Irreducible Green Functions Research Articles

Covariant tetraquark equations in quantum field theory

We derive general covariant coupled equations of QCD describing the tetraquark in terms of a mix of four-quark states $2q2\bar q$, and two-quark states $q\bar q$. The coupling of $2q2\bar q$ to $q\bar q$ states is achieved by a simple contraction of a four-quark $q\bar q$-irreducible Green function down to a two-quark $q\bar q$ Bethe-Salpeter kernel. The resulting tetraquark equations are expressed in an exact field theoretic form, and are in agreement with those obtained previously by consideration of disconnected interactions; however, despite being more general, they have been derived here in a much simpler and more transparent way.

The combinatorics of Green’s functions in planar field theories

The aim of this work is to outline in some detail the use of combinatorial algebra in planar quantum field theory. Particular emphasis is given to the relations between the different types of planar Green's functions. The key object is a Hopf algebra which is naturally defined on non-commuting sources, and the fact that its genuine unshuffle coproduct splits into left- and right unshuffle half-coproduts. The latter give rise to the notion of unshuffle bialgebra. This setting allows to describe the relation between planar full and connected Green's functions by solving a simple linear fixed point equation. A modification of this linear fixed point equation gives rise to the relation between planar connected and one-particle irreducible Green's functions. The graphical calculus that arises from this approach also leads to a new understanding of functional calculus in planar QFT, whose rules for differentiation with respect to non-commuting sources can be translated into the language of growth operations on planar rooted trees. We also include a brief outline of our approach in the framework of non-planar theories.

Irreducible Green Functions Method Applied to Nanoscopic Systems

The equation of motion method for the Green functions is one of the tools used in the analysis of quantum dot system coupled with the metallic and superconducting leads. We investigate modified equation of motion, based on dierentiation of double-time temperature dependent Green functions both over the primary time t and secondary time t 0 . Our equation of motion approach allows us to obtain the Abrikosov‐Suhl resonance in the particle‐hole symmetric case and also in the asymmetric cases. We will apply the irreducible Green functions technique to analyses of the equation of motion applied to the dot system. This method gives a workable decoupling scheme breaking the infinite set of the Green function equations. We apply this technique to calculate the density of the states and the dierential conductance of single-level quantum dot with the Coulomb repulsion attached to one metallic and one superconducting leads (N‐QD‐SC). Our results are compared with the previous calculations. DOI: 10.12693/APhysPolA.130.551 PACS/topics: 73.63.Kv, 73.23.Hk, 74.45.+c

A model for massless higher spin field interacting with a geometrical background

We study a very general four-dimensional field theory model describing the dynamics of a massless higher spin N symmetric tensor field particle interacting with a geometrical background. This model is invariant under the action of an extended linear diffeomorphism. We investigate the consistency of the equations of motion, and the highest spin degrees of freedom are extracted by means of a set of covariant constraints. Moreover, the highest spin equations of motions (and in general all the highest spin field 1-PI irreducible Green functions) are invariant under a chain of transformations induced by a set of N - 2 Ward operators, while the auxiliary fields equations of motion spoil this symmetry. The first steps to a quantum extension of the model are discussed on the basis of the algebraic field theory. Technical aspects are reported in Appendices, in particular, one of them is devoted to illustrate the spin-2 case.

Stationary property of the thermodynamic potential of the Hubbard model in strong coupling diagrammatic approach for superconducting state

Diagrammatic analysis for normal state of Hubbard model proposed in our previous paper is generalized and used to investigate superconducting state of this model. We use the notion of charge quantum number to describe the irreducible Green's function of the superconducting state. As in the previous paper we introduce the notion of tunneling Green's function and of its mass operator. This last quantity turns out to be equal to correlation function of the system. We proved the existence of exact relation between renormalized one-particle propagator and thermodynamic potential which includes integration over auxiliary interaction constant. The notion of skeleton diagrams of propagator and vacuum kinds were introduced. These diagrams are constructed from irreducible Green's functions and tunneling lines. Identity of this functional to the thermodynamic potential has been proved and the stationarity with respect to variation of the mass operator has been demonstrated.

