Development of the self-consistent approximation and its application to the problem of magnetoelastic resonance in an inhomogeneous medium

Our previously proposed approximation involving both the first and second terms of the expansion of the vertex function is generalized to the system of two interacting wavefields of different physical nature. A system of self-consistent equations for the matrix Green’s function and matrix vertex function is derived. On the basis of this matrix generalization of the new self-consistent approximation, a theory of magnetoelastic resonance is developed for a ferromagnetic model, where the magnetoelastic coupling parameter ε(x) is inhomogeneous. Equations for magnetoelastic resonance are analyzed for one-dimensional inhomogeneities of the coupling parameter. The diagonal and off-diagonal elements of the matrix Green’s function of the system of coupled spin and elastic waves are calculated with the change in the ratio between the average value ε and rms fluctuation Δε of the coupling parameter between waves from the homogeneous case (ε ≠ 0, Δε = 0) to the extremely randomized case (ε = 0, Δε ≠ 0) at various correlation wavenumbers of inhomogeneities k _c. For the limiting case of infinite correlation radius (k _c = 0), in addition to approximate expressions, exact analytical expressions corresponding to the summation of all diagrams of elements of the matrix Green’s function are obtained. The results calculated for an arbitrary k _c value in the new self-consistent approximation are compared to the results obtained in the standard self-consistent approximation, where only the first term of the expansion of the vertex function is taken into account. It is shown that the new approximation corrects disadvantages of the Green’s functions calculated in the standard approximation such as the dome shape of resonances and bends on the sides of resonance peaks. The appearance of a fine structure of the spectrum in the form of a narrow resonance on the Green’s function of spin waves and a narrow antiresonance on the Green’s function of elastic waves, which was previously predicted in the standard self-consistent approximation, is confirmed. With an increase in the parameter k _c, the Green’s functions calculated in the standard and new approximations approach each other and almost coincide with each other at k _c/k ≥ 0.5. At the same time, the results of this work indicate that the new self-consistent approximation has a certain advantage for studying the problems of stochastic radiophysics in media with long-wavelength inhomogeneities (small k _c values), because it describes both the shape and width of peaks much better than the standard approximation.     

Self-consistent approximations for correlation functions in the theory of triplet pairing superfluidity in3He

A self-consistent and “spin conserving” approximation scheme is developed for the two-particle correlation functions in superfluid³He. It is shown that the irreducible vertex part occuring in the Bethe-Salpeter equation for the generalized correlation function (having 4⁴ components) must be equal to the functional derivative of the approximate self-energy part with respect to the Green's function. The theory is specialized to the Hartree-Fock approximation yielding the RPA susceptibility belowT _c . An important feature of the resulting diagrammatic expansion is that the spin-fluctuation effects in the superfluid phase arise from infinite iteration of pairs of anomalous propagators inboth the particle-hole and in the particle-particle channels.     

The method of unitary transformations in the theory of nuclear matter

The theory of nuclear matter is investigated by means of the method of unitary transformations in the special case of point transformations. The general formula for the energy per particle as a function of the density is given in Hartree-Fock approximation being neglected the induced three and more body forces for reasonable correlation functions. This function always shows saturation due to a term proportional tok _F ⁵ in the direct part of the approximation. The physical connexion of this term with the scattering amplitudes of the potential is shown. We point out the equivalence to orderk _F ³ of the energy per particle function of the unitary method with Jastrow and separation methods in their simplest form. The saturation properties are calculated for certain classes of correlations using the realistic potential of Gammel, Christian and Thaler.     

Critical indices for the charged Bose gas

A self-consistent version of the static random phase approximation leads to a quasi-particle energy satisfying ?(k)=Ak^v for small k, where v ≈ 1.8. The critical indices are those of an ideal Bose gas with this spectrum.     

Logarithmic correlation functions in two-dimensional turbulence

We consider the correlation functions of two-dimensional turbulence in the presence and absence of a three-dimensional perturbation, by means of conformal field theory. In the presence of three-dimensional perturbation, we show that in the strong coupling limit of a small scale random force, there is some logarithmic factor in the correlation functions of velocity stream functions. We show that the logarithmic conformal field theory c_8,1 describes the 2D-turbulence both in the absence and in the presence of the perturbation. We obtain the energy spectrum E(k) ∼ k^−5.125 ln(k) for perturbed 2D-turbulence and E(k) ∼ k⁻⁵ ln(k) for unperturbed turbulence. Recent numerical simulation and experimental results confirm our prediction.     

