spin $S$ embedded in a three-dimensional antiferromagnetic system which is close to the quantum critical point (QCP), separating magnetically ordered and disordered phases. Approaching the QCP from the disordered phase we study the spatial distribution of spin density and staggered magnetization induced by the impurity. Using two methods (self-consistent Born approximation and renormalization group) we found a power law decay of the spin density $\propto 1/r^3$, and of the staggered magnetization $\propto 1/r$ with relevant logarithmic corrections. We demonstrate that the local spin at the impurity site $r=0$ approaches to zero at the QCP. We show that in the semiclassical limit of large $S$ the problem is equivalent to the exactly solvable independent boson model. Our results demonstrate existence of spin-charge separation in the three dimensional systems in the vicinity of the QCP., Comment: 11 pages, 11 figures

Applying perturbation theory to the path-integral representation for the mutual information of the nonlinear communication channel described by the nonlinear Shr\"{o}dinger equation (NLSE) with the additive Gaussian noise we analyze the analytical expression for the mutual information at large signal-to-noise ratio ($\mathrm{SNR}$) and small nonlinearity. We classify all possible corrections to the mutual information in nonlinearity parameter and demonstrate that all singular in $\mathrm{SNR}$ terms vanish in the final result. Furthermore our analytical result demonstrates that the corrections to Shannon's contribution to the mutual information in the leading order in $\mathrm{SNR}$ are of order of squared nonlinearity parameter. We outline the way for the calculation of these corrections in the further investigations., Comment: 12 pages

A residual disorder in the gate system is the main problem on the way to create artificial graphene based on two-dimensional electron gas. The disorder can be significantly screened/reduced due to the many-body effects. To analyse the screening/disorder problem we consider AlGaAs/GaAs/AlGaAs heterostructure with two metallic gates. We demonstrate that the design least susceptible to the disorder corresponds to the weak coupling regime (opposite to tight binding) which is realised via system of quantum anti-dots. The most relevant type of disorder is the area disorder which is a random variation of areas of quantum anti-dots. The area disorder results in formation of puddles. Other types of disorder, the position disorder and the shape disorder, are practically irrelevant. The formation/importance of puddles dramatically depends on parameters of the nanopatterned heterostructure. A variation of the parameters by 20--30\% can change the relative amplitude of puddles by orders of magnitude. Based on this analysis we formulate criteria for the acceptable design of the heterostructure aimed at creation of the artificial graphene., Comment: 9 pages, 12 figures

The method for computation of conditional probability density function for the nonlinear Schr\"odinger equation with additive noise is developed. We present in a constructive form the conditional probability density function in the limit of a small noise and analytically derive it in a weakly nonlinear case. The general theory results are illustrated using fibre-optic communications as a particular, albeit practically very important, example.

A quantum mechanical model of two interacting electrons in graphene is considered. We concentrate on the case of zero total momentum of the pair. We show that the dynamics of the system is very unusual. Both stationary and time-dependent problems are considered. It is shown that the complete set of the wave functions with definite energy includes the new functions, previously overlooked. The time evolution of the wave packet, corresponding to the scattering problem setup, leads to the appearance of the localized state at large time. The asymptotics of this state is found analytically. We obtain the lower bound of the life time of this state, which is connected with the breakdown of the continuous model on the lattice scale. The estimate of this bound gives one a hope to observe the localized states in the experiment., Comment: 10 pages, 2 figures

Using the operator method, the Green's functions of the Dirac and Klein-Gordon equations in the Coulomb potential $-Z\alpha/r$ are derived for the arbitrary space dimensionality $d$. Nonrelativistic and quasiclassical asymptotics of these Green's functions are considered in detail., Comment: 9 pages

We apply the DRA method to the calculation of the four-loop `QED-type' tadpoles. For arbitrary space-time dimensionality D the results have the form of multiple convergent sums. We use these results to obtain the epsilon-expansion of the integrals around D=3 and D=4., Comment: References added, some typos corrected. Results unchanged

