Thermodynamic Green's Functions Research Articles

Renormalization of theσmodel at finite temperature

The temperature dependence of a many-body theory with the dynamics defined by the relativistic linear $\ensuremath{\sigma}$ model is studied. The model has SU(2) \ifmmode\times\else\texttimes\fi{} SU(2) chiral symmetry with fermions belonging to a (\textonehalf{},0) + (0,\textonehalf{}) representation interacting with the $\ensuremath{\sigma}$ and $\ensuremath{\pi}$ mesons belonging to the (\textonehalf{}, \textonehalf{}) representation of the chiral symmetry group. The dimensional-regularization technique together with the renormalization procedure of 't Hooft is used to reproduce the well-known result that the counterterms of the symmetric theory remove the divergences of the theory with spontaneous symmetry breaking, without however the need for invoking auxiliary fermion fields. Renormalizability is maintained at finite temperatures by the cancellation of temperature-dependent infinities which appear at the two-loop level. This is shown explicitly for the ground-state expectation value of the scalar $\ensuremath{\sigma}$ field at the two-loop level. When the symmetry is explicitly broken by the term ${f}_{\ensuremath{\pi}}{{m}_{\ensuremath{\pi}}}^{2}\ensuremath{\sigma}$ the symmetry of the original Lagrangian is never restored. In the absence of such a term a symmetry change with temperature is realized and the persistence of the Goldstone mode up to a critical temperature ${T}_{c}$, above which the original symmetry is restored, is verified. Thus below $T={T}_{c}$ the low-energy theorems of current algebra associated with the existence of the Goldstone pions would be valid except that all parameters of the theory develop finite, temperature-dependent, corrections. A parallel discussion for density dependence of the symmetry is included. All calculations are done in the real-time formalism for the thermodynamic Green's functions.

Perturbation theory for classical thermodynamic Green's functions

Perturbation theory for classical thermodynamic Green's functions

Eikonal approximations in quantum statistical mechanics

Eikonal approximations to thermodynamic Green's functions are introduced into the functional representations of quantum statistical mechanics. In terms of summations and quadratures over an instantaneous interparticle potential, one obtains explicit representations at reasonably large temperatures for quantum corrections to classical cluster integrals.

Phonon relaxation time due to impurities with internal degrees of freedom

Application of thermodynamic Green's functions is made to the study of phonon scattering by impurities having two internal energy eigenstates over which transitions take place in the presence of lattice waves. The equation-of-motion method is used to solve the Green's functions. The inverse relaxation time for phonons is found to be essentially temperature dependent and is peaked at a frequency slightly different from the energy gap between the levels. The low-temperature thermal conductivity of Li-doped KCl is finally studied in the light of the present results.

Quantum theory of longitudinal dielectric response properties of a two-dimensional plasma in a magnetic field

An analysis of dynamic and nonlocal longitudinal dielectric response properties of a two-dimensional Landau-quantized plasma is carried out, using a thermodynamic Green's function formulation of the RPA with a two-dimensional thermal Green's function for electron propagation in a magnetic field developed in closed form. The longitudinal-electrostatic plasmon dispersion relation is discussed in the low wavenumber regime with nonlocal corrections, and Bernstein mode structure is studied for arbitrary wavenumber. All regimes of magnetic field strength and statistics are investigated. The class of integrals treated here should have broad applicability in other two-dimensional and finite slab plasma studies. The two-dimensional static shielding law in a magnetic field is analyzed for low wavenumber, and for large distances we find V( r) ∼ Q k 0 2r 3 . The inverse screening length k 0 = 2πe 2∂ϱ ∂ξ (ϱ = density, ξ = chemical potential) is evaluated in all regimes of magnetic field strength and all statistical regimes. k 0 exhibits violent DHVA oscillatory behavior in the degenerate zero-temperature case at higher field strengths, and the shielding is complete when ξ = r′ l z.shtsls; ω , but there is no shielding when ξ ≠ r′ l z.shtsls; ω c . A careful analysis confirms that there is no shielding at large distances in the degenerate quantum strong field limit l z.shtsls; ω c > ξ . Since shielding does persist in the nondegenerate quantum strong field limit l z.shtsls; ω c > KT , there should be a pronounced change in physical properties that depend on shielding if the system is driven through a high field statistical transition. (It should be noted that the static shielding law of semiclassical and classical models has no dependence on magnetic field in two dimensions, as in three dimensions.) Finally, we find that the zero field two-dimensional Freidel-Kohn “wiggle” static shielding phenomenon is destroyed by the dispersal of the zero field continuum of electron states into the discrete set of Landau-quantized orbitals due to the imposition of the magnetic field.

