Asymptotics of the susceptibility for the Ising model on the hierarchical lattices

Using the GKS inequality Griffiths and Kaufman showed in [12] that the susceptibility in the Ising model on the hierarchical lattices is infinite forT>T _c . In the present work we give an analytic calculation of the dominant singularity of the susceptibility whenh0 forT>T _c . Moreover we analytically find the spontaneous magnetization forT<T _c and prove that forT<T _c the susceptibility is finite.     

Numerical calculation of the fidelity for the Kondo and the Friedel-Anderson impurities

The fidelities of the Kondo and the Friedel-Anderson (FA) impurities are calculated numerically. The ground states of both systems are calculated with the FAIR (Friedel artificially inserted resonance) theory. The ground state in the interacting systems is compared with a nullstate in which the interaction is zero. The different multi-electron states are expressed in terms of Wilson states. The use of N Wilson states simulates the use of a large effective number N _eff of states. A plot of ln(F) versus N ∝ ln(N _eff ) reveals whether one has an Anderson orthogonality catastrophe at zero energy. The results are at first glance surprising. The ln(F) – ln(N _eff ) plot for the Kondo impurity diverges for large N _eff . On the other hand, the corresponding plot for the symmetric FA impurity saturates for large N _eff when the level spacing at the Fermi level is of the order of the singlet-triplet excitation energy. The behavior of the fidelity allows one to determine the phase shift of the electron states in this regime.     

Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces

LetZ(s, R) be the Selberg zeta function of a compact Riemann surfaceR. We study the behavior ofZ(s, R) asR tends to infinity in the moduli space of stable curves. The main result is an estimate forZ(s, R) valid fors in a neighborhood, depending only on the genus, ofs=1. Our analysis gives an alternate proof of the Belavin-Knizhnik double pole result, [5].Partially supported by the National Science Foundation and the Institute for Physical Science and Technology, University of Maryland, College Park, MD, USA     

On the static permittivity of the Coulomb system in the long-wavelength limit

The general expression for the static permittivity ε(q, 0) of the Coulomb system in the region of small wave vectors was derived based on exact limit relations. The relation obtained describes the function ε(q, 0) in both “metal” and “dielectric” states of the Coulomb system. On this basis, the concept of the “true” dielectric is introduced and the definition of the “true” screening length was discussed. Exact relations were derived for the function ε(q, 0) in the region of small wave vectors q within the random phase approximation at an arbitrary degeneracy.     

A Perturbative Approach for the Asymptotic Evaluation of the Neumann Value Corresponding to the Dirichlet Datum of a Single Periodic Exponential for the NLS

Boundary value problems for the nonlinear Schrödinger equation formulated on the half-line can be analyzed by the Fokas method. For the Dirichlet problem, the most difficult step of this method is the characterization of the unknown Neumann boundary value. For the case that the Dirichlet datum consists of a single periodic exponential, namely, a exp(iωt), a, ω real, it has been shown in [2–4] that if one assumes that the Neumann boundary value is given for large t by c exp(iωt), then c can be computed explicitly in terms of a and ω. Here, using the perturbative approach introduced in [16], it is shown that for typical initial conditions, it is indeed the case that at least up to third order in a perturbative expansion the Neumann boundary value is given by c exp(iωt) and the value of c is at least up to this order the value found in [2–4].     

Proposal of the experiment on the investigation of the (d, p) reaction at the extracted deuteron beam of the U-70 accelerator

The experiment on the investigation of the A(d, p)X reaction at the extracted deuteron beam created at the Serpukhov U-70 accelerator is proposed. Precise measurements of the momentum spectra of protons emitted in the (d, p) reaction at an initial deuteron momenta of 20–40 GeV/c can give unique information on the deuteron short-range structure. A manifestation of contribution of nonnucleonic degrees of freedom in the deuteron is entirely possible over this energy range. The text was submitted by the authors in English.     

Dependence of the Thomas-Fermi energy on the nuclear coordinates

LetE(R), respectivelye(R), denote the total energy, respectively the electronic contribution to the energy, in the Thomas-Fermi theory for a system of two fixed nuclei a distanceR apart. We prove thate(R) and –E(R) increase asR does. For the case ofN fixed nuclei, we prove the monotonicity ofe andE under certain displacements of the coordinates of the nuclei. The analogous result for the electronic contribution to the Born-Oppenheimer energy is proved.Research supported by U.S. National Science Foundation under Grant MCS 80-17781     

On the superconductivity of the tin-hydrogen system

Conductivity and superconductivity studies of amorphous [Sn_1–y Cu _y–]_1–x H _x samples in connection with¹¹⁹Sn Mössbauer effect experiments on¹¹⁹Sn_1–x H _x give strong evidence that the observed increase of the superconducting transition temperatureT _c in the Sn–H-system is caused by the stabilization of an amorphous structure. Thus the Sn–H-system is very similar to the Sn–Cu-system and no H-specific effect is needed to explain the increase ofT _c.     