Self-consistent calculation of the optical potential and ground-state properties of nuclear matter

On the basis of the Green-function formalism, we performed a self-consistent calculation of the self-energy ∑(k, ω) of a particle interacting with the infinite nuclear medium. The function ∑(k, ω) was mapped out in the energy-momentum plane, and the single-particle energy ω(k), momentum distribution ?(k) and the “on-shell” part of the self-energy, ∑(k, ω(k)), were defined, from which all physical properties followed. In particular we investigated the ground-state properties of nuclear matter in two Λ-approximations of the T-matrix. In one, the intermediate two-particle propagator, Λ₀₀, represented free-particle propagation; in the other, called Λ₁₁, intermediate states included both interacting particles and holes. Pauli principle effects were included in both approximations. The second approximation was expected to be conserving because it included a large part of the rearrangement effects which, we found, contributed ～6 MeV per particle to the average energy and ～28 MeV to the singleparticle energy at zero momentum. The Hugenholtz-van Hove theorem was nearly satisfied, with only 1 MeV separating the chemical potential from the average energy. We also studied, in the Λ₀₀-approximation, the optical potential for the scattering of a particle by a large nucleus; it was directly related to the “on-shell” part of the self-energy. It was found that, below 100 MeV, the real part varied as (?90 + 0.584E) [MeV], and the imaginary part as (2.4 + 0.009 E) [MeV].     

Debye-waller factor for electron scattering by surface atoms

The contribution of anharmonic atomic vibrations to the EXAFS-spectrum is described by the third cumulantσ(³) which contributes with a term proportional tok ³ to the phase in the single scattering plane wave approximation. Due to the lattice expansion and enhancedσ(²) in the surface region we get additional temperature dependent contributions, which are evaluated in a curved wave theory. Transforming this expression into the basic formula we get an additional term proportional tok ³ in the phase of the generalized scattering amplitude. This valueΔα(³) has to be subtracted to get a value forσ(³). Numerical results at the O-K-edge of NiO give a value forΔσ(³) of about 10^?4 ų.     

Exact four-body calculation of the 4ΛHe-4ΛH binding energy difference

The coupled, two-variable integral equations that determine the ⁴_ΛHe and ⁴_ΛH bound states, when the NN and ΛN interactions are represented by separable potentials, are derived from the Schrödinger equation. The integral equations are solved numerically for simple s-wave potentials and for tensor potentials in the truncated t-matrix approximation without resort to separable expansion of the kernels. The Λ-separation energy difference ΔB_Λ resulting from the genuine four-body model is shown to be approximately twice as large as that coming from an “effective two-body” model calculation, when identical central potentials are used. The four-body model estimate of ΔB_Λ made with tensor forces is consistent with the experimental value, indicating that charge symmetry breaking implied by the low energy Λ N scattering parameters is compatible with that suggested by the known binding energy difference in the A = 4 hypernuclear isodoublet.     

The electronic Raman spectrum of cobalt doped MgO

The energies of the ⁴T_1g states of Co²⁺ as a dilute substitutional impurity in MgO relative to the ground Λ_6g doublet have been found by low temperature Raman spectroscopy to be two Λ_8g states at 305 ± 3 cm^?1 and 930 ± 3 cm^?1; the remaining Λ_7g energy is predicted to be in the 980–1010 cm^?1 range the corresponding Λ_6g → Λ_7g Raman transition being weak and buried in the extensive two-phonon background. A second-order perturbation calculation which couples the spin-orbit states to both E_g and T_2g modes of vibration gives a weak but important Jahn-Teller stabilization energy for the Λ_8g states.     

Accurate energy levels for the anharmonic oscillator and a summable series for the double-well potential in perturbation theory