Raman scattering from two-phonon resonance states in SrTiO 3

Two-phonon Raman spectroscopy has been performed on SrTiO 3 from 13.3 to 295 K. In the low frequency region, softenings of two bands are observed. With decreasing temperature, they showed sharpening of structure and intensity transfer. It is concluded that one band comes from two-phonon resonance state of ferroelectric soft mode. Theoretical investigations are performed by using thermodynamic Green's function and by taking into account quartic anharmonicity of the lattice force.

Ground State Energy of the Truncated Anderson Hamiltonian

AbstractThe ground state energy of the Anderson Hamiltonian for infinite dd correlation energy is calculated using the method of thermodynamic Green's functions. The quantum mechanical ground state energy has a term proportional to exp (−α · ed/rv2) for rv2 ⪅ 1.5 where ed, r, v denote the energy of the d‐electron, the density of state and strength of interaction between the d‐electron and the conduction band electrons, all expressed in dimensionless units. The value of α remains almost constant over a wide range of the physical parameter rv2.

Effect of Biquadratic Exchange on Two Magnon States of Heisenberg Ferromagnets: Other Neighbor Exchange

We have investigated the two magnon localized states of a one dimensional Heisenberg ferromagnet the Hamiltonian of which is made up of nearest neighbor and next nearest neighbor isotropic bilinear and biquadratic exchange terms, and a single ion anisotropy term. We have restricted our choice of parameters so that the ground state at T = 0 is the fully aligned ferromagnetic state and we have used the thermodynamic Green functions where the averages have been evaluated in the ground state so that our results are good for [Formula: see text]. We have evaluated the probabilities of finding two spin deviations a distance n apart when the system is in a localized state described by total wave vector q. We have (a) compared the effects of ferromagnetic and antiferromagnetic next nearest neighbor exchange, (b) found that localized modes can lie below or above the two free magnon band depending upon the sign and magnitude of the biquadratic exchange, (c) found that in certain cases two spin deviations appear to behave like objects interacting only via a soft core, and (d) found that modes can have a large single ion component when the single ion anisotropy is zero.

Theory of phonon relaxation due to multilevel impurities

AbstractA theory is developed for the calculation of the single‐mode relaxation time of phonons in a crystal containing impurities that have a possibly degenerate spectrum of energy levels. Transitions of the impurities from one level to another under the effect of the lattice waves can result in the scattering of the latter. Following earlier calculations corresponding to the case of nondegenerate two‐level impurities, certain operators are defined to describe these transitions, which allow to describe the phonon‐impurity system with a model Hamiltonian that is suited to the diagrammatic technique of the thermodynamic Green's functions. The relaxation time obtained exhibits resonant behaviour with peaks at frequencies equal to the energy gaps between various pairs of energy levels. The relaxation time is also found to be quadratic in the degeneracy factors of the impurity levels and shows a dependence on temperature through the occupation probability of these levels.

Scattering of phonons by two-level impurities

An application of imaginary-time thermodynamic Green functions has been made to derive an expression for the relaxation time of phonons in the presence of impurities that have two energy levels corresponding to some sort of internal degree of freedom. The result differs slightly from those available in literature inasmuch as it contains a temperature-dependent term that relates to the difference of the probabilities of the two levels being occupied.

