Equations For Green Functions Research Articles

Bound States and Bootstraps in Field Theory

We present a general, model-independent theory of composite particles. Starting from a general local Lagrangian containing an elementary particle, we show how this particle becomes composite (in a precisely defined sense) as its wave-function renormalization constant tends to zero. As this limit is taken, the usual dynamical type of field equation changes its form. We show that the Green's functions of the theory still possess the usual physical interpretation in the limit; in particular, for renormalizable theories, if a certain coupling constant does not tend to zero, the "composite" propagator possesses a pole at the composite mass and with nonzero residue. We show that the operator form of the Green's-function equations describing the composite particle can be manipulated by natural approximations to explain a wide variety of known modeldependent results on composite particles, and in particular to obtain all the usual types of bootstrap equations. We present a preliminary classification of the operator equations, showing that many of them cannot possess physically meaningful solutions. Our results agree qualitatively with earlier, heuristic discussions. Finally we make some remarks about the possibility of "bootstrapped symmetry schemes."

Theory of Localized Magnetic States in Metals

The problem of determining the conditions for the occurrence of localized moments in dilute alloys on the basis of Anderson's model is re-examined. From a coupled set of Green's-function equations an approximate solution is found which includes the electron correlation in the impurity levels and which passes over to the exact solution in the limiting case of an isolated impurity. The Hartree-Fock theory is obtained only if the Coulomb repulsion $U$ is relatively small. When $U$ is large the correlation effects are important, and magnetic states are found only if the energy shift caused by the impurity-band electron scattering satisfies certain fairly restrictive conditions. Possible modifications when the degeneracy of the impurity levels, the intra-atomic exchange interactions, and the electron interactions in the host metal are taken into account are qualitatively discussed.

Thermal Properties of Spin-Wave Impurity States

The effect of dilute magnetic impurities on the thermal properties of an ideal simple cubic spin-\textonehalf{} Heisenberg ferromagnet has been investigated using the thermal-Green's-function procedure with a simple randomphase decoupling scheme. It is shown that for a small ratio $\ensuremath{\epsilon}$ of impurity-host to host-host exchange, low-lying $s$-type virtual spin-wave states result which cause a large density of states to occur at low energies. These low-energy states lead to an accumulation of spin disorder at and near the impurity site. Consequently the impurity magnetization decreases far more rapidly than that of the host. This effect is accompanied by a large increase in the low-temperature spin-wave specific heat. Analytic solutions to the Green's-function equations are calculated for temperatures near 0 and near ${T}_{c}$ the Curie temperature. Self-consistent numerical solutions are presented for both the magnetization and the spin-wave specific heat as a function of temperature. For small $\ensuremath{\epsilon}$ the impurity magnetization is approximated by the Brillouin function ${m}_{I}={\ensuremath{\mu}}_{\ensuremath{\beta}}tanh{{\ensuremath{\mu}}_{\ensuremath{\beta}}H+\frac{6{J}^{\ensuremath{'}}〈{{S}_{1}}^{z}〉}{\mathrm{kT}}},$ where ${J}^{\ensuremath{'}}$ is host-impurity exchange and $〈{{S}_{1}}^{z}〉$ is the thermal average for the impurity nearest-neighbor spins. $〈{{S}_{1}}^{z}〉$ is found to be depressed from the bulk value by an amount which increases with temperature and is about 0.84 of the bulk value as $T\ensuremath{\rightarrow}{T}_{c}$.

First-Order Green's-Function Theory of the Heisenberg Ferromagnet

Coupled Green function equations for Heisenberg ferromagnet approximated via differential equations

Thermal Properties of Spin-Wave Impurity States

The effect of dilute magnetic impurities on the thermal properties of an ideal simple cubic spin ½ Heisenberg ferromagnet has been investigated. It is shown that for a small ratio ε of impurity—host to host—host exchange low-lying ``s-type'' virtual spin-wave states result which cause a large density of states to occur at low energies. These low-energy states lead to an accumulation of spin disorder at and near the impurity site. Consequently the impurity magnetization decreases far more rapidly than that of the host. This effect is accompanied by a large increase in the low-temperature spin-wave specific heat. Analytic solutions to the Green's function equations are calculated for temperatures near T = 0 and near T = Tc, the Curie temperature. Self-consistent numerical solutions are presented for both the magnetization and spin-wave specific heat as a function of temperature. For small ε the impurity magnetization is approximated by the Brillouin function mI=μβ tanh [(μβH+6J′〈S1z〉)/kT],where J′ is the host-impurity exchange and 〈S1z〉 is the thermal average for the impurity nearest-neighbor spins. 〈S1z〉 is found to be depressed from the bulk value by an amount which increases with temperature and is about 0.84 of the bulk value as T→Tc. A detailed account of these calculations will be published elsewhere.