Canonical transformations and renormalization group invariance in the presence of nontrivial backgrounds

We show that for a SU(N) Yang-Mills theory the classical background-quantum splitting is non-trivially deformed at the quantum level by a canonical transformation with respect to the Batalin-Vilkovisky bracket associated with the Slavnov-Taylor identity of the theory. This canonical transformation acts on all the fields (including the ghosts) and antifields; it uniquely fixes the dependence on the background field of all the one-particle irreducible Green's functions of the theory at hand. The approach is valid both at the perturbative and non-perturbative level, being based solely on symmetry requirements. As a practical application, we derive the renormalization group equation in the presence of a generic background and apply it in the case of a SU(2) instanton. Finally, we explicitly calculate the one-loop deformation of the background-quantum splitting in lowest order in the instanton background.

Pseudogap state of two-dimensional Kondo lattice

The pseudogap behavior of spectral function A(k, ω) of charge carriers is considered in the weak doping regime for a 2D Kondo lattice with a strong spin-hole antiferromagnetic interaction. The scattering of carriers is described in terms of a local polaron according to the irreducible Green functions. The behavior of the carrier spectrum in the nodal and antinodal domains is considered. The resultant value of the pseudogap is in conformity with experimental data on photoemission with angular resolution.

Quasiaverages, symmetry breaking and irreducible Green functions method

The development and applications of the method of quasiaverages to quantum statistical physics and to quantum solid state theory and, in particular, to quantum theory of magnetism, were considered. It was shown that the role of symmetry (and the breaking of symmetries) in combination with the degeneracy of the system was reanalyzed and essentially clarified within the framework of the method of quasiaverages. The problem of finding the ferromagnetic, antiferromagnetic and superconducting “symmetry broken” solutions of the correlated lattice fermion models was discussed within the irreducible Green functions method. A unified scheme for the construction of generalized mean fields (elastic scattering corrections) and self-energy (inelastic scattering) in terms of the equations of motion and Dyson equation was generalized in order to include the “source fields”. This approach complements previous studies of microscopic theory of antiferromagnetism and clarifies the concepts of Neel sublattices for localized and itinerant antiferromagnetism and “spin-aligning fields” of correlated lattice fermions.

Recursion and Growth Estimates in Renormalizable Quantum Field Theory

In this paper we show that there is a Lipatov bound for the radius of convergence for superficially divergent one-particle irreducible Green functions in a renormalizable quantum field theory if there is such a bound for the superficially convergent ones. In the nonnegative case the radius of convergence turns out to be min{ρ,1/b 1}, where ρ is the bound on the convergent ones, the instanton radius, and b 1 the first coefficient of the β-function, while in general it is bounded by the above.

INVESTIGATION OF THE DENSITY OF STATES IN THE NON-HALF-FILLED TWO BAND PERIODIC ANDERSON MODEL

We study density of states in the symmetrical and asymmetrical two-band periodic Anderson models at various band fillings with self-consistent calculation of the orbital occupancies. The application of the improved truncation approximation for irreducible Green functions that takes into account resonance broadening and band shifting inter-orbital exchange effects, resulted in the appearance of four spectral density moments and four- or five-subbands in the density of states depending upon the parameters of the model. It is shown that closing of the hybridization gap can occur as the result of doping, applied pressure, or change of the f-band width.

The investigation of the band-filling and pressure effects in the two-band periodic Anderson model

The evolution of density of states and specific heat with band-filing and pressure changes has been investigated for asymmetric two-band periodic Anderson model. The equation of motion method for the irreducible Green functions with the extended Hubbard I approximation contributing to the appearance of “band-shifting” correction to the self-energy has been applied. It is shown that the energy spectrum varies between four- and five-band structures, while specific heat changes from one- to two-peak temperature dependencies, with the variation of parameters of the model. It is also revealed that the application of pressure can lead to the opening of the hybridization gap and subsequent metal–semiconductor–insulator transition.

