Functional Integral Method Research Articles

Gauge dependence of matrix elements of the Chern-Simons current K micro in the Schwinger model.

The gauge dependence of matrix elements of the Chern-Simons current ${\mathit{K}}^{\mathrm{\ensuremath{\mu}}}$ can be computed exactly in the (1+1)-dimensional Schwinger model. The matrix elements are gauge dependent even in the forward direction, i.e., when the momentum transfer vanishes. The Kogut-Susskind dipole mechanism is derived using functional-integral methods.

Derivation of one-way wave equations for vector wave propagation problems

The recent application of pseudodifferential and Fourier integral operators and functional integral methods to the factored scalar Helmholtz equation has resulted in extended parabolic wave theories, corresponding path integral solutions, and a numerical marching algorithm for a variety of acoustic wave propagation problems. These known techniques are applied here to vector wave equations (Maxwell's and elasticity), resulting in new first-order Weyl pseudodifferential equations, which are recognized as exact one-way wave equations for transversely inhomogeneous environments. Perturbation treatments of the appropriate Weyl composition equations for the operator symbol matrix yield high-frequency and other asymptotic wave theories. Unlike the scalar Helmholtz equation case, the one-way vector equations (and a scalar analog provided by the Klein-Gordon equation of relativistic physics) require the solution of generalized quadratic operator equations. While these operator solutions do not have a simple formal representation as in the straightforward (acoustic) square root case, they are conveniently constructed in the Weyl pseudodifferential operator calculus. This is an exact formulation at the level of the wave field—no special symmetry and/or far-field assumptions are made. [Work supported by NSF, AFOSR, ONR.]

Effects of charging-energy fluctuations on an anomalous behavior of mesoscopic Josephson junctions

A theory is proposed for mesoscopic Josephson junctions in the low-temperature intermediate quantum regime in which such junctions are characterized by a large single-electron charging energy {ital E}{sub {ital c}} that is of the order of the Josephson energy, {ital E}{sub {ital J}}. This contrasts with the classical Josephson junctions where the Josephson energy {ital E}{sub {ital J}} is usually dominant. In this intermediate regime the phase difference of the order parameters for left and right superconductors is no longer localized. In particular, it is believed that at low temperature the quantum phase fluctuations play a very important role. By employing functional-integral methods to the junction Hamiltonian, the resistive regimes of the mesoscopic Josephson junctions below the critical current are studied. It is shown that the Josephson current acquires a spectral decomposition, characterized by the spectral width {Delta}{sub {omega}}. The relative values of {Delta}{sub {omega}} and the voltage {ital V} are shown to determine the resistive regimes in the junction.

Determination of covariant Schwinger terms in anomalous gauge theories

A functional integral method is used to determine equal time commutators between the covariant currents and the covariant Gauss-law operators in theories which are affected by an anomaly. By using a differential geometrical setup we show how the derivation of consistent- and covariant Schwinger terms can be understood on an equal footing. We find a modified consistency condition for the covariant anomaly. As a by-product the Bardeen-Zumino functional, which relates consistent and covariant anomalies, can be interpreted as connection on a certain line bundle over the space of all gauge potentials. Finally the convariant commutator anomalies are calculated for the two- and four dimensional case.

Currents and energy-momentum tensor in the chiral Schwinger model with general vector and axial-vector couplings.

Using the point-splitting procedure and the method of functional integration, we define currents in the chiral Schwinger model and compute the correlation funtions of currents with themselves and with the fundamental fields. We show that the ambiguities in the choice of the phase factor employed in the point-splitting procedure can be compensated by mixing of the currents with the gauge potential ${\mathit{A}}_{\mathrm{\ensuremath{\mu}}}$ and ${\mathrm{\ensuremath{\epsilon}}}_{\mathrm{\ensuremath{\mu}}\ensuremath{\nu}}$${\mathit{A}}^{\ensuremath{\nu}}$. A three-parameter family of conserved currents is found and the transformations they generate are identified. In order to construct the conserved energy-momentum tensor, it is necessary to allow for mixings with ${\mathit{A}}_{\mathrm{\ensuremath{\mu}}}$${\mathit{A}}_{\ensuremath{\nu}}$ and ${\mathit{g}}_{\mathrm{\ensuremath{\mu}}\ensuremath{\nu}}$${\mathit{A}}_{\mathrm{\ensuremath{\alpha}}}$${\mathit{A}}^{\mathrm{\ensuremath{\alpha}}}$. We compute the two-point function of the energy-momentum tensor and the correlation functions of it with the fundamental fields. The physics of the chiral model is discussed in comparison with the vector model.

Relaxations, Fluctuations, Phase Transitions and Chemical Reactions in Liquid Water

Fluctuations and collective motions in liquid water and their effects on chemical reactions dynamics are analyzed. Liquid water is a ‘frustrated’ system with multiple random hydrogen bond network structures, and has anomalous microscopic and macroscopic properties. Rearrangement dynamics of the hydrogen bond network induces collective motions of water molecules and energy fluctuations. Vibrational motions of photoexcited molecules strongly resonate to these water fluctuations, and thus energy dissipation processes in liquid water are extremely fast. Time scale, spatial scale and energy scale of the collective motions are analyzed by examining potential energy surfaces involved. A model of hydrogen bond network based on the functional integral method is presented and spatial and energy scales of fluctuations are discussed. Instability of hydrogen bond network is studied to understand the physical origin of these microscopic and macroscopic anomalies of liquid water.