The theory of the ferromagnetism of conduction electrons at low temperatures

A theory is presented in which the effect of spin waves on the single-particle states of conduction electrons is obtained as well as the effect of the conduction electrons on the spin waves. Green function techniques are employed. The Hamiltonian is taken to contain the single-particle energies of the conduction electrons in the absence of interactions, the Coulomb interaction between electrons in Wannier states centred on the same lattice site C , and the interatomic exchange terms J ij . Interband integrals are neglected. The chain of equations for the single-particle Green functions is decoupled in such a way as to include the effects of the spin waves in the single-particle Green functions. The theory is worked out on the assumption that C is very much greater than the band width and the J ij so that at T ═ 0 the double occupation of Wannier orbital states is the minimum possible. The resulting single-particle occupation numbers are linear combinations of Fermi-Dirac functions. The low temperature spontaneous magnetization ξ is found to be a product of a spin-wave magnetization and a single-particle magnetization ξ s.p ., and so contains terms varying as T 1 and T 1 , and T 2 if both spin sub-bands are partially occupied in the ground state. The low temperature specific heat contains T and T 1 terms. The results of the Heisenberg model are obtained in the appropriate limit. Expressions for the spin-wave energy and its temperature dependence are discussed.

A model of the insulator-metal transition Part I. Two-body correlations

A simple cubic hydrogenic lattice in the extreme tight binding approximation with only one energy band is set up as a model for an insulator. A two particle Green's function equation for one full and one empty state is derived with a mutual interaction potential. The model is a generalization of one by Hubbard applying only one particle Green's functions to electrons in narrow energy bands. The model is also a generalization of one by Slater and Koster which applies to electron-impurity interactions. It is shown that the energy of a pair of full and empty states depends on the orientation and magnitude of the relative crystal momentum of the pair. Kohn's criterion for the insulating state is satisfied by all bound pairs. A qualitative discussion of this lattice as a superposition of pairs is also given on the basis of the quasi-chemical equilibrium theory.

Green's Function Method in the Theory of Ferromagnetism

AbstractThe limits of applicability are established for the different decoupling procedures in the Green's function equations of the spin system of an Heisenberg ferromagnet.

Coupled Electron-Phonon System

The coupled electron-phonon system is considered for phonon spectra of Einstein and Debye forms. The single-particle electron Green's function $G$ is calculated in a nonperturbative manner in both models, and its spectral weight function is examined to determine the validity of a quasiparticle picture. The weight function and the poles of $G$ both lead to several branches of excitations rather than a single "dressed" electron. The asymptotic time dependence of the $G$ is found, and the effect of multiphonon processes on the electron decay rate is discussed. The electronic polarizability, $P$ of the interacting system is calculated with the aid of a generalized Ward's identity for the electron-phonon vertex. This identity, which is a consequence of electronic charge conservation, is derived in an Appendix. The calculation of $P$ is carried out in the limit that the Fermi velocity is small compared with the phase velocity of the polarization field. An Appendix on the formal development of the Green's function equations is included.

Green's functions for fermion systems with pairing correlations

The Bogoliubov-Valatin canonical transformation is used to transform the Hamiltonian for a system of interacting fermions, to that for a system of quasi particles in which an important part of the pairing interaction has been absorbed into the free quasi particle. Field theoretic Green's functions, which do not in general conserve the number of quasi particles, are introduced to treat the interaction between the quasi particles. These Green's functions are shown to satisfy an infinite set of coupled integral equations, which can be conveniently represented graphically. The infinite set of equations can be terminated by neglecting the interaction Hamiltonian above some arbitrarily chosen equation. The Green's function equations are then solved in the special case where the sum of quasi particles created and annihilated is four, neglecting self-energy terms, and using a pairing plus Coulomb interaction of the type important in superconductivity. A secular equation for the energy eigenvalues is obtained which is examined for the cases of no pairing interaction and no Coulomb interaction. When the Coulomb interaction alone is present the secular equation becomes that obtained by Sawada et al. for plasmons. When the pairing interaction alone is present, the energy spectrum for excitons, bound pairs of quasi particles, is obtained. It is shown that one of the excitons is a phonon.

Functional Derivative Techniques in the Theory of Superconductivity

The equations for the Green's functions of the theory of superconductivity are developed in an iterative scheme in which the two-particle function is rewritten as the functional derivative of the single-particle function with respect to external-source terms. When a source coupled to electron pairs in addition to an external potential coupled to the charge density is included in the Hamiltonian, the possibility for a gap in the single-particle excitations is found. The first order in the iteration scheme for the solution to the Green's function equations is independent of the way in which the two-particle function is generated, as the final result must be. Equations for the vertex functions are obtained and used to find the linear response of the current to an applied electromagnetic field. An exact solution to the vertex equation of interest is used to show that no current will be induced by a static longitudinal vector potential. In addition, all the functions calculated are found to have translationally invariant solutions. Thus, the original invariances of the Hamiltonian are restored when the source terms are set equal to zero. Corrections to the self-energy are estimated by using another of the solutions to the vertex equations.