On the range of validity of a semiclassical relation between the critical exponents at the Anderson localization transition

Using Wegner's result relating critical exponents s and ν for conductivity and localization length, respectively, via dimensionality d and that for ν given by García-García, we derive what we term a semiclassical (sc) relation for ν in terms of s, which is independent of dimensionality. Forming the ratio s/ν versus d from the above relations, s/ν=0 at d=2 is due to a singularity in the sc relation for ν. We argue that, in reality, s/ν=0 results from s being zero at d=2. Finally we conjecture that (i) Wegner's prediction s/ν=1 when d=3 and (ii) ν tends to 1/2 at large s, are both insensitive to interactions.     

Decays of the mesons and the singlet photon

In an attempt to explain the recent measurements on the radiative decays of the vector-mesons (V→Pγ), we study the consequences of introducing a small admixture of SU (3) singlet piece in the electromagnetic current. We find that this leads to an excellent fit of the theory with the new measurements on theV→Pγ decays. However, this addition adversely affects the fit of the leptonic decays of the vector mesons (V→e ⁺ e ⁻) and of the radiative decay of the pion (π→2γ). We conclude that the overall fit to the available data does not favour a large (>10%) admixture of the SU(3) singlet. The decay rates have been calculated in the vectormeson dominance model. At the hadronic vertex (VVP), we assume asymptotic nonet symmetry. The electromagnetic couplings (V−γ) are the ones appropriate to vector-mixing.     

On the Stability of the O(N)-Invariant and the Cubic-Invariant Three-Dimensional N-Component Renormalization-Group Fixed Points in the Hierarchical Approximation

We compute renormalization-group fixed points and their spectrum in an ultralocal approximation. We study a case of two competing nontrivial fixed points for a three-dimensional real N-component field: the O(N)-invariant fixed point vs. the cubic-invariant fixed point. We compute the critical value N _c of the cubic ⁴-perturbation at the O(N)-fixed point. The O(N)-fixed point is stable under a cubic ⁴-perturbation below N _c; above N _c it is unstable. The Critical value comes out as 2.219435<N _c<2.219436 in the ultralocal approximation. We also compute the critical value of N at the cubic invariant fixed point. Within the accuracy of our computations, the two values coincide.     

On the spectrum of the hydrogen atom in an ultrastrong magnetic field

Various approaches to computing the energies of the ground state and excited levels of the hydrogen atom in an ultrastrong magnetic field B that considerably exceeds the field B _a = m _e ² e ³ c/ħ ³ ∼ 10⁹ G are considered. The effects of polarization of vacuum and anomalous magnetic moment of the electron on the position of the atomic levels are discussed. The vacuum polarization effects are negligibly weak for B < 10¹⁵ G but become significant in fields B ≫ 10¹⁶ G, in which these effects qualitatively modify the atomic spectrum in this range. The difference in the behaviors of the even and odd energy levels for B ≫ B _a is analyzed and the formulas for the energies of odd levels as a function of field B are refined.     

Study of the pion trajectory in the photoproduction of leading neutrons at HERA

Energetic neutrons produced in ep collisions at HERA have been studied with the ZEUS detector in the photoproduction regime at a mean photon–proton center-of-mass energy of 220 GeV. The neutrons carry a large fraction 0.64<x_L<0.925 of the incoming proton energy, and the four-momentum transfer squared at the proton–neutron vertex is small, |t|<0.425 GeV². The x_L distribution of the neutrons is measured in bins of t. The (1−x_L) distributions in the t bins studied satisfy a power law dN/dx_L∝(1−x_L)^a(t), with the powers a(t) following a linear function of t: . This result is consistent with the expectations of pion-exchange models, in which the incoming proton fluctuates to a neutron–pion state, and the electron interacts with the pion.     

Variation of the galactic force law in the region of the Sun

From the radial velocity data for O-B5 stars within 3 kpc of the Sun the variation of the semi-axesΣ ₁ andΣ ₂ of the velocity ellipsoid in the solar region is estimated. It is found that both the semi-axes as well as their ratioΣ ₂/Σ ₁ decrease away from the galactic centre. The value ofΣ ₂/Σ ₁ changes from 1.17 atR=9 kpc to 0.48 atR=11 kpc passing through a value of 0.86 at the solar distance ofR ₀=10 kpc. These results are consistent with the usually assumed inverse square law of force in the outer regions of the galaxy.     

