Renormalization Group Functions for Two-Dimensional Phase Transitions: To the Problem of Singular Contributions

According to the available publications, the field theoretical renormalization group (RG) approach in the two-dimensional case gives the critical exponents that differ from the known exact values. This fact was attempted to explain by the existence of nonanalytic contributions in the RG functions. The situation is analysed in this work using a new algorithm for summing divergent series that makes it possible to analyse dependence of the results for the critical exponents on the expansion coefficients for RG functions. It has been shown that the exact values of all the exponents can be obtained with a reasonable form of the coefficient functions. These functions have small nonmonotonities or inflections, which are poorly reproduced in natural interpolations. It is not necessary to assume the existence of singular contributions in RG functions.Comment: PDF, 11 page

Gell-Mann - Low Function for QCD in the strong-coupling limit

The Gell-Mann - Low function \beta(g) in QCD (g=g0^2/16\pi^2 where g0 is the coupling constant in the Lagrangian) is shown to behave in the strong-coupling region as \beta_\infty g^\alpha with \alpha\approx -13, \beta_\infty\sim 10^5.Comment: 5 pages, PD

Renormalons and Analytic Properties of the \beta function

The presence or absense of renormalon singularities in the Borel plane is shown to be determined by the analytic properties of the Gell-Mann - Low function \beta(g) and some other functions. A constructive criterion for the absense of singularities consists in the proper behavior of the \beta function and its Borel image B(z) at infinity, \beta(g)\sim g^\alpha and B(z)\sim z^\alpha with \alpha\le 1. This criterion is probably fulfilled for the \phi^4 theory, QED and QCD, but is violated in the O(n)-symmetric sigma model with n\to\infty.Comment: 6 pages, PD

The Minimum-Uncertainty Squeezed States for for Atoms and Photons in a Cavity

We describe a six-parameter family of the minimum-uncertainty squeezed states for the harmonic oscillator in nonrelativistic quantum mechanics. They are derived by the action of corresponding maximal kinematical invariance group on the standard ground state solution. We show that the product of the variances attains the required minimum value 1/4 only at the instances that one variance is a minimum and the other is a maximum, when the squeezing of one of the variances occurs. The generalized coherent states are explicitly constructed and their Wigner function is studied. The overlap coefficients between the squeezed, or generalized harmonic, and the Fock states are explicitly evaluated in terms of hypergeometric functions. The corresponding photons statistics are discussed and some applications to quantum optics, cavity quantum electrodynamics, and superfocusing in channeling scattering are mentioned. Explicit solutions of the Heisenberg equations for radiation field operators with squeezing are found.Comment: 27 pages, no figures, 174 references J. Phys. B: At. Mol. Opt. Phys., Special Issue celebrating the 20th anniversary of quantum state engineering (R. Blatt, A. Lvovsky, and G. Milburn, Guest Editors), May 201

The Degenerate Parametric Oscillator and Ince's Equation

We construct Green's function for the quantum degenerate parametric oscillator in terms of standard solutions of Ince's equation in a framework of a general approach to harmonic oscillators. Exact time-dependent wave functions and their connections with dynamical invariants and SU(1,1) group are also discussed.Comment: 10 pages, no figure

Divergent Perturbation Series

Various perturbation series are factorially divergent. The behavior of their high-order terms can be found by Lipatov's method, according to which they are determined by the saddle-point configurations (instantons) of appropriate functional integrals. When the Lipatov asymptotics is known and several lowest order terms of the perturbation series are found by direct calculation of diagrams, one can gain insight into the behavior of the remaining terms of the series. Summing it, one can solve (in a certain approximation) various strong-coupling problems. This approach is demonstrated by determining the Gell-Mann - Low functions in \phi^4 theory, QED, and QCD for arbitrary coupling constants. An overview of the mathematical theory of divergent series is presented, and interpretation of perturbation series is discussed. Explicit derivations of the Lipatov asymptotic forms are presented for some basic problems in theoretical physics. A solution is proposed to the problem of renormalon contributions, which hampered progress in this field in the late 1970s. Practical schemes for summation of perturbation series are described for a coupling constant of order unity and in the strong-coupling limit. An interpretation of the Borel integral is given for 'non-Borel-summable' series. High-order corrections to the Lipatov asymptotics are discussed.Comment: Review article, 45 pages, PD

On the Critical Exponents for the \Lambda-Transition in Liquid Helium

The use of a new method for summing divergent series makes it possible to significantly increase the accuracy of determining the critical exponents from the field theoretical renormalization group. The exponent value \nu=0.6700\pm 0.0006 for the \lambda-transition in liquid helium is in good agreement with the experiment, but contradicts the last theoretical results based on using high-temperature series, the Monte Carlo method, and their synthesis.Comment: PDF, 7 page

Large Magnetoresistance of a Dilute pp-Si/SiGe/Si Quantum Well in a Parallel Magnetic Field

We report the results of an experimental study of the magnetoresistance $\rho_{xx}$ in two samples of $p$-Si/SiGe/Si with low carrier concentrations $p$=8.2$\times10^{10}$ cm$^{-2}$ and $p$=2$\times10^{11}$ cm$^{-2}$. The research was performed in the temperature range of 0.3-2 K in the magnetic fields of up to 18 T, parallel to the two-dimensional (2D) channel plane at two orientations of the in-plane magnetic field $B_{\parallel}$ against the current $I$: $B_{\parallel} \perp I$ and $B_{\parallel} \parallel I$. In the sample with the lowest density in the magnetic field range of 0-7.2 T the temperature dependence of $\rho_{xx}$ demonstrates the metallic characteristics ($d \rho_{xx}/dT>$0). However, at $B_{\parallel}$ =7.2 T the derivative $d \rho_{xx}/dT$ reverses the sign. Moreover, the resistance depends on the current orientation with respect to the in-plane magnetic field. At $B_{\parallel} \cong$ 13 T there is a transition from the dependence $\ln(\Delta\rho_{xx} / \rho_{0})\propto B_{\parallel}^2$ to the dependence $\ln(\Delta\rho_{xx} / \rho_{0})\propto B_{\parallel}$. The observed effects can be explained by the influence of the in-plane magnetic field on the orbital motion of the charge carriers in the quasi-2D system.Comment: revised: included AC conductivity experiments to study the role of localized state in transport; total 6 pages, 7 figures. Accepted PRB; to appear vol.79, Issue 2