Fermion-boson Vertex Research Articles

Reply to “Comment on ‘Towards exact solutions for the superconducting Tc induced by electron-phonon interaction' ”

In a series of papers, we have proposed a nonperturbative field-theoretic approach to deal with strong electron-phonon and strong Coulomb interactions. The key ingredient of such an approach is to determine the full fermion-boson vertex corrections by solving a number of self-consistent Ward-Takahashi identities. Palle argued that our Ward-Takahashi identities failed to include some important additional terms and thus are incorrect. We agree that our Ward-Takahashi identities have ignored some potentially important contributions and here give some remarks on the role played by the additional terms. Published by the American Physical Society 2024

Dressed quark-gluon vertex form factors from gauge symmetry


 
 
 We present preliminary results on the longitudinal and transverse form factors of the quark-gluon vertex as functions of the incoming and outgoing quark momenta and an angle θ = 2π/3 between them. The expressions for these form factors were previously derived from Slavnov-Taylor identities, gauge covariance and multiplicative renormalizability that firmly constrain the fermion-boson vertex.
 
 

The impact of transverse Slavnov-Taylor identities on dynamical chiral symmetry breaking

We extend earlier studies of transverse Ward-Fradkin-Green-Takahashi identities in QED, their usefulness to constrain the transverse fermion-boson vertex and their importance for multiplicative renormalizability, to the equivalent gauge identities in QCD. To this end, we consider transverse Slavnov-Taylor identities that constrain the transverse quark-gluon vertex and derive its eight associated scalar form factors. The complete vertex can be expressed in terms of the quark’s mass and wave-renormalization functions, the ghost-dressing function, the quark-ghost scattering amplitude and a set of eight form factors. The latter parametrize the hitherto unknown nonlocal tensor structure in the transverse Slavnov-Taylor identity which arises from the Fourier transform of a four-point function involving a Wilson line in coordinate space. We determine the functional form of these eight form factors with the constraints provided by the Bashir-Bermudez vertex and study the effects of this novel vertex on the quark in the Dyson-Schwinger equation using lattice QCD input for the gluon and ghost propagators. We observe significant dynamical chiral symmetry breaking and a mass gap that leads to a constituent mass of the order of 500 MeV for the light quarks. The flavor dependence of the mass and wave-renormalization functions as well as their analytic behavior on the complex momentum plane is studied and as an application we calculate the quark condensate and the pion’s weak decay constant in the chiral limit. Both are in very good agreement with their reference values.

Nonperturbative Dyson-Schwinger equation approach to strongly interacting Dirac fermion systems

Studying the strong correlation effects in interacting Dirac fermion systems is one of the most challenging problems in modern condensed matter physics. The long-range Coulomb interaction and the fermion-phonon interaction can lead to a variety of intriguing properties. In the strong-coupling regime, weak-coupling perturbation theory breaks down. The validity of $1/N$ expansion with $N$ being the fermion flavor is also in doubt since $N$ equals to 2 or 4 in realistic systems. Here, we investigate the interaction between $(1+2)$- and $(1+3)$-dimensional massless Dirac fermions and a generic scalar boson, and develop an efficient nonperturbative approach to access the strong-coupling regime. We first derive a number of self-consistently coupled Ward-Takahashi identities based on a careful symmetry analysis and then use these identities to show that the full fermion-boson vertex function is solely determined by the full fermion propagator. Making use of this result, we rigorously prove that the full fermion propagator satisfies an exact and self-closed Dyson-Schwinger integral equation, which can be solved by employing numerical methods. A major advantage of our nonperturbative approach is that there is no need to employ any small expansion parameter. Our approach provides a unified theoretical framework for studying strong Coulomb and fermion-phonon interactions. It may also be used to approximately handle the Yukawa coupling between fermions and order-parameter fluctuations around continuous quantum critical points. Our approach is applied to treat the Coulomb interaction in undoped graphene. We find that the renormalized fermion velocity exhibits a logarithmic momentum dependence but is nearly energy independent, and that no excitonic gap is generated by the Coulomb interaction. These theoretical results are consistent with experiments in graphene.

Fermion Propagator in QED - Landau-Khalatnikov-Fradkin Transformations

Gauge theories such as quantum electrodynamics (QED) and quantum chromodynamics (QCD) describe the physical world accurately at the level of fundamental particles. They possess gauge symmetry reflected in terms of several identities and transformation laws which impose tight constraints on all conceivable Green functions which define the theory. In this article, we describe and summarize the role played by the Landau-Khalatnikov-Fradkin (LKF) transformations in this context. Within the set of covariant gauges, these transformations tell us how to construct a Green function in an arbitrary gauge, starting from its explicit expression in a particular gauge. In perturbation theory, these transformations are satisfied at every order of approximation. A non-perturbative description of QED and QCD in the continuum is provided by the Schwinger-Dyson Equations (SDEs). These are the fundamental equations of motion encoding the dynamics of Green functions. These equations provide a unified description of weak and strong coupling regimes and are thus increasingly employed to study strongly interacting theories and their transition to the perturbative limit. As these equations are an infinite set of coupled non-linear equations, a truncation is essential to reduce them to a solvable number. LKF transformations provide a stringent constraint on the acceptable truncations which preserve the original symmetries of the gauge theory involved. Most of these truncations consist in cleverly constructing an Anstaz for the electron-photon vertex in QED and the quark-gluon vertex in QCD. In this article, we review the LKF transformations for the fermion propagator. Very importantly, they imply the gauge invariance of the chiral fermion condensate and the pole mass of a fermion. We provide the first demonstration of the latter in this article. Moreover, we also describe how the LKF transformations of the fermion propagator provide gauge-symmetry constraints on a non-perturbative construction of the three-point fermion-boson vertex.

