Functional Integral Formalism Research Articles

Existence of mesons and mass splitting in strong coupling lattice quantum chromodynamics

We consider one flavor lattice quantum chromodynamics in the imaginary time functional integral formulation for space dimensions d=2, 3 with 4×4 Dirac spin matrices, small hopping parameter κ, 0<κ≪1, and zero plaquette coupling. We determine the energy-momentum spectrum associated with four-component gauge invariant local meson fields which are composites of a quark and an antiquark field. For the associated correlation functions, we establish a Feynman–Kac formula and a spectral representation. Using this representation, we show that the mass spectrum consists of two distinct masses ma and mb, given by mc=−2 ln κ+rc(κ), c=a,b, where rc is real analytic. For d=2, ma and mb have multiplicity two and the mass splitting is κ4+O(κ6); for d=3, one mass has multiplicity one and the other three, with mass splitting 2κ4+O(κ6). In the subspace of the Hilbert space generated by an even number of fermion fields the dispersion curves are isolated (upper gap property) up to near the two-meson threshold of asymptotic mass −4 ln κ.

One- and two-loop corrections in the periodic Coqblin–Schrieffer model

We investigate the periodic Coqblin–Schrieffer model in terms of functionalintegral formalism, taking account of one- and two-loop corrections to themean-field solution. First, we take account of the one-loop (1L) diagrams todetermine the order parameter corresponding to the heavy-fermion state for thesymmetric case with isotropic hybridization. It is shown that the mean-field orderparameter is enhanced by the radial fluctuations, while it is reduced by the phasefluctuations. Therefore, if the phase fluctuations are gauge-fixed by theAnderson–Higgs mechanism, the heavy-fermion state is more stabilized due to thecorrections. Second, we give the analytical expressions for the two-loop correctionsto the free energy, which consist of the 1L corrections to the fermionself-energy, the boson self-energy, and the fermion–boson vertex function.

F-wave versusP-wave superconductivity in organic conductors

Current experimental results suggest that some organic quasi-one-dimensional superconductors exhibit triplet pairing symmetry. Thus, we discuss several potential triplet order parameters for the superconducting state of these systems within the functional integral formulation. We compare weak spin-orbit coupling $f_{xyz}$, $p_x$, $p_y$ and $p_z$ symmetries via several thermodynamic quantities. For each symmetry, we analyse the temperature dependences of the order parameter, condensation energy, specific heat, and superfluid density tensor.

Superconducting transition in a mixture of bosons and fermions

By means of the functional-integral formulation we discuss the occurrence of superconductivity resulting from Bose-Einstein condensation within a model describing preformed pairs which coexist in thermal equilibrium with unpaired fermions. At saddle-point level we study the thermodynamic properties of the model as functions of the boson excitation energy $\ensuremath{\nu},$ discussing the results in connection with the phenomenology of the pseudogap state of the superconducting cuprates. For temperatures above the Bose-Einstein condensation temperature ${T}_{c}$ we also analyze the effects of Gaussian fluctuations on the critical temperature and on the boson spectrum. Finally, we compare the bosonic effective action which controls thermal fluctuations of the mixture to the time-dependent Ginzburg-Landau functional of an interacting fermionic system at intermediate coupling.

Currents, chemical potential and boundary conditions in lattice QCD

A connection between the operator fermionic currents Jˆ and corresponding ‘Grassmannian’ currents J in the functional integral formalism is studied. The QCD action with non-zero chemical potential μ is derived. A connection between the fermionic Fock space and boundary conditions along the forth direction is discussed.

One-loop corrections to the susceptibility and the specific heat in the periodic Coqblin–Schrieffer model

We investigate the periodic Coqblin–Schrieffer model in terms of the functional integral formalism, taking into account the one-loop corrections. We determine the order parameter corresponding to heavy fermion state for the symmetric case with isotropic hybridization, and calculate the thermodynamic quantities such as specific heat and susceptibility. We show how the one-loop corrections modify the thermodynamic quantities.

Functional integral formulation of classical wave equations

It is shown in this paper that classical wave equations admit path integral formulations. For this, the evolution of the system is first set-up in terms of a fundamental solution or propagator. We choose this last name because it suggests a connection with functional integrals, which are exploited in this work. A functional integral in terms of non-singular functions is then proposed and shown to converge to the propagator in the appropriate limit for the case of scalar wave equations. One of the advantages of such formulation is that it provides an adequate framework for mesh-free numerical methods. This is demonstrated through a computational implementation that combines a simple second-degree polynomial local approximation of the continuous field and an approximate statement of the exact evolution equations. Numerical simulations of modal analysis and transient dynamics indicate the feasibility of the technique.