The induced current density, corresponding to the massless Dirac equation in (2+1) dimensions in a magnetic flux tube of small radius is considered. This problem is important for graphene. In the case, when an electron can not penetrate the region of nonzero magnetic field, this current is the odd periodical function of the magnetic flux. If the region inside the magnetic tube is not forbidden for penetration of electron, the induced current is not a periodical function of the magnetic flux. However in the limit $R\to 0$, where $R$ is the radius of magnetic flux tube, this function has the universal form which is independent of the magnetic field distribution inside the magnetic tube at fixed value of the magnetic flux., Comment: 5 pages, 1 figure

The induced charge density, $\rho_{ind}(\bm r)$, generated in graphene by the potential well of the finite radius $R$ is considered. The result for $\rho_{ind}(\bm r)$ is derived for large distances $r\gg R$. We also obtained the induced charges outside of the radius $r\gg R$ and inside of this radius for subcritical and supercritical regimes. The consideration is based on the convenient representation of the induced charge density via the Green's function of electron in the field., Comment: 12 pages, 2 figures, version published in Phys.Rev. B

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 finite periodical function of the magnetic flux $\Phi$. 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 $\Phi/\Phi_0$ as well as at half-integer values of this ratio, where $\Phi_0=2\pi\hbar c/e$ 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., Comment: 3 pages, 1 figure, version accepted to Phys. Rev. B

A spin-dependent emission of optically oriented electrons from p-GaAs(Cs,O) into vacuum was experimentally observed in a magnetic field normal to the surface. This phenomenon is explained within the model which takes into account the jump in the electron g factor at the semiconductor-vacuum interface. Due to this jump, the effective electron affinity on the semiconductor surface depends on the mutual direction of optically oriented electron spins and the magnetic field, resulting in the spin-dependent photoemission. It is demonstrated that the observed effect can be used for the determination of spin diffusion length in semiconductors., Comment: 7 pages, 6 figures, published version

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 $Z\alpha$, where $\alpha$ 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, $Z_{eff} \alpha < 1/2$, for any $\alpha$, while impurities with $Z=2,3$ and higher can become supercritical at certain values of $\alpha$., Comment: 4 pages, 2 figures

In this article, we study the Coulomb corrections to the Delbrueck scattering amplitude. We consider the limit when the energy of the photon is much less than the electron mass. The calculations are carried out in the coordinate representation using the exact relativistic Green function of an electron in a Coulomb field. The resulting relative corrections are of the order of a few percent for scattering on for a large charge of the nucleus. We compare the corrections with the corresponding ones calculated through the dispersion integral of the pair production cross section and also with the magnetic loop contribution to the g-factor of a bound electron. The last one is in a good agreement with our results but the corrections calculated through the dispersion relation are not., Comment: 8 pages, 6 figures

A high-statistics measurement of bremsstrahlung emitted in the alpha decay of 210Po has been performed, which allows to follow the photon spectra up to energies of ~ 500 keV. The measured differential emission probability is in good agreement with our theoretical results obtained within the quasi classical approximation as well as with the exact quantum mechanical calculation. It is shown that due to the small effective electric dipole charge of the radiating system a significant interference between the electric dipole and quadrupole contributions occurs, which is altering substantially the angular correlation between the alpha particle and the emitted photon., Comment: 10 pages, 5 figures, v2: fix of small typos

We present a quasiclassical theory of alpha decay accompanied by bremsstrahlung with a special emphasis on the case of 210Po, with the aim of finding a unified description that incorporates both the radiation during the tunneling through the Coulomb wall and the finite energy E_gamma of the radiated photon up to E_gamma on the order of Q_alpha/sqrt(eta), where Q_alpha is the alpha-decay Q-value and eta is the Sommerfeld parameter. The corrections with respect to previous quasiclassical investigations are found to be substantial, and excellent agreement with a full quantum mechanical treatment is achieved. Furthermore, we find that a dipole-quadrupole interference significantly changes the alpha-gamma angular correlation. We obtain good agreement between our theoretical predictions and experimental results., Comment: 8 pages, 4 figures