On the Residual Entropy of the Blume-Emery-Griffiths Model

By means of a transfer matrix method, we show that the residual entropy S of the two dimensional square lattice Blume-Emery-Griffiths model on the antiquadrupolar-disordered and ferromagnetic-antiquadrupolar-disordered phase boundaries satisfies the inequalities (ln λ _1,n )/(n+1)≤S≤(ln λ _1,n )/n, where λ _1,n is the largest eigenvalue of a transfer matrix F _n on a strip of width n. These bounds imply the existence of a O(1/n) correction in the approximation of S by (ln λ _1,n )/n. Using these bounds, we calculate numerically the value of S, with precise estimates on the errors.     

On the harmonic oscillator on the Lobachevsky plane

We introduce the harmonic oscillator on the Lobachevsky plane with the aid of the potential V (ϱ) = (a ² ω ²/4) sinh(ϱ/a)², where a is the curvature radius and ϱ is the geodesic distance from a chosen center. Thus, the potential is rotationally symmetric and unbounded, as in the Euclidean case. The eigenvalue equation leads to the differential equation of spheroidal functions. We provide a basic numerical analysis of eigenvalues and eigenfunctions provided that the value of the angular momentum, m, is equal to 0. Dedicated to the memory of V. A. Geyler     

On the operator equivalents and the crystal-field and spin Hamiltonian parameters

Operator equivalents used in magnetic resonance (electron paramagnetic resonance [EPR], electron-nuclear double resonance, nuclear magnetic resonance, etc.) are considered. The known non-Hermitian tensor operators O _±q ^(k) (0 ≤ q ≤ k) defined by D. Smith and J. H. M. Thornley (Proc. Phys. Soc. 89, 779–781, 1966) after H. A. Buckmaster (Can. J. Phys. 40, 1670–1677, 1962) are expressed in terms of Hermitian cosine and sine operators O _kq ^c and O _kq ^s. These new normalized operators are related to the nonnormalized extended Stevens operators O _k ^q and O _k ^−q ≡ Ω _k ^q defined by C. Rudowicz (J. Phys. C.: Solid State Phys. 18, 1415–1430, 1985), which are commonly used in EPR of transition ions. The relations between O _kq ^c,s and some other normalized operators occurring in the literature are also given. New crystal-field or spin Hamiltonian parameters B _kq ^c and B _kq ^s associated with the corresponding operators O _kq ^c and O _kq ^s are discussed.     

New search for T violation in the decays of the charged kaon

We report the results of the measurement of T-violating transverse muon polarization in the decays K ⁺→μ ⁺ ν _μ π ⁰ (K _μ3) and K ⁺→μ ⁺ ν _μ γ(K _μ2γ) performed in experiment E246 at KEK. The preliminary results obtained for the entire data set taken in the period 1996–2000 are consistent with no T violation in both decays.

     

On the p,q-binomial distribution and the Ising model

We employ p, q-binomial coefficients, a generalisation of the binomial coefficients, to describe the magnetisation distributions of the Ising model. For the complete graph this distribution corresponds exactly to the limit case p = q. We apply our investigation to the simple d-dimensional lattices for d = 1, 2, 3, 4, 5 and fit p, q-binomial distributions to our data, some of which are exact but most are sampled. For d = 1 and d = 5, the magnetisation distributions are remarkably well-fitted by p,q-binomial distributions. For d = 4 we are only slightly less successful, while for d = 2, 3 we see some deviations (with exceptions!) between the p, q-binomial and the Ising distribution. However, at certain temperatures near T _c the statistical moments of the fitted distribution agree with the moments of the sampled data within the precision of sampling. We begin the paper by giving results of the behaviour of the p, q-distribution and its moment growth exponents given a certain parameterisation of p, q. Since the moment exponents are known for the Ising model (or at least approximately for d = 3) we can predict how p, q should behave and compare this to our measured p, q. The results speak in favour of the p, q-binomial distribution's correctness regarding its general behaviour in comparison to the Ising model. The full extent to which they correctly model the Ising distribution, however, is not settled.     

On the Problem of the Amenability of the Gauge Group

Let G be one of the local gauge groups C(X, U(n)),C (X, U(n)), C(X, SU(n)) or C (X, SU(n)) where X is a compact Riemannian manifold. We observe that G has a nontrivial group topology, coarser than its natural topology, w.r.t. which it is amenable, viz. the relative weak topology of C(X, M(n)). This topology seems more useful than other known amenable topologies for G. We construct a simple fermionic model containing an action of G, continuous w.r.t. this amenable topology.