BOUND AND SCATTERING STATES OF ITINERANT CHARGE CARRIERS IN COMPLEX MAGNETIC MATERIALS

The concept of magnetic polaron is analyzed and developed to elucidate the nature of itinerant charge carrier states in magnetic semiconductors and similar complex magnetic materials. By contrasting the scattering and bound states of carriers within the s–d exchange model, the nature of bound states at finite temperatures is clarified. The free magnetic polaron at certain conditions is realized as a bound state of the carrier (electron or hole) with the spin wave. Quite generally, a self-consistent theory of a magnetic polaron is formulated within a nonperturbative many-body approach, the Irreducible Green Functions (IGF) method which is used to describe the quasiparticle many-body dynamics at finite temperatures. Within the above many-body approach we elaborate a self-consistent picture of dynamic behavior of two interacting subsystems, the localized spins and the itinerant charge carriers. In particular, we show that the relevant generalized mean fields emerges naturally within our formalism. At the same time, the correct separation of elastic scattering corrections permits one to consider the damping effects (inelastic scattering corrections) in the unified and coherent fashion. The damping of magnetic polaron state, which is quite different from the damping of the scattering states, finds a natural interpretation within the present self-consistent scheme.

Three-Center Interactions and Magnetic Mechanism of Superconductivity with dx2-y2-Symmetry in the t - J*-Model

The effect of three-center interactions on the formation of a superconducting phase with dx2-y2-symmetry is considered using the diagram technique for Hubbard operators and irreducible Green's functions. It is shown that these interactions lead to a decrease in Tc by a factor of several tens.

LONGITUDINAL DYNAMICAL SPIN SUSCEPTIBILITY IN HEISENBERG FERROMAGNETS

We study the longitudinal dynamical spin susceptibility [Formula: see text] within the framework of Heisenberg model in the ferromagnetic phase, where no external field exists. On the basis of memory function formalism within the framework of the irreducible Green's functions we obtained the self-consistent equations for [Formula: see text] in mode coupling approximation. The expression for the longitudinal spin susceptibility is valid in the whole range of system frequencies. Its validity in the range of low and high frequencies was discussed in particular for the magnetic systems in ferromagnetic phase.

Irreducible Green functions method and many-particle interacting systems on a lattice

The Green-function technique, termed the irreducible Green functions (IGF) method, that is a certain reformulation of the equation-of motion method for double-time temperature dependent Green functions (GFs) is presented. This method was developed to overcome some ambiguities in terminating the hierarchy of the equations of motion of double-time Green functions and to give a workable technique to systematic way of decoupling. The approach provides a practical method for description of the many-body quasi-particle dynamics of correlated systems on a lattice with complex spectra. Moreover, it provides a very compact and self-consistent way of taking into account the damping eects and finite lifetimes of quasi-particles due to inelastic collisions. In addition, it correctly defines the Generalized Mean Fields (GMF), that determine elastic scattering renormalizations and , in general, are not functionals of the mean particle densities only. The purpose of this article is to present the foundations of the IGF method. The technical details and examples are given as well. Although some space is devoted to the formal structure of the method, the emphasis is on its utility. Applications to the lattice fermion models such as Hubbard/Anderson models and to the Heisenberg ferro- and antiferromagnet, which manifest the operational ability of the method are given. It is shown that the IGF method provides a powerful tool for the construction of essentially new dynamical solutions for strongly interacting many-particle systems with complex spectra.

Fermionic Renormalization Group Flows: Technique and Theory

We give a self-contained derivation of the differential equations for Wilson’s renormalization group for the one-particle irreducible Green functions in fermionic systems. The application of this equation to the (t, t � )-Hubbard model appears in Ref. 9). Here we focus on theoretical aspects. After deriving the equations, we discuss the restrictions imposed by symmetries on the effective action. We discuss scaling properties due to the geometry of the Fermi surface and give precise criteria to determine when they justify the use of one-loop flows. We also discuss the relationship of this approach to other RG treatments, as well as aspects of the practical treatment of truncated equations, such as the projection to the Fermi surface and the calculation of susceptibilities. Renormalization group (RG)studies of fermionic models are important for understanding models of solid-state theory and high-energy physics. In this paper, we set up a system of equations suitable for studying flows for general fermionic systems, in particular those with a Fermi surface at any density, and discuss general, but nontrivial, properties of their solutions. The system of equations applies both to “normal” and to symmetry-broken situations of fermionic systems with short-range interactions for d ≥ 1. By “short-range” we mean that the two-particle interaction decays so fast that its Fourier transform is a regular function. In Ref. 9)we have, together with Furukawa and Rice, applied the RG equations that we derive here to the two-dimensional (t, t � )-Hubbard model in the regime relevant for high-temperature superconductivity. The purpose of the present paper is to give more background on the theoretical aspects of the equations, in particular on a detailed argument which justifies the one-loop flow in a certain scale range, even if the scale-dependent interaction is no longer small. In the following, we give an overview; the details are filled in in the later sections. 1.1. The RG and the three scale regimes The Wilsonian RG organizes the functional integration corresponding to the grand canonical trace as an iterated integral over degrees of freedom with energies in a certain range, i.e. those that belong to a corresponding length scale. As such, it is simply a rewriting of the generating functional of the correlation functions as a function of an energy scale � . In contrast to the situation in other RG schemes, this approach does not require any assumptions about perfect scaling laws as a