Resonant Scattering of Phonons by Two‐Level Impurities

AbstractApplication of the thermodynamic Green's functions is made to the study of phonen scattering by impurities that have two distinct integral configurations corresponding to different energy eigenvalues. Operators are defined which do the function of transferring the impurities from one state to another, and the Hamiltonian of the system is phenomenologically expressed in terms of these. Imaginary phonon Green's functious are calculated, using finite‐temperature S‐matrix expansion and diagrammatic representation. It is found that the phonon relaxation time, which is obtained from these Green's functions, explicitly involves a temperature dependent factor.

The thermodynamic one-particle Green function in the renormalized spin wave approximation for isotropic cubic ferromagnets

The thermodynamic one-particle Green function in the renormalized spin wave approximation for isotropic cubic ferromagnetic insulators with Dyson's spin wave theory as a base is derived. In quantitative respect, dynamic and kinematic effects of spin waves are approximated by the graphs deficient in the energy denominators, wherefore at low temperature kinematic interaction turns out to be too strong. As against the one-particle Green function for independent spin waves, dynamic interaction of ferromagnons is shown to effect the renormalization of the spin wave energy, whereas kinematic interaction directly modifies the average ferromagnon population numbers. In the matter of magnetization, its formula based on the Green function assumes a similar form as in the spin wave theory without interactions on the understanding that it remains valid within the entire range of temperatures from absolute zero up to the critical point.

The theory and properties of randomly disordered crystals and related physical systems

We review the methods which have been developed over the past several years to determine the behavior of solids whose properties vary randomly at the microscopic level, with principal attention to systems having composition variation on a well-defined structure (random "alloys"). We begin with a survey of the various elementary excitations and put the dynamics of electrons, phonons, magnons, and excitons into one common descriptive Hamiltonian; we then review the use of double-time thermodynamic Green's functions to determine the experimental properties of systems. Next we discuss these aspects of the problem which derive from the statistical specification of the microscopic parameters; we examine what information can and cannot be obtained from averaged Green's functions. The central portion of the review concerns methods for calculating the averaged Green's function to successively better approximation, including various self-consistent methods, and higher-order cluster effects. The last part of the review presents a comparison of theory with the experimental results of a variety of properties---optical, electronic, magnetic, and neutron scattering. An epilogue calls attention to the similarity between these problems and those of other fields where random material heterogeneity has played an essential role.

Surface excitations in magnetic systems with a singlet ground state

We have investigated surface excitations in a system wherein the ionic ground state is a magnetic singlet. The pseudospin formalism is employed to account for the crystal-field and exchange interactions between the ions in a Heisenberg ferromagnet with the singlet-triplet crystal-field-only level scheme. The Hamiltonian was studied in the molecular field approximation to find the possible ground states. Surface excitations for the simple cubic structure were investigated for the (001) surface in the random phase approximation. Analytic expressions have been obtained for the thermodynamic Green functions. For a fully magnetized molecular field ground state, there are in general two bulk bands, the spin-wave and the excitonic. Surface modes were found to exist below the spin-wave band, above the excitonic band and between the bands. The dispersion curves can exist only over one or two limited regions of the two-dimensional wave-vector parallel to the crystal surface.

Effect of finite f level linewidth on the theory of the α-γ transition in Ce

The Ramirez-Falicov theory of the alpha - gamma transition in Ce is extended to include the effect of hybridization of the localized 4f states with the band. The physical properties of the alpha and gamma phases are calculated using thermodynamic Green functions which are determined by a decoupling procedure that accounts for the highly correlated character of the f levels. The decoupling procedure is exact to second order in the hybridization parameter. Satisfactory values are found for the noninteger valency, electronic specific heat and magnetic susceptibility of the alpha phase. A Curie law for the gamma phase is also obtained.

Bounds for thermodynamic Green's functions from the application of path integral methods

Symanzik's inequalities for thermodynamic Green's functions are applied to correlation functions occurring in the quantum statistical mechanics of dilute gases. The application is based on the use of the constant-force Green's function. The results are compared with the results of classical path approximations. The Green's function inequality corresponding to the Feynman-Hibbs partition function inequality is derived and discussed. Applications to a model of hydrogen vapor and to the statistical mechanics of the hydrogen atom are presented.

