Functional Integral Method Research Articles

Functional approach to the periodic Anderson model

The method of functional integration is applied to the study of the behavior of the electron subsystem of adsorbed films with a regular structure. The effective action functional is calculated; equations for its parameters have been derived and investigated.

Finite-temperature theory of amorphous magnetic alloys.

A finite-temperature theory of magnetism in amorphous magnetic alloys with strong environment effects is developed on the basis of the functional integral method for thermal spin fluctuations and the distribution-function method for random distribution of local magnetic moments. The theory drastically simplifies numerical calculations by means of the geometrical-mean model for amorphous structure as well as electronic structure, and allows us to investigate the magnetism in amorphous transition-metal alloys with large difference in atomic size via average coordination number which depends on the type of central atom. Numerical example is given of the amorphous Fe-Zr alloys. It is demonstrated that the theory reproduces quantitatively the local densities of states obtained from the first-principles calculations, and describes the magnetic phase diagram, in particular, the itinerant-electron spin glass. It is also shown that the atomic-size effects play an important role in the formation of ferromagnetism in concentrated Fe-Zr amorphous alloys. The reentrant spin-glass behavior around 90 at. % Fe is shown to be due to the thermal spin fluctuations of amplitudes of local magnetic moments.

Finite-temperature band calculations of Fe/Cr multilayers

The magnetic and electronic properties of Fe/Cr multilayers at finite temperatures are discussed within the framework of the itinerant-electron theory developed previously by the present author using the static functional integral method. The distribution of the local magnetic moment on each layer in the Fe/Cr multilayer is calculated as a function of the temperature. An implication for the temperature dependence of the giant magnetoresistance is also discussed.

Theory of the temperature-dependent giant magnetoresistance in magnetic multilayers.

The giant magnetoresistance (GMR) in magnetic multilayers at finite temperatures is studied with the use of the functional-integral method in the static approximation. Numerical calculations are performed by using a simple model. The effect of spin fluctuations is shown to play important roles in discussing the temperature-dependent GMR. The temperature dependence of the calculated GMR is much larger than that of the magnetization, and it shows an almost linear decrease near the Curie temperature when the temperature is raised. It is also shown that the temperature dependence of the GMR is more significant in a multilayer with a larger ground-state GMR. Our calculation accounts well for the GMR feature observed in many transition-metal multilayers.

Wiener and Volterra analyses applied to the auditory system

The application of a particular branch of non-linear system analysis, the functional series expansion or integral method, to the auditory system is reviewed. Both the Volterra and Wiener approach are discussed and an extension of the Wiener method from its traditional white-noise stimulus approach to that of Poisson distributed clicks is presented. This type of analysis has been applied to compound and single-unit responses from the auditory nerve, cochlear nucleus, auditory midbrain and medial geniculate body. Most studies have estimated only first-order Wiener kernels but in recent years second-order Wiener and Volterra kernels have been estimated, particularly with reference to dynamic non-linearities. A particular form of second-order analysis, the Spectro Temporal Receptive Field, offers an alternative to first-order cross-correlation when phase-lock is absent. The correlation method has revealed that neural synchronization is less affected by intensity changes and damage to the hair cells than is neural firing rate. Although the presence of the static cochlear non-linearity could be demonstrated on the basis of the intensity dependence of the first-order Wiener kernel, the identification of the exact form of the nonlinearity of the peripheral auditory system on basis of higher-order Wiener kernels has so far been inconclusive. However, successes of the method can be found in the description of the dynamic non-linearities and non-linear neural interactions.