Pseudogaps in incommensurate charge density waves and one-dimensional semiconductors

We consider pseudogap effects for electrons interacting with gapless modes. We study generic 1D semiconductors with acoustic phonons and incommensurate charge density waves. We calculate the subgap absorption as it can be observed by means of photoelectron or tunneling spectroscopy. Within the formalism of functional integration and adiabatic approximation, the probabilities are described by nonlinear configurations of an instanton type. Particularities of both cases are determined by the topological nature of stationary excited states (acoustic polarons or amplitude solitons) and by the presence of gapless phonons that change the usual dynamics to the quantum dissipation regime. Below the free-particle edge, the pseudogap starts with an exponential (stretched exponential for gapful phonons) decrease of the transition rates. Deeply within the pseudogap, they are dominated by a power law, in contrast to a nearly exponential law for gapful modes.

THE FERMION-BOSON MAPPING APPLIED TO LAGRANGIAN MODELS FOR CHARGE-DENSITY-WAVES IN ONE-DIMENSIONAL SYSTEMS

We use the two-dimensional Fermion-Boson mapping to perform a field theory analysis of the effective Lagrangian model for incommensurate charge-density waves (ICDW) in one-dimensional systems. We consider an approach in which both the phase of the complex phonon field and the electron field are dynamical degrees of freedom contributing to the quantum dynamics and symmetry-related features of the ICDW phenomenon. We obtain the bosonized and fermionized versions of the effective electron–phonon Lagrangian. The phase of the phonon field and the phase of the bosonized chiral density of the electron field condense as a soliton order parameter, carrying neither the charge nor the chirality of the electron–phonon system, leading to a periodic sine-Gordon potential. The phonon field is fermionized in terms of a chiral fermionic condensate and the effective model is mapped into the chiral Gross–Neveu (GN) model with two Fermi field species. The linked electron–phonon symmetry of the ICDW system is mapped into the chiral symmetry of the GN model. Within the functional integral formulation, we obtain for the vacuum expectation value of the phonon field < ϕ > = 0 and < ϕ ϕ* > ≠ 0, due to the charge selection rule associated with the chiral electron–phonon symmetry. We show that the two-point correlation function of the phonon field satisfies the cluster decomposition property, as required by the chiral symmetry of the underlying GN model. The quantum description of the ICDW corresponds to charge transport through the lattice, due to the propagation of a "Goldstone mode" carrying the effective charge of the electron–phonon system, is accomplished by an electron–lattice energy redistribution. This accounts for a dynamical Peierls's energy gap generation.

Collective modes in thed-density-wave state of the cuprates

Using a functional integral formulation, we analyze the collective modes in the $d$-density wave state. Since only discrete symmetries are broken, no massless phase mode is present. The only relevant fluctuation is the amplitude fluctuation of the order parameter, which is treated at the gaussian level above the long range order. These fluctuations give rise to a massive mode, the analog of the U(1) Higgs, or amplitude mode at wave-vector $(\pi,\pi)$. Neutrons in inelastic neutron scattering experiments couple to this mode via the orbital currents and are likely to produce direct measurements of this amplitude boson.

Optimized Rayleigh–Schrödinger expansion of the effective potential

An optimized Rayleigh–Schrödinger expansion scheme of solving the functional Schrödinger equation with an external source is proposed to calculate the effective potential beyond the Gaussian approximation. For a scalar field theory whose potential function has a Fourier representation in a sense of tempered distributions, we obtain the effective potential up to the second order, and show that the first-order result is just the Gaussian effective potential. Its application to the λφ 4 field theory yields the same post-Gaussian effective potential as obtained in the functional integral formalism.

Functional integral approach: a third formulation of quantum statistical mechanics.

Quantum statistical mechanics has developed primarily through two approaches, pioneered by Gibbs and Feynman, respectively. In Gibbs' method one calculates partition functions from phase-space integrations or sums over stationary states. Alternatively, in Feynman's approach, the focus is on the path-integral formulation. The Hubbard-Stratonovich transformation leads to a functional-integral formulation for calculating partition functions. We outline here the functional integral approach to quantum statistical mechanics, including generalizations and improvements to Hubbard's formulation. We show how the dimensionality of the integrals is reduced exactly, how the problem of assuming an unknown canonical transformation is avoided, how the reality of the partition function in the complex representation is guaranteed, and how the extremum conditions are simplified. This formulation can be applied to general systems, including superconductors.

