Self-consistent Approximation Scheme Research Articles

Nonperturbative finite-temperature Yang-Mills theory

We present nonperturbative correlation functions in Landau-gauge Yang-Mills theory at finite temperature. The results are obtained from the functional renormalisation group within a self-consistent approximation scheme. In particular, we compute the magnetic and electric components of the gluon propagator, and the three- and four-gluon vertices. We also show the ghost propagator and the ghost-gluon vertex at finite temperature. Our results for the propagators are confronted with lattice simulations and our Debye mass is compared to hard thermal loop perturbation theory.

Critical exponents from the two-particle irreducible1/Nexpansion

We calculate the critical exponent $\ensuremath{\nu}$ of the $O(N)$ symmetric ${\ensuremath{\varphi}}^{4}$ model within the $1/N$ expansion of the two-particle-irreducible effective action, which provides us with a self-consistent approximation scheme for the correlation function. The exponent $\ensuremath{\nu}$ controls the behavior of a two-point function $⟨\ensuremath{\varphi}\ensuremath{\varphi}⟩$ near the critical point $T\ensuremath{\ne}{T}_{c}$ through the correlation length $\ensuremath{\xi}\ensuremath{\sim}|T\ensuremath{-}{T}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}$, but we notice that it appears also in the scaling form of the three-point vertex function ${\ensuremath{\Gamma}}^{(2,1)}\ensuremath{\sim}⟨\ensuremath{\varphi}\ensuremath{\varphi}{\ensuremath{\varphi}}^{2}⟩$ at the critical point $T={T}_{c}$; in the momentum space, ${\ensuremath{\Gamma}}^{(2,1)}\ensuremath{\sim}{k}^{2\ensuremath{-}\ensuremath{\eta}\ensuremath{-}1/\ensuremath{\nu}}$. We derive a self-consistent equation for ${\ensuremath{\Gamma}}^{(2,1)}$ from the two-particle-irreducible effective action including the skeleton diagrams up to the next-leading-order in the $1/N$ expansion, and solve it to the leading-log accuracy (i.e., keeping the leading $\mathrm{ln}k$ terms) to obtain $\ensuremath{\nu}$. Our results turn out to improve those obtained in the standard one-particle-irreducible calculation at the next-leading-order.

Self-consistent perturbation expansion for Bose-Einstein condensates satisfying Goldstone’s theorem and conservation laws

Quantum-field-theoretic descriptions of interacting condensed bosons have suffered from the lack of self-consistent approximation schemes satisfying Goldstone's theorem and dynamical conservation laws simultaneously. We present a procedure to construct such approximations systematically by using either an exact relation for the interaction energy or the Hugenholtz-Pines relation to express the thermodynamic potential in a Luttinger-Ward form. Inspection of the self-consistent perturbation expansion up to the third order with respect to the interaction shows that the two relations yield a unique identical result at each order, reproducing the conserving-gapless mean-field theory [T. Kita, J. Phys. Soc. Jpn. 74, 1891 (2005)] as the lowest-order approximation. The uniqueness implies that the series becomes exact when infinite terms are retained. We also derive useful expressions for the entropy and superfluid density in terms of Green's function and a set of real-time dynamical equations to describe thermalization of the condensate.

Primordial non-Gaussianities in single-field inflation

We calculate the three-point function for primordial scalar fluctuations in a single-fieldinflationary scenario where the scalar field Lagrangian is a completely general function ofthe field and its first derivative. We obtain an explicit expression for the three-pointcorrelation function in a self-consistent approximation scheme where the expansion ratevaries slowly, analogous to the slow-roll limit in standard, single-field inflation. Thethree-point function can be written in terms of the familiar slow-roll parameters and threenew parameters which measure the non-trivial kinetic structure of the scalar field, thedeparture of the speed of sound from the speed of light, and the rate of change of the speedof sound.

Magnetism of finite-n Kondo-lattice films

Abstract We investigate the magnetic properties of thin ferromagnetic Kondo-lattice model films. The presence of mobile charge carriers leads to a coupling of the localized magnetic moments (e.g., in Gd). Within a self-consistent approximation scheme, the critical temperature is determined. It exhibits two regimes: an RKKY-like in the weak-coupling region and a strong-coupling double exchange regime.

Renormalization in self-consistent approximation schemes at finite temperature. III. Global symmetries

We investigate the symmetry properties for Baym's $\Phi$-derivable schemes. We show that in general the solutions of the dynamical equations of motion, derived from approximations of the $\Phi$-functional, do not fulfill the Ward-Takahashi identities of the symmetry of the underlying classical action, although the conservation laws for the expectation values of the corresponding Noether currents are fulfilled exactly for the approximation. Further we prove that one can define an effective action functional in terms of the self-consistent propagators which is invariant under the operation of the same symmetry group representation as the classical action. The requirements for this theorem to hold true are the same as for perturbative approximations: The symmetry has to be realized linearly on the fields and it must be free of anomalies, i.e., there should exist a symmetry conserving regularization scheme. In addition, if the theory is renormalizable in Dyson's narrow sense, it can be renormalized with counter terms which do not violate the symmetry.