Coherent-state functional-integral approach to high-field transport in coupled electron-phonon systems

Coherent-state functional-integral methods are used to develop approximation schemes for the description of coupled electron-phonon systems far from equilibrium. We consider an interacting-electron system which is coupled to a system of phonons and subjected to an external---not necessarily weak---perturbation. For systems where the electron-phonon interaction is linear in the phonon variables, the phonon degrees of freedom can be accounted for exactly. From a formally exact representation of a suitable generating function for the expectation value of electronic observables, approximation schemes based on mean-field solutions can be derived. Within the lowest order of approximation, the coupled electron-phonon system may be treated as a self-consistent system of indepenent electrons. Higher-order corrections are shown to give a random-phase approximation based on self-consistent mean fields. This approach is applied to electronic two-level systems coupled to phonons under a large external bias. We find that the coupling to acoustical-phonon modes leads to a damping of the quantum beats and drives the system into a new equilibrium state. The coupling to LO phonons is shown to account correctly for resonance effects which arise when the level splitting is close to the LO-phonon energy.

1/ N expansion and the correlation energy of nuclear matter in the relativistic σω model

We propose the 1/ N expansion method, where N can be identified with 2 due to the isospin SU(2), as a systematic expansion scheme for the relativistic meson-nucleon many-body theory. We derive a general formula for the computation of the effective action in the σω model based on the 1/ N expansion, utilizing the functional integral method. In this scheme the leading contribution coincides with the familiar Hartree approximation. The elimination of the Landau ghost from the meson propagators in the subgraphs, which appears to be essential for our formulation, is also established. As an application we compute the energy density of nuclear matter beyond the Hartree approximation including the next-to-leading order contributions in the 1/ N expansion. In this order, one should include the ring energy contribution as a correlation energy addition to the exchange energy contribution. We find that we can describe the nuclear matter saturation properties without drastic changes of the overall physical picture which emerged in the Hartree approximation. The important role of the repulsive contribution due to the vacuum polarization effects in the ring energy is emphasized.

Monte Carlo approach to the dynamical coherent-potential approximation in metallic magnetism.

A Monte Carlo approach to the dynamical coherent-potential approximation (CPA) has been proposed on the basis of the functional-integral method to deal with the dynamical spin fluctuations in the Gutzwiller-Hubbard model. The functional integral on a site in the effective medium is replaced by the ${\mathit{N}}_{\ensuremath{\xi}}$-fold integral, which is evaluated by the Monte Carlo method. Numerical calculations have been performed up to ${\mathit{N}}_{\ensuremath{\xi}}$=64 for intermediate Coulomb interaction strength in the paramagnetic state. It is demonstrated that the present theory recovers the amplitude of local moment, reduces the effective Coulomb interaction, and leads to more Fermi-liquid-like momentum distribution when they are compared with the static approximation. Furthermore, the single-particle excitation spectra calculated by a numerical analytic continuation are shown to have some shoulders due to many-body excitations.

Calculation of the Spin System Correlation Functions in the Frames of the Functional Integration Method. II. The Heisenberg Model

New equations for the correlation functions obtained in the frames of the functional integration method are applied to investigate the magnetic ordering in the Heisenberg isotropic quantum model. The Curie temperature of this model is calculated from the pole of the longitudinal Green function. Good agreement with known data is obtained in the crystalline case. It is shown that the magnetic order can be destroyed by the amorphization of the crystal. [Russian Text Ignored].

Theory of magnetic and thermal properties of the weak itinerant ferromagnet Ni 3Al

Results of band structure calculations are used to calculate the temperature dependence of magnetization, fluctuation of the local moment, paramagnetic susceptibility and the specific heat of the weak itinerant ferromagnet Ni 3Al. Calculations of these quantities are performed over a broad temperature range from 0 K to about 5 T c. Arrott plots are presented. The functional integral method in the static approximation is used. The model with the intra-atomic Coulomb integral as the only interaction parameter is applied, its value is determined from the ground state magnetic moment. Results compare favourably with experimental data.

Electron correlation on metallic surfaces

Electron correlation on metallic surfaces is studied for the first time with the use of the slave-boson functional integral method. The ground-state properties of the semi-infinite simple-cubic model in the non-magnetic state are investigated. The double occupancy, delta n, and the band narrowing factor, qn, on layer n are calculated as a function of the electron interaction. It is shown that delta 1 and q1 on the surface show a peculiar behaviour depending on the ratio U1/Ub where U1(Ub) denotes the interaction on the surface (in the bulk).