We introduce a generalization of Wick-ordering which maps the anharmonic oscillator (AO) Hamiltonian for mass m and coupling λ exactly into a “Wick-ordered” Hamiltonian with an effective mass M which is a simple analytic function of λ and m. The effective coupling $\text{Λ =}\text{λ}\text{M}{}^3$ is bounded. We transform the AO perturbation series in λ into one in Λ. This series may then be summed using Borel summation methods. We also introduce a new summation method for the AO series (which is a practical necessity to obtain accurate energy levels of the excited states). We obtain a numerical accuracy for $\text{(E}{}_{\mathrm{PT}}\text{? E}{}_{\mathrm{exact}}\text{)}\text{E}{}_{\mathrm{exact}}$ of at least 10^?7 (using 20 orders of perturbation theory) and 10^?3 (using only 2 orders of perturbation theory) for all couplings and all energy levels of the anharmonic oscillator. The methods are applicable also to the double-well potential (DWP, the AO with a negative mass-squared). The only change is that now the effective coupling is unbounded as λ → 0. The series in Λ is, however, still summable. The relative accuracy in the energy levels for 20 orders of perturbation theory varies from 10^?7 for large coupling to 1% at λ = 0.1 and to 10% at λ = .05. We also present results for the sextic oscillator.     

Breakdown of renormalizability in the Kondo problem in the high-temperature expansion of the free energy

It is shown that there are two energy scales in the Kondo problem: T _k and T ₀, one of which (T _k) is exponentially small in the coupling constant g. The second scale T ₀is proportional to the squared coupling constant. Perturbation theory is valid only in the region T? T ₀. The point T ₀ is apparently the crossover from weak to strong coupling. The first indications of the breakdown of the hypothesis of only one energy scale in the Kondo problem appear in fourth order of perturbation theory.     

Variational methods for the ground state of liquid helium4

Assuming a Bijl-Jastrow-type wave function for the ground state of liquid He⁴, one can express the energy as a functional of the pair distribution functiong(r) when use is made of one of several “cluster approximations” known from the theory of classical fluids. The applicability of these approximations, and especially an integrodifferential equation forg(r) derived byAbe andHiroike, are discussed. It is shown that both the HNC and the PY approximations, when used consistently, yield the phonon behaviour of the liquid-structure factorS(k) for smallk. In the HNC approximation the energy as a function of density is calculated by a variational procedure. The velocity of sound following from \(\mathop {\lim }\limits_{k \to 0} \) S(k) is in good agreement with experiments and, at the equilibrium density, also with that calculated from the energy-versus-density curve. In the PY approximation a minimum of the energy expectation value does not exist without further restrictions on the trial wave function.     

Catoptric tadpoles

Fermionic string perturbation theory is known to suffer from an ambiguity in the form of a total derivative in the moduli space. For a class of backgrounds (including R¹⁰, orbifolds and theories with no U(1) factors in gauge group) we show that these ambiguities for the partition function of heterotic string theory at any genus are proportional to massless physical tadpoles in the theory at lower genera and hence vanish in stable vacua. We also find that in R¹⁰ the cosmological constant at a given genus is proportional to the cosmological constant at lower genera. This enables us to give an inductive argument for the vanishing of the cosmological constant in R¹⁰ to all orders in string perturbation theory. We also address the ambiguity and finiteness of n-point functions. Our results indicate that in R¹⁰ the ambiguity can be absorbed by a renormalization of the string coupling constant and the string tension. The expected sources of divergence in the n-point function in arbitrary tachyon-free backgrounds, besides the usual infrared divergences for d ≤ 4, are shown to be proportional to tadpoles of physical massless fields. For type II strings in arbitrary backgrounds, we show by explicit calculations that the ambiguity vanishes at g = 2.     

Theory of the local field correction in an electron gas

The wave-vector- and frequency-dependent dielectric function ?(k,ω) of an electron gas can be expressed in terms of Lindhard's function and a complex local field correctionG(k,ω) which incorporates all the effects of dynamic exchange and correlation in the system. The general properties ofG(k,ω) are discussed, in particular the static and high-frequency limits. It is shown that for smallk, bothG(k, 0) andG(k, ∞) vary ask ², with different coefficients, but both determined by the average kinetic and potential energies per particle. For largek,G(k, ∞) varies again ask ² and it is argued that the same holds true forG(k, 0), with both coefficients (though different) determined by the average kinetic energy per particle. General formulas for the plasma dispersion relation and damping, involving, respectively, the real and imaginary parts ofG(k,ω), are given. The term in the plasma frequency which is proportional tok ² is given directly in terms of the average kinetic and potential energies per particle, a result true at all temperatures. A calculation of the frequency dependence ofG(k,ω), starting from the exact equation of motion for the particle-hole operator and employing a decoupling approximation introduced previously by Toigo and Woodruff, is presented. Explicit results forG(k,ω) are obtained for smallk and allω. The complete expressions forG(k, 0) andG(k, ∞) in this approximation have been obtained and are plotted.     