On the theory of dissipative phenomena in viscous liquids

The method of non-equilibrium density matrix has been used to express the viscosity of liquid substances resulting in a simple computation algorithm of η as a function of temperature and several other parameters of the system. An adequate transition from the correlator to the tensor of the energy-momentum flux connected to the viscosity η to a simple integral relationship is proposed. The developed formula enables one to obtain qualitatively consistent results and significantly simplifies abundant and complex computations. The method of functional integration is used to analyse the possibility to describe the behaviour of η( T − T g) in the domain of freezing T g.

Effect of finite hole mass on edge singularities in optical spectra.

A theory of optical transitions from a valence state to the conduction band taking into account the dynamical response of the Fermi sea is given using the functional-integral method. The method yields an approximation for the effects of the finite hole mass and finite temperature on the Mahan edge singularities. Magneto-optical spectra are calculated for n-doped quantum wells, which explain the range of observed behavior corresponding to finite-to-infinite hole mass at various temperatures.

Analytical properties of a (Wannier) exciton-phonon system: On the exclusion of self-trapping and overscreening.

We perform a critical analysis of the concept of phonon-induced phase transitions in connection with (Wannier) exciton-phonon systems. Particular attention is paid to the frequently discussed phenomena of self-trapping and overscreening. We demonstrate for a large class of (generalized Fr\"ohlich) models that a phase-transition-like behavior can be excluded: The ground-state energy, ground-state wave function, and the formal free energy are smooth functions of the electron-phonon and electron-hole coupling parameters. Analogous results hold for a magnetoexciton-phonon system, moving unrestricted or, e.g., within a quantum well. Our proof makes use of functional-analytic and functional-integral methods, which have been successfully applied in previous studies of polaron systems.

Some properties of far field patterns of acoustic waves in an inhomogeneous medium

In this paper, by using functional analysis and integral equation method, we obtain some results about the properties of far field of acoustic waves in an inhomogeneous medium. And we also discuss some ill-posed inverse scattering problems by Tikhonov regularization method.

Two-level system coupled to a boson mode.

The thermodynamics of a two-level system coupled to a boson field is investigated by means of a functional-integral method. The model can be used to describe the motion of a small polaron between two sites on a molecule, or a spin-1/2 magnetic moment coupled to a phonon bath. The model is also relevant to the problem of dissipation in macroscopic quantum phenomena. We use a functional-integral method to investigate a two-level electronic system coupled linearly to a harmonic oscillator. The density matrix is cast as a functional integral over paths obeying one-sided boundary conditions at either 0 or \ensuremath{\beta}=1/${\mathit{k}}_{\mathit{B}}$T. The integration variables are c numbers for the oscillator and Grassmann numbers for the electrons. The oscillator variables are integrated out, leaving an effective density matrix for the electrons containing retarded interactions. The partition function is then calculated up to second order of an expansion about the exact fermionic classical paths. Then, the free energy, specific heat, and average hopping energy are obtained. We find that the most significant reduction in the hopping energy is manifested at intermediate temperatures.

Theory of semiclassical expansions for many-particle systems.

We present a functional-integral method that is suitable for obtaining nonperturbative solutions to the partition function of fermion systems. The density matrix is cast as an expansion about the classical path contributions, and the quantum corrections can be calculated systematically by using retarded propagators in the space holomorphic functions. It is shown that the semiclassical expansion amounts to a high-temperature expansion, where the first nonvanishing quantum correction is of O(${\mathrm{\ensuremath{\beta}}}^{2}$). Each term of the semiclassical expansion may contain the coupling parameter in all orders of perturbation theory. The method can also be extended to boson systems.

Finite-temperature theory of local environment effects in amorphous and liquid magnetic alloys.

A finite-temperature theory of magnetism that takes into account the fluctuations of local magnetic moments due to structural disorder is presented on the basis of the functional-integral method and the method of the distribution function. The theory describes qualitatively or semiquantitatively the finite-temperature magnetism of liquid and amorphous metals and alloys as well as the substitutional alloys in a wide range of electron-electron interaction strength from metal to insulator. The results of numerical calculations are presented for amorphous iron. The local environment effects on the density of states, local magnetic moment, susceptibility, and amplitude of local moment are examined. It is found that amorphous iron forms an itinerant-electron spin glass at low temperatures because of the nonlinear magnetic coupling between Fe local moments and the local environment effect on the amplitude of the Fe local moment due to the structural disorder. The calculated spin-glass temperature (100 K) is in good agreement with the value extrapolated from experimental data on Fe-rich amorphous alloys.