We investigate quantum electrodynamic effects under the influence of an external, time-dependent electromagnetic field, which mediates dynamic modifications of the radiative corrections. Specifically, we consider the quantum electrodynamic vacuum-polarization tensor under the influence of two external background fields: a strong laser field and a nuclear Coulomb field. We calculate the charge and current densities induced by a nuclear Coulomb field in the presence of a laser field. We find the corresponding induced scalar and vector potentials. The induced potential, in first-order perturbation theory, leads to a correction to atomic energy levels. The external laser field breaks the rotational symmetry of the system. Consequently, the induced charge density is not spherically symmetric, and the energy correction therefore leads to a "polarized Lamb shift." In particular, the laser generates an additional potential with a quadrupole moment. The corresponding laser-dressed vacuum-polarization potential behaves like 1/r**3 at large distances, unlike the Uehling potential that vanishes exponentially for large r. Our investigation might be useful for other situations where quantum field theoretic phenomena are subjected to external fields of a rather involved structure., Comment: 13 pages, RevTeX

The contribution of the light-by-light diagram to the g factor of electron and muon bound in Coulomb field is obtained. For electron in a ground state, our results are in good agreement with the results of other authors obtained numerically for large Z. For relatively small Z our results have essentially higher accuracy as compared to the previous ones. For muonic atoms, the contribution is obtained for the first time with the high accuracy in whole region of Z., Comment: 10 pages, 3 figures, RevTeX

We consider one-loop self-energy and vacuum polarization radiative corrections to the shift of atomic energy level due to finite nuclear size. Analytic expressions for vacuum polarization corrections are derived. For the self-energy of p1/2 and p3/2 states in addition to already known terms we derive next-to-leading nonlogarithmic Z\alpha-terms. Together with contributions obtained earlier the terms derived in the present work give explicit analytic expressions for s1/2 and p1/2 corrections which agree with results of previous numerical calculations up to Z=100 (Z is the nuclear charge number). We also show that the finite nuclear size radiative correction for a p3/2 state is not small compared to the similar correction for a p1/2 state at least for small Z., Comment: 12 pages, 7 figures

We calculate the self-energy and the vertex radiative corrections to the effect of parity nonconservation in heavy atoms. The sum of the corrections is of the form ${\cal A}\ln(\lambda_C/r_0)+{\calB}$, where ${\cal A}$ and ${\cal B}$ are functions of $Z\alpha$, and $\lambda_C$ and $r_0$ are the Compton wavelength and the nuclear radius, respectively. The function ${\cal A}$ is calculated exactly in $Z\alpha$ and the function ${\cal B}$ is calculated in the leading order. In the leading order ${\cal A}\propto \alpha(Z\alpha)^2$ and ${\cal B}\propto \alpha(Z\alpha)$. The sum of the corrections is -0.85% for Cs and -1.48% 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, $Q_W=-72.81(28)_{exp}(36)_{theor}$ for Cs and $Q_W=-116.8(1.2)_{exp}(3.4)_{theor}$ for Tl, agree with predictions of the standard model within $0.6\sigma$. As an application of our approach we have also calculated dependence of the Lamb shift on the finite nuclear size., Comment: 14 pages, 7 figures

We consider corrections to the Lamb shift of p-wave atomic states due to the finite nuclear size (FNS). In other words, these are radiative corrections to the atomic isotop shift related to FNS. It is shown that the structure of the corrections is qualitatively different from that for s-wave states. The perturbation theory expansion for the relative correction for a $p_{1/2}$-state starts from $\alpha\ln(1/Z\alpha)$-term, while for $s_{1/2}$-states it starts from $Z\alpha^2$ term. Here $\alpha$ is the fine structure constant and $Z$ is the nuclear charge. In the present work we calculate the $\alpha$-terms for $2p$-states, the result for $2p_{1/2}$-state reads $(8\alpha/9\pi)[\ln(1/(Z\alpha)^2)+0.710]$. Even more interesting are $p_{3/2}$-states. In this case the ``correction'' is by several orders of magnitude larger than the ``leading'' FNS shift., Comment: 4 pages, 2 figures

The self-energy and the vertex radiative corrections to the effect of parity nonconservation in heavy atoms are calculated analytically in orders Z alpha^2 and Z^2 alpha^3 ln(lambda_C/r_0), where lambda_C and r_0 being the Compton wavelength and the nuclear radius, respectively. The value of the radiative correction 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 obtained values of 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 dependence of the Lamb shift on the finite nuclear size., Comment: 4 pages, 4 figures

The analytic expression for the cross section of low-energy electron scattering in a strong Coulomb field is obtained. It is shown that in a wide energy region this cross section differs essentially from that obtained in the first Born approximation., Comment: 5 pages, 1 figure, RevTex