Energy levels of 1s~2n d(n≤9)states for lithium-like ions

The full-core plus correlation method with multi-configuration interaction wave functions is extended to the calculation of the non-relativistic energies of 1s²nd (n ≤ 9) states for the lithium isoelectronic sequence from Z = 11 to 20. Relativistic and mass-polarization effects on the energy are calculated as the first-order perturbation correction. The quantum-electrodynamics correction is also included. The fine structure splittings are determined from the expectation values of spin—orbit and spin—other-orbit interaction operators in the Pauli—Breit approximation. Combining the term energies of lowly excited states obtained with the quantum defects calculated by the single channel quantum defect theory, each of which is a smooth function of energy and approximated by a weakly varying function of energy, the ion potentials of highly excited states (n ≤ 6) are obtained with the semi-empirical iteration method. The results are compared with experimental data in the literature and found to be closely consistent with the regularity.     

Advection of a passive vector field by the Gaussian velocity field with finite correlations in time

Using the field theoretic renormalization group technique the model of a passive vector field advected by an incompressible turbulent flow is investigated up to the second order of the perturbation theory (two-loop approximation). The turbulent environment is given by statistical fluctuations of the velocity field that has a Gaussian distribution with zero mean and defined noise with finite correlations in time. Two-loop analysis of all possible scaling regimes in general d-dimensional space is done in the plane of exponents ? ? η, where ? characterizes the energy spectrum of the velocity field in the inertial range E ∝ k ^{1 ? 2ε}, and η is related to the correlation time at the wave number k which is scaled as k ^{?2 + η}. It is shown that the scaling regimes of the present model of vector advection have essentially different properties than the scaling regimes of the corresponding model of passively advected scalar quantity. The results demonstrate the fact that within the present model of passively advected vector field the internal tensor structure of the advected field can have nontrivial impact on the diffusion processes deep inside in the inertial interval of given turbulent flow.     

Thomas-Fermi theory for atomic nuclei revisited

The recently developed semiclassical variational Wigner-Kirkwood (VWK) approach is applied to finite nuclei using external potentials and self-consistent mean fields derived from Skyrme interactions and from relativistic mean field theory. VWK consists of the Thomas-Fermi part plus a pure, perturbative ?² correction. In external potentials, VWK passes through the average of the quantal values of the accumulated level density and total energy as a function of the Fermi energy. However, there is a problem of overbinding when the energy per particle is displayed as a function of the particle number. The situation is analyzed comparing spherical and deformed harmonic oscillator potentials. In the self-consistent case, we show for Skyrme forces that VWK binding energies are very close to those obtained from extended Thomas-Fermi functionals of ?⁴ order, pointing to the rapid convergence of the VWK theory. This satisfying result, however, does not cure the overbinding problem, i.e., the semiclassical energies show more binding than they should. This feature is more pronounced in the case of Skyrme forces than with the relativistic mean field approach. However, even in the latter case the shell correction energy for e.g., ²⁰⁸Pb turns out to be only ∼−6 MeV what is about a factor two or three off the generally accepted value. As an ad hoc remedy, increasing the kinetic energy by 2.5%, leads to shell correction energies well acceptable throughout the periodic table. The general importance of the present studies for other finite Fermi systems, self-bound or in external potentials, is pointed out.     

Random-phase approximation for the grand-canonical potential of composite fermions in the half-filled lowest Landau level

We reconsider the theory of the half-filled lowest Landau level using the Chern-Simons formulation and study the grand-canonical potential in the random-phase approximation (RPA). Calculating the unperturbed response functions for current- and charge-density exactly, without any expansion with respect to frequency or wave vector, we find that the integral for the ground-state energy converges rapidly (algebraically) at large wave vectors k, but exhibits a logarithmic divergence at small k. This divergence originates in the k^-2 singularity of the Chern-Simons interaction and it is already present in lowest-order perturbation theory. A similar divergence appears in the chemical potential. Beyond the RPA, we identify diagrams for the grand-canonical potential (ladder-type, maximally crossed, or a combination of both) which diverge with powers of the logarithm. We expand our result for the RPA ground-state energy in the strength of the Coulomb interaction. The linear term is finite and its value compares well with numerical simulations of interacting electrons in the lowest Landau level. Received: 19 February 1998 / Revised: 25 March 1998 / Accepted: 17 April 1998