Quantum tunneling through a fluctuating barrier: Enhancement of cold-fusion rate

Abstract Using the method of functional integration we calculate the rate of tunneling of a quantum particle weakly coupled to a single mode of the heat reservoir. The extremal paths of the steepest descent method are obtained in the adiabatic limit, valid when the reservoir mode frequency is well below the attempt frequency of the tunneling particle. In this limit, the tunneling rate is shown to exhibit a considerable enhancement due to the barrier fluctuation induced by the thermally excited reservoir mode. The results suggest that the rate of cold fusion in deuterated metals can be enhanced by fluctuations of the deuterium-pair distance, induced by thermal phonons of the host metal.

Temperature dependence of the autolocalization rate of excitons in rare gas solids

Abstract To analyze the self-trapping process of excitons in a deformable lattice we use a simple Hamiltonian, which yields the adiabatic potential energy surface by variation with regard to the excitonic wave function. To study the tunneling through the resulting potential barrier between metastable itinerant and stable self-trapped states we evaluate the partition function in a semiclassical approximation employing usual functional integral methods. For low temperatures the instanton solution dominates the self-trapping rates, for high temperatures the Arrhenius law is obtained. The results are compared with experimental data for Kr and Xe.

Interplay of superconductivity and antiferromagnetism in the Hubbard model: A functional-integral study of the slave bosom formulation

An effective quasi-2D antiferromagnetic Hamiltonian has been studied as a large U limit of the Hubbard model. The slave-boson language has been used and a functional-integral method has been employed. By a gauge transformation it has been shown that the phase of the complex field ei corresponding to an empty site-holon may be absorbed into the Lagrange multiplier λi representing the restriction of exclusion of double occupancy. It has been proved that the average number of holons per site is equal to the fractional difference of the number of particles-n from the half-filled case-δ and that the effective spinon-hopping term is proportional to δ. Superconducting and antiferromagnetic order parameters have been introduced by using the Hartree-Fock procedure. The free energy has been calculated and AF-SC coexistence has been confirmed. A phase diagram (temperature versus doping) has been constructed. The relevance of these results to experimental facts for copper oxides, especially to coexistence reported by Mezei et al.1, Petigrand et al.2 and Jonhston3 has been discussed.

Electron correlations in narrow-band systems at finite temperatures: Gutzwiller-type variational approach

A theory is presented for finite-temperature band magnetism which takes into account the effect of electron correlations using the Gutzwiller-type variational approach. The functional-integral method recently proposed by Kotliar and Ruckenstein (1986), is combined with the alloy-analogy approximation to include the effect of local spin fluctuations at finite temperatures. The theory at T=0 K is equivalent to Gutzwiller's approximation for the correlated ground state, while in the high-temperature limit it reduces to the local spin-fluctuation theory developed previously by Hasegawa and Hubbard. Numerical calculations for the half-filled simple cubic Hubbard model demonstrate that both electron correlations and local spin fluctuations play important roles at finite temperatures. The Brinkman-Rice-type metal-insulator transition is critically discussed on the basis of the model calculation.

ON ANTIFERROMAGNETIC INSTABILITY IN THE HUBBARD MODEL

A functional integral method applied to the Hubbard model has been used to construct the magnetic phase diagram (temperature versus doping) for large U/t. The antiferromagnetic order parameter has been represented in terms of fermion operators which correspond to neutral, spin one-half quasi-particles i.e. spinons. Static approximation has been applied. Numerical calculations have been performed for the two-dimensional square lattice. A stable antiferromagnetic phase for the half-filled band has been found below the critical temperature TN=J where J=4t2/U is the superexchange interaction. It has been shown that antiferromagnetism is destroyed by doping with about 15% of holes for t/J=10. The results are discussed in the light of recent experiments for high Tc oxides.

Linear mobility for coherent quantum tunneling in a periodic potential

The dynamics of a quantum particle moving in a tight binding lattice and coupled to an ohmic heat bath is investigated. Imaginary-time functional integral methods are utilized to determine the self-energy of the velocity-velocity correlation function for weak damping and arbitrary temperatures. Recent results for the linear mobility atT=0 and high temperatures are confirmed and extended to intermediate temperatures. The dominant corrections to the zero temperature mobility are given in terms of a power law.

Generalised master equation: dissipative dynamics of the double-well system

The author applied the generalised master equation (GME) to the analysis of the object function, s(t), which describes the motion of the tunnelling particle in a double-well system coupled to a reservoir consisting of independent bosonic-type elementary excitation. First, using the 'unrelaxed' initial condition and a suitable projection operator we give the exact formal solution of the GME. Then, the memory operator in the GME is explicitly calculated in the weak-tunnelling regime and the function s(t) is obtained. An independent derivation is given which enables us to obtain the exact expansion of the function s(t) by means of a purely algebraic method. This expansion forms a basis on which this GME method and the well known functional-integral approach are compared: the GME-weak-tunnelling approximation is shown to be identical to the commonly used non-interacting-blip approximation ensuing from the functional-integral method. A new approximation is analysed which is weaker than the non-interacting-blip one.