Asymptotic behavior in the scalar field theory

An asymptotic solution of the system of Schwinger-Dyson equations for four-dimensional Euclidean scalar field theory with interaction $\frac{\lambda}{2}(\phi^*\phi)^2$ is obtained. For $\lambda>\lambda_{cr}=16\pi^2$ the two-particle amplitude has the pathology-free asymptotic behavior at large momenta. For $\lambda<\lambda_{cr}$ the amplitude possesses Landau-type singularity.Comment: 16 pages; journal version; references adde

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

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

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

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

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

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

Quantum Abacus for counting and factorizing numbers

We generalize the binary quantum counting algorithm of Lesovik, Suslov, and Blatter [Phys. Rev. A 82, 012316 (2010)] to higher counting bases. The algorithm makes use of qubits, qutrits, and qudits to count numbers in a base 2, base 3, or base d representation. In operating the algorithm, the number n < N = d^K is read into a K-qudit register through its interaction with a stream of n particles passing in a nearby wire; this step corresponds to a quantum Fourier transformation from the Hilbert space of particles to the Hilbert space of qudit states. An inverse quantum Fourier transformation provides the number n in the base d representation; the inverse transformation is fully quantum at the level of individual qudits, while a simpler semi-classical version can be used on the level of qudit registers. Combining registers of qubits, qutrits, and qudits, where d is a prime number, with a simpler single-shot measurement allows to find the powers of 2, 3, and other primes d in the number n. We show, that the counting task naturally leads to the shift operation and an algorithm based on the quantum Fourier transformation. We discuss possible implementations of the algorithm using quantum spin-d systems, d-well systems, and their emulation with spin-1/2 or double-well systems. We establish the analogy between our counting algorithm and the phase estimation algorithm and make use of the latter's performance analysis in stabilizing our scheme. Applications embrace a quantum metrological scheme to measure a voltage (analog to digital converter) and a simple procedure to entangle multi-particle states.Comment: 23 pages, 15 figure

Unusual Shubnikov-de Haas oscillations in BiTeCl

We report measurements of Shubnikov-de Haas (SdH) oscillations in single crystals of BiTeCl at magnetic fields up to 31 T and at temperatures as low as 0.4 K. Two oscillation frequencies were resolved at the lowest temperatures, $F_{1}=65 \pm 4$ Tesla and $F_{2}=156 \pm 5$ Tesla. We also measured the infrared optical reflectance $\left(\cal R(\omega)\right)$ and Hall effect; we propose that the two frequencies correspond respectively to the inner and outer Fermi sheets of the Rashba spin-split bulk conduction band. The bulk carrier concentration was $n_{e}\approx1\times10^{19}$ cm$^{-3}$ and the effective masses $m_{1}^{*}=0.20 m_{0}$ for the inner and $m_{2}^{*}=0.27 m_{0}$ for the outer sheet. Surprisingly, despite its low effective mass, we found that the amplitude of $F_{2}$ is very rapidly suppressed with increasing temperature, being almost undetectable above $T\approx4$ K

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