Renormalization of self-consistent approximation schemes at finite temperature. II. Applications to the sunset diagram

The theoretical concepts for the renormalization of self-consistent Dyson resummations, deviced in the first paper of this series, are applied to first example cases for the $\phi^4$-theory. Besides the tadpole (Hartree) approximation as a novel part the numerical solutions are presented which includes the sunset self-energy diagram into the self-consistent scheme based on the $\Phi$-derivable approximation or 2PI effective action concept.

Renormalization in self-consistent approximation schemes at finite temperature: Theory

Within finite temperature field theory, we show that truncated non-perturbative self-consistent Dyson resummation schemes can be renormalized with local counter-terms defined at the vacuum level. The requirements are that the underlying theory is renormalizable and that the self-consistent scheme follows Baym''s $\Phi$-derivable concept. The scheme generates both, the renormalized self-consistent equations of motion and the closed equations for the infinite set of counter terms. At the same time the corresponding 2PI-generating functional and the thermodynamical potential can be renormalized, in consistency with the equations of motion. This guarantees the standard $\Phi$-derivable properties like thermodynamic consistency and exact conservation laws also for the renormalized approximation schemes to hold. The proof uses the techniques of BPHZ-renormalization to cope with the explicit and the hidden overlapping vacuum divergences.

Dynamical theory of strongly coupled two-dimensional Coulomb fluids in the weakly degenerate quantum domain.

We study the problem of dynamical response and plasma mode dispersion in strongly coupled two-dimensional Coulomb fluids (2DCFs) in the weakly degenerate quantum domain. Adapting the nonlinear response function approach of Golden and Kalman [Phys. Rev. A 19, 2112 (1979)] to the 2DCF, we construct a self-consistent approximation scheme for the calculation of the density response functions and plasma mode dispersion at long wavelengths. The basic ingredients in the construction are (i). the first kinetic equation in the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, (ii). the velocity-average-approximation (VAA) hypothesis, (iii.) the quadratic fluctuation-dissipation theorem, and (iv). the dynamical superposition approximation (DSA) closure hypothesis. The reliability of the VAA-DSA theory can be assessed by observing that the principal coupling correction to the 2D temperature-dependent Lindhard function is identified as being precisely the part of the third-frequency-moment sum-rule coefficient proportional to the potential energy.

Electronic states of a two-dimensional electron system in a lateral superlattice and a perpendicular magnetic field

The electronic state of a two-dimensional electron system (2DES) in the presence of a perpendicular uniform magnetic field and a lateral superlattice (LS) is investigated theoretically. A comparative study is made between a LS induced by a spatial electrostatic potential modulation (referred to as a PMLS) and that induced by a spatial magnetic-field modulation (referred ro asa MMLS). By utilizing a finite-temperature self-consistent Hartree-Fock approximation scheme; the dependence of the electronic state on different system parameters (e.g., the modulation period, the modulation strength, the effective electron-electron interaction strength, the averaged electron density, and the system temperature) is studied in detail. The inclusion of exchange effect is found to bring qualitative changes to the electronic state of a PMLS, leading generally to a nonuniform spin splitting, and consequently the behavior of the electronic state becomes similar to that of a MMLS. The Landau-level coupling is taken into account, and is found to introduce some interesting features not observed before. It is also found that, even in the regime of intermediate modulation strength, the density dependence of the spin splitting of energy levels, either for a PMLS or a MMLS, can be qualitatively understood within the picture of a 2DES in a perpendicular magnetic field with the modulation viewed as a perturbation. [S0163-1829(97)02248-0].

Chirally Symmetric [formula omitted](1/ NC) Corrections to the Nambu-Jona-Lasinio Model

We develop an extended chirally symmetric self-consistent approximation scheme to the Nambu-Jona-Lasinio model, that corresponds to O (1/ N C ) corrections to the usual Hartree + random phase approximations. This scheme amounts to adding "meson cloud" contributions self-consistently to the quark self-energy and the meson polarization functions in a manner suggested by the weakly interacting nature of the quark and collective meson degrees of freedom of the NJL model in the large N C limit. We demonstrate explicitly that this scheme fulfills all the chiral symmetry theorems, namely the Goldstone theorem, the Goldberger-Treiman relation, and the conservation of the quark axial current. We explore the corrections to the quark self-energy and scalar condensate, as well as to the pion polarization function and the weak decay constant ƒ π. The numerical evaluation of these corrections is presented and discussed.

