Quantum Electrodynamics at Extremely Small Distances

The asymptotics of the Gell-Mann - Low function in QED can be determined exactly, \beta(g)= g at g\to\infty, where g=e^2 is the running fine structure constant. It solves the problem of pure QED at small distances L and gives the behavior g\sim L^{-2}.Comment: Latex, 6 pages, 1 figure include

Spin Polarization Phenomena and Pseudospin Quantum Hall Ferromagnetism in the HgTe Quantum Well

The parallel field of a full spin polarization of the electron gas in a \Gamma8 conduction band of the HgTe quantum well was obtained from the magnetoresistance by three different ways in a zero and quasi-classical range of perpendicular field component Bper. In the quantum Hall range of Bper the spin polarization manifests in anticrossings of magnetic levels, which were found to strongly nonmonotonously depend on Bper.Comment: to be published in AIP Conf. Proc.: 15-th International Conference on Narrow Gap Systems (NGS-15

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

Finite-size scaling from self-consistent theory of localization

Accepting validity of self-consistent theory of localization by Vollhardt and Woelfle, we derive the finite-size scaling procedure used for studies of the critical behavior in d-dimensional case and based on the use of auxiliary quasi-1D systems. The obtained scaling functions for d=2 and d=3 are in good agreement with numerical results: it signifies the absence of essential contradictions with the Vollhardt and Woelfle theory on the level of raw data. The results \nu=1.3-1.6, usually obtained at d=3 for the critical exponent of the correlation length, are explained by the fact that dependence L+L_0 with L_0>0 (L is the transversal size of the system) is interpreted as L^{1/\nu} with \nu>1. For dimensions d\ge 4, the modified scaling relations are derived; it demonstrates incorrectness of the conventional treatment of data for d=4 and d=5, but establishes the constructive procedure for such a treatment. Consequences for other variants of finite-size scaling are discussed.Comment: Latex, 23 pages, figures included; additional Fig.8 is added with high precision data by Kramer et a

Magnetotransport in Double Quantum Well with Inverted Energy Spectrum: HgTe/CdHgTe

We present the first experimental study of the double-quantum-well (DQW) system made of 2D layers with inverted energy band spectrum: HgTe. The magnetotransport reveals a considerably larger overlap of the conduction and valence subbands than in known HgTe single quantum wells (QW), which may be regulated by an applied gate voltage $V_g$. This large overlap manifests itself in a much higher critical field $B_c$ separating the range above it where the quantum peculiarities shift linearly with $V_g$ and the range below with a complicated behavior. In the latter case the $N$-shaped and double-$N$-shaped structures in the Hall magnetoresistance $\rho_{xy}(B)$ are observed with their scale in field pronouncedly enlarged as compared to the pictures observed in an analogous single QW. The coexisting electrons and holes were found in the whole investigated range of positive and negative $V_g$ as revealed from fits to the low-field $N$-shaped $\rho_{xy}(B)$ and from the Fourier analysis of oscillations in $\rho_{xx}(B)$. A peculiar feature here is that the found electron density $n$ remains almost constant in the whole range of investigated $V_g$ while the hole density $p$ drops down from the value a factor of 6 larger than $n$ at extreme negative $V_g$ to almost zero at extreme positive $V_g$ passing through the charge neutrality point. We show that this difference between $n$ and $p$ stems from an order of magnitude larger density of states for holes in the lateral valence band maxima than for electrons in the conduction band minimum. We interpret the observed reentrant sign-alternating $\rho_{xy}(B)$ between electronic and hole conductivities and its zero resistivity state in the quantum Hall range of fields on the basis of a calculated picture of magnetic levels in a DQW.Comment: 15 pages, 13 figure

Scaling near the upper critical dimensionality in the localization theory

The phenomenon of upper critical dimensionality d_c2 has been studied from the viewpoint of the scaling concepts. The Thouless number g(L) is not the only essential variable in scale transformations, because there is the second parameter connected with the off-diagonal disorder. The investigation of the resulting two-parameter scaling has revealed two scenarios, and the switching from one to another scenario determines the upper critical dimensionality. The first scenario corresponds to the conventional one-parameter scaling and is characterized by the parameter g(L) invariant under scale transformations when the system is at the critical point. In the second scenario, the Thouless number g(L) grows at the critical point as L^{d-d_c2}. This leads to violation of the Wegner relation s=\nu(d-2) between the critical exponents for conductivity (s) and for localization radius (\nu), which takes the form s=\nu(d_c2-2). The resulting formulas for g(L) are in agreement with the symmetry theory suggested previously [JETP 81, 925 (1995)]. A more rigorous version of Mott's argument concerning localization due topological disorder has been proposed.Comment: PDF, 7 pages, 6 figure

Analytical realization of finite-size scaling for Anderson localization. Does the band of critical states exist for d>2?

An analytical realization is suggested for the finite-size scaling algorithm based on the consideration of auxiliary quasi-1D systems. Comparison of the obtained analytical results with the results of numerical calculations indicates that the Anderson transition point is splitted into the band of critical states. This conclusion is supported by direct numerical evidence (Edwards and Thouless, 1972; Last and Thouless, 1974; Schreiber, 1985; 1990). The possibility of restoring the conventional picture still exists but requires a radical reinterpretetion of the raw numerical data.Comment: PDF, 11 page

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