Calculation of the Spin System Correlation Functions in the Frames of the Functional Integration Methol. I. The Ising Model

The equations for the spin system correlation functions and Green functions as well as the way of their self-consistent solution are suggested in the frames of functional integration method. As an example the Ising model is considered. The Curie temperature of this model is calculated from the pole of the pair correlation function. Calculations are performed for the cases of the crystalline and amorphous system. The value of critical temperature in the crystalline case is in good agreement with known data. It is shown that the magnetic order can be destructed by amorphization of the crystal. [Russian Text Ignored].

Quantum Fluctuations in Equilibrium Superradiance

Abstract The role of the quantum fluctuations of the order parameter in a second-order phase transition is discussed, for a superradiant model. The mean field equations and the equation of the critical temperature are obtained. The change of the value of the critical exponent of the specific heat α is connected with the supplementary dimension introduced by the frequency dependence of the field variables. The functional integral method is used, most of the calculations being performed within the Gaussian approximation.

Functional integration method for quantum magnets and 1D-random potential problem, with application to the infrared dynamics

The functional integral approach to the statistical physics of the Quantum Heisenberg Ferromagnet is described. Slow relaxation degrees of freedom in the low-temperature phase are studied. A path integral interpretation of the so-called phase phormalism for the 1D random potential problem is presented.

Electron transfer induced by liquid defect motion. Exact solution

An exact solution for the probability of electron transfer caused by one-dimensional liquid defect motion and polarization fluctuations has been obtained by the method of functional integration over stochastic trajectories. The nonlinear coupling of a reactant electronic state with the defect is considered.

Quantum and classical integrability: new approaches in statistical mechanics

The present status of the statistical mechanics (SM), quantum and classical, of integrable models is reviewed by reporting new results for their partition functions Z obtained for anyon type models in one space and one time (1 + 1) dimensions. The methods of functional integration developed already are extended further. Bose-Fermi equivalence and anyon descriptions are natural parts of the quantum theory and the anyon phase is quantised. The classical integrability is exploited throughout and both classical and quantum integrability theory are reviewed this way, and related to underlying algebraic structures - notably the Hopf algebras (“quantum groups”). A new “ q-boson” lattice gas is solved in this connection. Further results to be realised within 10 years concerning integrability, integrable systems coupled to large dimensional chaos provided by a heat bath, and their application in physics e.g. to high- T c superconductivity, are suggested.

Functional integral method for gibbs systems with many-body potentials. I

Functional integral method for gibbs systems with many-body potentials. I

Hubbard model: Functional-integral approach and diagrammatic perturbation theory.

We present a careful analysis of the functional-integral approach as applied to the Hubbard model in the framework of diagrammatic perturbation theory. It is demonstrated that a proper comprehension of the formalism can be attained and that the functional-integral method is a very systematic technique, free of ambiguity. In particular, additional exact relations among physical quantities of interest are derived in the context of the formalism.

Brownian motion constrained to enclose a given area

The probability of a planar Brownian closed curve enclosing a fixed area is rederived using a simple method of functional integration. The mean square distance between two points of the ring is calculated. When the constraint on the enclosed area increases, one passes from a diffusion regime of the Brownian ring to a normal regime, where the curve approaches a circle.

On the Distribution of the Integrated Square of the Ornstien–Uhlenbeck Process

Using functional integral methods, the Laplace transform of the square of the Ornstein-Uhlenbeck process $X( t )$ integrated over $0\leqq t\leqq T$ is calculated. Via the complex inversion integral, the asymptotic behavior as $T \to \infty $ of the density and distribution functions, as well as these functions conditioned on the event $X( T ) = 0$, are studied. It is found that the approximation by an inverse Gaussian distribution, introduced earlier by Grenander, Pollak, and Slepian, is asymptotically correct (to within a constant factor) in the conditional case, but not in the unconditional case.