Self-consistent local-density approximation with model Coulomb pair-correlation functions for electronic systems.

The model Coulomb pair-correlation functions proposed several years ago by Gritsenko, Bagaturyants, Kazansky, and Zhidomirov are incorporated into the self-consistent local-density approximation (LDA) scheme for electronic systems. Different correlation functions satisfying well-established local boundary conditions and integral conditions have been tested by performing LDA calculations for closed-shell atoms. Those correlation functions contain a single parameter which can be optimized by fitting the atomic correlation energies to empirical data. In this way, a single (universal) value of the parameter is found to give a very good fit for all the atoms studied

Effect of repulsion among fermion pairs on the problem of crossover between Cooper-pair and Bose-Einstein condensation

Interpolation of the chemical potential μ in the Cooper-pair and the Bose-Einstein condensation, as increasing the strength of attraction among fermions, is discussed in 2D by extending Nozières and Schmitt-Rink's theory so as to include the repulsion among fermion pairs along the self-consistent renormalization approximation scheme. It is shown in the weak attraction limit that the repulsion makes the solution with negative chemical potential μ < 0 and T c = 0 unstable and pushes up μ toward the Fermi energy of non-interacting system.

Self-consistent approximation to the solution of the Bethe-Salpeter equation.

A technique for the approximate solution of the Bethe-Salpeter equation is examined. The technique requires the solution of a pair of coupled equations for the relative-momentum and relative-energy dependence of the relativistic T matrix. The solutions obey a self-consistency requirement as well as the usual elastic-unitarity constraint. It is also shown that the approximate T matrix is stable under a single iteration in the exact four-dimensional equation at certain kinematic points, including the fully on-shell point. A model problem with an exactly solvable separable interaction is examined and exact, approximate, as well as three-dimensional reduction results are compared. The phase shifts calculated in this self-consistent approximation scheme are found to be in excellent agreement with the exact phase shifts.

Nonlinear-response-function approach to binary ionic mixtures: Dynamical theory.

The authors generalize the Golden-Kalman one-component-plasma (OCP) nonlinear-response-function approach to the formulation of a dynamical theory for binary-ionic-mixture plasmas. The principal result of the new dynamical theory is a self-consistent approximation scheme for the calculation of linear ionic polarizabilities and collective-mode structure at long wavelengths and arbitrary coupling strengths. The approximation scheme is constructed from the dynamical nonlinear fluctuation-dissipation theorem and the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) kinetic equation prepared in the velocity-average approximation (VAA). Equilibrium three-point correlations, quadratic response functions, and the dynamical superposition approximation are all central elements of the theory. The theory is exact at zero frequency and exactly reproduces the coefficients of the high-frequency-moment-sum-rule expansion through order 1/..omega../sup 4/. Collective-mode calculations based on the new approximation scheme indicate that the (positive) shift in the plasma frequency is temperature dependent at weak coupling and temperature independent at very strong coupling. These calculations, moreover, reproduce the qualitative features of the Hansen-McDonald-Vieillefosse molecular-dynamics data for the dispersion of the optical mode in the strong-coupling regime, while at the same time very nearly reproducing the weak-coupling frequency shift predicted by Baus's microscopic theory. The authors conclude that the temperature-dependent broadening and shifting of the plasma made at weak coupling is controlled primarily bymore » ionic interdiffusion transport.« less

Dynamical theory of classical surface plasmas

The authors present a self-consistent approximation scheme for the calculation of the dynamical polarizability \ensuremath{\alpha}(k\ensuremath{\rightarrow},\ensuremath{\omega}) and plasmon dispersion at long wavelengths in electron films trapped on the free surface of liquid helium. The principal building blocks to the construction of the approximation scheme are the nonlinear fluctuation-dissipation theorem and linearized equations for the plasma density, fluid velocity, pressure tensor, and heat-flow tensor moments. Equilibrium three-point correlations, quadratic polarizability response functions, and the Navier-Stokes hypothesis linking the pressure tensor to its trace are all central elements in the development of the theory. At frequencies high compared with the collision frequency, the Golden-Lu (GL) formula for \ensuremath{\alpha}(k\ensuremath{\rightarrow},\ensuremath{\omega}) exactly satisfies the ${\ensuremath{\omega}}^{\mathrm{\ensuremath{-}}4}$ moment sum rule. Analysis of ${\ensuremath{\alpha}}_{\mathrm{GL}(\mathrm{k}\ensuremath{\rightarrow}}$,\ensuremath{\omega}) at \ensuremath{\omega}\ensuremath{\simeq}(2\ensuremath{\pi}${\mathrm{ne}}^{2}$k/m${)}^{1/2}$\ensuremath{\gg}${\mathrm{kv}}_{\mathrm{th}}$ leads to the description of the long-wavelength plasmon structure over the range of couping strengths spanning the entire fluid regime. The GL theory predicts that the transition from plasmonlike to longitudinal-phonon-like dispersion occurs at the critical value of the coupling parameter ${\ensuremath{\Gamma}}_{\mathrm{crit}}$=3.22; this transition is inextricably linked to the onset of a ``liquid-state'' short-range order signaled by the development of oscillations in the equilibrium pair correlation function somewhere in the range 2.2\ensuremath{\Gamma}2.9. In the infinite coupling (\ensuremath{\Gamma}\ensuremath{\rightarrow}\ensuremath{\infty}) limit, the GL long-wavelength dispersion relation very nearly reproduces the zero-temperature Bonsall-Maradudin longitudinal-phonon dispersion for the two-dimensional hexagonal lattice.