Generalized Hartree-Fock Method

A variational principle is formulated to determine the single-particle states, their pairing, and the occupation number distribution for a trial state vector of the Bardeen, Cooper, Schrieffer type. The equations which are derived generalize those of the Hartree-Fock method obtained with a Slater determinant trial wave function. It is pointed out that in a suitable representation the vacuum state of a general quasi-particle transformation has such a trial form which exhibits directly the pairing of single-particle states. Another variational principle determines the excitation energies.Two coupling cases are distinguished: the commutative case in which the self-consistent densities and energies are related to quantities which all commute, and the more general noncommutative case. The latter is of importance in critical-field phenomena. The equations for the commutative case can be written in a matrix form which retains its validity in the more general noncommutative case. The simple matrix commutator equations appear as direct generalizations of the density matrix form of the Hartree-Fock equations.The equations for small oscillations have an equally simple form. Their connection with a diagonal representation of the quasi-particle energies is exhibited in a way which remains valid in the general coupling case. The "unphysical" solutions are excluded by the supplementary condition. The contact with the Green's function approach is established. The generalized matrix form of the Green's function equations shows especially clearly the symmetry properties of the method.

Theory of Many-Particle Systems. I

This is the first of a series of papers dealing with many-particle systems from a unified, nonperturbative point of view. It contains derivations and discussions of various field-theoretical techniques which will be applied in subsequent papers. In a short introduction the general method of approach is summarized, and its relationship to other field-theoretic problems indicated. In the second section the macroscopic properties of the spectra of many-particle systems are described. Asymptotic evaluations are performed which characterize these macroscopic features in terms of intensive parameters, and the relationship of these parameters to thermodynamics is discussed. The special characteristics of the ground state are shown to follow as a limiting case of the asymptotic evaluations. The third section is devoted to the time-dependent field correlation functions, or Green's functions, which describe the microscopic behavior of a multiparticle system. These functions are defined, and related to intensive macroscopic variables when the energy and number of particles are large. Spectral representations and other properties of various one-particle Green's functions are derived. In the fourth section the treatment of non-equilibrium processes is considered. As a particular example, the electromagnetic properties of a system are expressed in terms of the special two-particle Green's function which describes current correlation. The discussion yields specifically a fluctuation-dissipation theorem, a sum rule for conductivity, and certain dispersion relations. The fifth section deals with the differential equations which determine the Green's functions. The boundary conditions that characterize the Green's function equations are exhibited without reference to adiabatic decoupling. A method for solving the equations approximately, by treating the correlations among successively larger numbers of particles, is considered. The first approximation in this sequence is shown to yield a generalized Hartree-like equation. A related, but rigorous, identity for the single-particle Green's function is then derived. A second approximation, which takes certain two-particle correlations into account, is shown to produce various additional effects: The interaction between particles is altered in a manner characterized by the intensive macroscopic parameters, and the modification and spread of the energy-momentum relation come into play. In the final section compact formal expressions for the Green's functions and other physical quantities are derived. Alternative equations and systematic approximations for the Green's functions are obtained.

Electrodynamic Displacement of Atomic Energy Levels. III. The Hyperfine Structure of Positronium

A functional integro-differential equation for the electron-positron Green's function is derived from a consideration of the effect of sources of the Dirac field. This equation contains an electron-positron interaction operator from which functional derivatives may be eliminated by an iteration procedure. The operator is evaluated so as to include the effects of one and two virtual quanta. It contains an interaction resulting from quantum exchange as well as one resulting from virtual annihilation of the pair. The wave functions of the electron-positron system are the solutions of the homogeneous equation related to the Green's function equation. The eigenvalues of the total energy of the system may be found by a four-dimensional perturbation technique. The system bound by the Coulomb interaction is here treated as the unperturbed situation. Numerical values for the spin-dependent change of the energy from the Coulomb value in the ground state are finally obtained accurate to order $\ensuremath{\alpha}$ relative to the hyperfine structure ${\ensuremath{\alpha}}^{2}$ Ry. The result for the singlet-triplet energy difference is $\ensuremath{\Delta}{W}_{\mathrm{ts}}=\frac{1}{2}{\ensuremath{\alpha}}^{2}{\mathrm{Ry}}_{\ensuremath{\infty}}[\frac{7}{3}\ensuremath{-}(\frac{32}{9}+2\mathrm{ln}2)\frac{\ensuremath{\alpha}}{\ensuremath{\pi}}]=2.0337\ifmmode\times\else\texttimes\fi{}{10}^{5}\frac{\mathrm{Mc}}{sec}.$ Theory and experiment are in agreement.