Gaussian effective potential and Coleman's normal-ordering prescription: the functional integral formalism

For a class of systems, the potential of whose bosonic Hamiltonian has a Fourier representation in the sense of tempered distributions, we calculate the Gaussian effective potential within the framework of the functional integral formalism. We show that Coleman's normal-ordering prescription can be formally generalized to the functional integral formalism.

Monte Carlo calculations of energies in the erbium region

We investigate the validity of the method of computing ground-state energies, at a given angular momentum, with Monte Carlo methods applied to the functional integral formulation containing pairing fields. Calculations have been performed for several erbium isotopes and a few dysprosium isotopes. It is found that some residual sign fluctuations, which seem to be associated with a small neutron number above the $N=82$ shell closure, are present, but that they do not affect the feasibility of the calculations. Such sign fluctuations disappear as the neutron number is increased. Also, an efficient improved method for the computation of the functional integral in the case of an odd number of particles is presented. The Monte Carlo method of Metropolis et al. [J. Chem. Phys. 21, 1087 (1953)] is used in the evaluation of expectation values.

Spin glass phase in Kondo lattice systems

A theory is presented that describes a spin glass phase at finite temperatures in Kondo lattice systems with random couplings among localized spins. The problem is studied with the functional integral formalism and we present a comparison with disordered cerium alloys.

Functional integral formulations for classical fluids

Various functional integral formalisms of classical monatomic fluids are considered with their applicabilities and limitations compared. For length scales large compared to the particle size, the density field theory, in which the action of the functional integral is simply given by the mean field free energy functional expression, is shown to be a well-defined and rigorous formulation. For short range properties of dense fluids, a different version of the functional integral method is developed by explicitly separating out the hard core part of the interaction. The resulting functional integral is seen to require correlation functions of the hard sphere fluid as the input. The generalized van der Waals equation and the random phase approximation of the cluster diagrammatic methods are recovered simply as the stationary-phase approximation and its Gaussian correction to the functional integral, respectively.

Some problems concerning interchange of integration order in functional integral formalism of U(1) gauge field theories

We show that in the functional integral formalism of U(1) gauge field theory some formal manipulation such as interchange of integration order can yield erroneous results. The example studied is analysed by Fubini theorem.

Spin-glass freezing in Kondo-lattice compounds

It is presented a theory that describes a spin glass phase at finite temperatures in Kondo lattice systems with an additional RKKY interaction represented by long range, random couplings among localized spins like in the Sherrington- Kirkpatrick (SK) spin glass model. The problem is studied within the functional integral formalism where the spin operators are represented by bilinear combinations of fermionic (anticommuting) Grassmann variables. The Kondo and spin glass transitions are both described with the mean field like static ansatz that reproduces good results in the two well known limits. At high temperatures and low values of the Kondo coupling there is a paramagnetic (disordered) phase with vanishing Kondo and spin glass order parameters. By lowering the temperature a second order transition line is found at Tsg to a spin glass phase. For larger values of the Kondo coupling there is a second order transition line at roughly Tk to a Kondo ordered state. For T<Tsg the transition between the Kondo and spin glass phases becomes first order.

不規則き裂進展に対する確率場を用いた新しいモデル

A new probabilistic model describing the random fatigue crack growth is proposed based upon a spatially random differential equation driven by a random field. First, a basic equation to describe the random crack growth is formulated as a system of spatially random differential equations by modeling the random propagation resistance in the well-known Paris law as a homogeneous random field. The probability of failure against the fatigue crack growth is then formulated as a functional integral form by the use of its solution. Next, validity of the proposed model is quantitatively verified by comparing with statistical experimental data reported by Ichikawa et. al. Next, an importance sampling simulation scheme is constructed by the use of the Girsanov theorem to make an efficient esimtation for the probability of failure. A numerical example is finally given, which shows that the proposed simulation scheme can give accurate estimation for extremely small probability of failure.

Functional integral formulation of the Thirring model with two fermion species

A two-dimensional Abelian model with quartic interaction among two species of Fermi fields is analyzed through a functional integral formulation. We consider the bosonization of the generating functional of the model in order to establish the fermion-boson mapping in the Hilbert space of states. The grand partition function of the statistical mechanical system associated with the effective bosonized theory is obtained and the corresponding exact equation of state exhibits a critical line characterizing a Kosterlitz-Thouless phase transition.