Self-consistent approximations and the Legendre transform

It is well-known in quantum field theory and statistical physics that the effective action (i.e., the Legendre transform of the free energy) is the generating function for one particle irreducible diagrams. We present an explicit diagrammatic proof of this result which permits a clear statement concerning its validity in the context of self-consistent approximation schemes. In particular, we find a necessary and sufficient condition on the approximate set of Green's functions for the Legendre transform theorem to hold. This condition is exactly what one expects on physical grounds. Our proof also lends itself to an examination of more extended diagrammatic schemes, such as those involving composite operators in renormalization theory.

Approximation scheme for strongly coupled plasmas: Dynamical theory

The authors present a self-consistent approximation scheme for the calculation of the dynamical polarizability $\ensuremath{\alpha}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},\ensuremath{\omega})$ at long wavelengths in strongly coupled one-component plasmas. Development of the scheme is carried out in two stages. The first stage follows the earlier Golden-Kalman-Silevitch (GKS) velocity-average approximation approach, but goes much further in its application of the nonlinear fluctuation-dissipation theorem to dynamical calculations. The result is the simple expression for $\ensuremath{\alpha}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},\ensuremath{\omega})$, ${\ensuremath{\alpha}}_{\mathrm{GKS}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},\ensuremath{\omega})={\ensuremath{\alpha}}_{\mathrm{RPA}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},\ensuremath{\omega})[1+v(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},\ensuremath{\omega})]$, where the dynamical screening function $v(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},\ensuremath{\omega})$ is expressed in terms of quadratic polarizabilities, and RPA stands for random-phase approximation. Its zero-frequency limit $v(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},0)$ has already been established and analyzed in the earlier GKS work. At high frequency, ${\ensuremath{\alpha}}_{\mathrm{GKS}} (\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},\ensuremath{\omega}\ensuremath{\rightarrow}\ensuremath{\infty})$ exactly satisfies the $\frac{1}{{\ensuremath{\omega}}^{4}}$ moment sum rule. In the second stage, the above dynamical expression is made self-consistent at long wavelengths by postulating that a decomposition of the quadratic polarizabilities in terms of linear ones, which prevails in the $k\ensuremath{\rightarrow}0$ limit for weak coupling, can be relied upon as a paradigm for arbitrary coupling. The result is a relatively simple quadratic integral equation for $\ensuremath{\alpha}$. Its evaluation in the weak-coupling limit and its comparison with known exact results in that limit reveal that almost all important correlational and long-time effects are reproduced by our theory with very good numerical accuracy over the entire frequency range; the only significant defect of the approximation seems to be the absence of the "dominant" $\ensuremath{\gamma}\mathrm{ln}{\ensuremath{\gamma}}^{\ensuremath{-}1}$ ($\ensuremath{\gamma}$ is the plasma parameter) contribution to $\mathrm{Im} \ensuremath{\alpha}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},\ensuremath{\omega})$.

Superconducting virtual bound state alloys

Static and dynamic properties of superconducting alloys containing resonant impurity scattering centers are considered. The formation of bound states within the energy gap is described and connected with locald-level correlations induced by superconductivity. The effect of the bound states on the exactly solubled-spin dynamics is investigated by evaluating the impurity atom's magnetic excitation spectrum. Finite impurity concentrations are treated within a self-consistent approximation scheme. For increasing impurity content the bound states merge to impurity bands which tend to suppress superconductivity. The relevance of the simple extra orbital model is discussed in connection with the interpretation of pressure-induced variations on the properties of superconductors alloyed with ambivalent rare earth ions.

Self-consistent approximation schemes for the treatment of a prototype ferroelectric

Abstract As a simple prototype of a ferroelectric crystal we treat a system of coupled anharmonic oscillators in order to determine the range of applicability of different self-consistent approximation schemes. In particular, we examine the self-consistent phonon approximation with an intercell coupling which possesses a variable range of interaction. In the limit where the interaction is of sufficiently long range, the density matrix of the system is obtained numerically exactly.