Recursive graphical construction of feynman diagrams in straight phi(4) theory: asymmetric case and effective energy

The free energy of a multicomponent scalar field theory is considered as a functional W[G,J] of the free correlation function G and an external current J. It obeys nonlinear functional differential equations which are turned into recursion relations for the connected Green's functions in a loop expansion. These relations amount to a simple proof that W[G,J] generates only connected graphs and can be used to find all such graphs with their combinatoric weights. A Legendre transformation with respect to the external current converts the functional differential equations for the free energy into those for the effective energy Gamma[G,Phi], which is considered as a functional of the free correlation function G and the field expectation Phi. These equations are turned into recursion relations for the one-particle irreducible Green's functions. These relations amount to a simple proof that Gamma[G,J] generates only one-particle irreducible graphs and can be used to find all such graphs with their combinatoric weights. The techniques used also allow for a systematic investigation into resummations of classes of graphs. Examples are given for resumming one-loop and multiloop tadpoles, both through all orders of perturbation theory. Since the functional differential equations derived are nonperturbative, they constitute also a convenient starting point for other expansions than those in numbers of loops or powers of coupling constants. We work with general interactions through four powers in the field.

SPECTRAL PROPERTIES OF THE GENERALISED SPIN-FERMION MODELS

In order to account for competition and interplay of localized and itinerant magnetic behaviour in correlated many body systems with complex spectra the various types of spin-fermion models have been considered in the context of the Irreducible Green's Functions (IGF) approach. Examples are generalised d–f model and Kondo–Heisenberg model. The calculations of the quasiparticle excitation spectra with damping for these models has been performed in the framework of the equation-of-motion method for two-time temperature Green's Functions within a non-perturbative approach. A unified scheme for the construction of Generalised Mean Fields (elastic scattering corrections) and self-energy (inelastic scattering) in terms of the Dyson equation has been generalised in order to include the presence of the two interacting subsystems of localised spins and itinerant electrons. A general procedure is given to obtain the quasiparticle damping in a self-consistent way. This approach gives the complete and compact description of quasiparticles and show the flexibility and richness of the generalised spin-fermion model concept.

Strong interaction of correlated electrons with phonons: A diagrammatic approach

We investigate the interaction of strongly correlated electrons with phonons in the frame of the Hubbard-Holstein model. The electron-phonon interaction is considered to be strong and is an important parameter of the model besides the Coulomb repulsion of electrons and band filling. This interaction with the nondispersive optical phonons has been transformed to the problem of mobile polarons by using the canonical transformation of Lang and Firsov. A new diagram technique is used in order to handle the strong Coulomb repulsion of the electrons, which leads to a closed set of integral equations for the irreducible on-site many-particle Green's functions. The coupling to the phonons leads to an essential renormalization of the one- and two-particle irreducible Green's functions. It also influences the parameters which control the metal-insulator or superconducting phase transition. In addition we find a collective mode of phonon clouds which surround the polarons. The physics of the emission and absorption of this mode by the polarons has been investigated.

Two-time temperature Green's functions in kinetic theory and molecular hydrodynamics: I. The chain of equations for the irreducible functions

We develop a scheme for constructing chains of equations for the irreducible Green's functions. The structure of the equations allows going beyond the usual perturbation theory in solving specific problems. We obtain general relations that allow any correlation function to be expressed through solutions of an infinite chain of equations for the irreducible functions.