Surface States of an Induced-Moment System and a Hydrogen-Bonded Ferroelectric

We have investigated the surface bound states and resonances that occur on a (001) surface of either a two-level induced-moment system or the tunneling model of a hydrogen-bonded ferroelectric. We have assumed (a) that there is a perfectly sharp surface and (b) that only parameters on the surface have values different from those in the bulk of the crystal, and we have used the random-phase approximation to solve the equations of motion for the thermodynamic Green's functions. Analytical expressions have been obtained for the complete Green's functions. Three phases can exist: (i) Both the surface and the bulk are disordered, (ii) the surface is ordered but the bulk is disordered, or (iii) both the surface and the bulk are ordered. In phases (i) and (ii) only one kind of localized mode can appear, while in phase (iii) two kinds, localized on the first and second layers, can exist. No resonances appear inside the bulk band when both surface and bulk are disordered, but resonances can appear in the other two phases. Some criteria for the appearance of bound states have been derived and numerical calculations have been carried out for the three phases at zero temperature. Some experiments are suggested.

Point-Group Symmetry and the Impurity Problem

It is shown that in a pure crystal the density of states of a given point-group symmetry (that is, belonging to a given row of a given irreducible representation of the point group) is simply $(\frac{{d}_{\ensuremath{\Gamma}}}{p}){g}^{0}(E)$, where ${g}^{0}(E)$ is the familiar over-all density of states, ${d}_{\ensuremath{\Gamma}}$ is the dimensionality of the representation, and $p$ is the order of the group. Point-group symmetry is incorporated into Green's-function theory to define Green's functions which propagate excitations of particular symmetries from shell to shell (rather than from site to site). Both simple crystal Green's functions and two-time thermodynamic Green's functions are considered, and the ferromagnetic-impurity problem is formulated in terms of such Green's functions (at $T=0$ and at arbitrary temperatures, respectively). The analysis is illustrated explicitly for a fcc ferromagnet with first- and second-nearest-neighbor exchange.

Correlation operators in many-particle physics

Correlation functions and thermodynamic Green's functions are expressed as matrix elements of corresponding superoperators. A connection between the super-operator formalism, which has been used extensively in relaxation problems and in magnetism in recent years, and the method of thermodynamic Green's functions is thus established. The introduction of correlation (super-) operators describing dynamical as well as statistical aspects of a physical system allows for an abstract formulation of the theory independent of any special representation which results in considerable formal simplifications.

Far-Infrared Optical Constants of KI

The far-infrared optical constants of KI were studied in the vicinity of the $\mathrm{TO}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}=0)$ phonon frequency ${\ensuremath{\nu}}_{0}=101$ ${\mathrm{cm}}^{\ensuremath{-}1}$ (in wave-number units). Experimental values of the optical constants at $T=300$\ifmmode^\circ\else\textdegree\fi{}K were determined from transmission and reflection measurements employing a Michelson interferometer operated in the asymmetric mode. Theoretical values were calculated from a theory which utilizes thermodynamic Green's functions to include the effects of the interaction of the optically active phonon with other phonons because of cubic anharmonicity. Phonon-dispersion data for the calculation were generated from a shell model, with parameters selected by Dolling to give the phonon frequencies as determined from neutron-diffraction experiments. Good agreement for the optical constants in the region approximately 20-175 ${\mathrm{cm}}^{\ensuremath{-}1}$ was obtained by using a nearest-neighbor central-force model and adjusting the third derivative of the ${\mathrm{K}}^{+}$-${\mathrm{I}}^{\ensuremath{-}}$ bond potential to ${\ensuremath{\Phi}}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}}({r}_{0})=\ensuremath{-}3.6\ifmmode\times\else\texttimes\fi{}{10}^{12}$ ${\mathrm{e}\mathrm{r}\mathrm{g}/\mathrm{c}\mathrm{m}}^{3}$.