Fermion-boson vertex within dynamical mean-field theory

In the study of strongly interacting systems, correlations on the two-particle level are receiving more and more attention. In this work, we study a particular two-particle correlation function: the fermion-boson vertex. It describes the response of the Green's function when an external field is applied and is an important ingredient of diagrammatic extensions of Dynamical Mean-Field Theory. We provide several perspectives on this object, using Ward identities, sum rules, perturbative analysis and asymptotic relations. We then use these tools to study the vertex across the doping-driven metal-insulator transition and find a divergence in the imaginary part.

How gauge covariance of the fermion and boson propagators in QED constrain the effective fermion-boson vertex

We derive the gauge covariance requirement imposed on the QED fermion-photon three-point function within the framework of a spectral representation for fermion propagators. When satisfied, such requirement ensures solutions to the fermion propagator Schwinger-Dyson equation (SDE) in any covariant gauge with arbitrary numbers of spacetime dimensions to be consistent with the Landau-Khalatnikov-Fradkin transformation (LKFT). The general result has been verified by the special cases of three and four dimensions. Additionally, we present the condition that ensures the vacuum polarization is independent of the gauge parameter. As an illustration, we show how the gauge technique dimensionally regularized in four dimensions does not satisfy the covariance requirement.

Beyond extended dynamical mean-field theory: Dual boson approach to the two-dimensional extended Hubbard model

The dual boson approach [Ann. Phys. 327, 1320 (2012)] provides a means to construct a diagrammatic expansion around the extended dynamical mean-field theory (EDMFT). In this paper, we present the numerical implementation of the approach and apply it to the extended Hubbard model with nearest-neighbor interaction $V$. We calculate the EDMFT phase diagram and study the effect of diagrams beyond EDMFT on the transition to the charge-ordered phase. Including diagrammatic corrections to the EDMFT polarization shifts the EDMFT phase boundary to lower values of $V$. The approach interpolates between the random phase approximation in the weak coupling limit and EDMFT for strong coupling. Neglecting vertex corrections leads to results reminiscent of the EDMFT+$GW$ approximation. We however find significant deviations from the dual boson results already for small values of the interaction, emphasizing the crucial importance of fermion-boson vertex corrections.

Discussion for the Solutions of Dyson-Schwinger Equations at <i>m</i> ≠ 0 in QED<sub>3</sub>

In the case of nonzero fermion mass, within a range of Ansatze for the full fermion-boson vertex, we show that Dyson-Schwinger equation for the fermion propagator in QED3 has two qualitatively distinct dynamical chiral symmetry breaking solutions. As the fermion mass increases and reaches to a critical value mc, one solution disappears, and the dependence of mc on the number of fermion flavors is also given.

Frequency-dependent vertex functions of the (t,t′) Hubbard model at weak coupling

We present a functional renormalization group calculation for the two-dimensional (t,t')-Hubbard model at Van Hove filling. Using a channel decomposition we describe the momentum and frequency dependence of the vertex function in the normal phase. Compared to previous studies that neglect frequency dependences we find higher pseudo-critical scales and a smaller region of d-wave superconductivity. A large contribution to the effective interaction is given by a forward scattering process with finite frequency exchange. We test different frequency parameterizations and in a second calculation include the frequency dependence of the imaginary self-energy. We also generalize the channel decomposition to frequency-dependent fermion-boson vertex functions.

Effects of anisotropy in (2+1 )-dimensional QED

We summarize our results for the impact of anisotropic fermionic velocities in (2+1)-dimensional QED on the critical number of fermion flavors, Nfc, and dynamical mass generation. We apply different approximation schemes for the gauge boson vacuum polarization and the fermion–boson vertex to analyze the Dyson–Schwinger equations in a finite volume. Our results point toward large variations of Nfc away from the isotropic point in agreement with other approaches.

Chiral symmetry breaking and confinement beyond rainbow-ladder truncation

A non-perturbative construction of the 3-point fermion-boson vertex which obeys its Ward-Takahashi or Slavnov-Taylor identity, ensures the massless fermion and boson propagators transform according to their local gauge covariance relations, reproduces perturbation theory in the weak coupling regime and provides a gauge independent description for dynamical chiral symmetry breaking (DCSB) and confinement has been a long-standing goal in physically relevant gauge theories such as quantum electrodynamics (QED) and quantum chromodynamics (QCD). In this paper, we demonstrate that the same simple and practical form of the vertex can achieve these objectives not only in 4-dimensional quenched QED (qQED4) but also in its 3-dimensional counterpart (qQED3). Employing this convenient form of the vertex \emph{ansatz} into the Schwinger-Dyson equation (SDE) for the fermion propagator, we observe that it renders the critical coupling in qQED4 markedly gauge independent in contrast with the bare vertex and improves on the well-known Curtis-Pennington construction. Furthermore, our proposal yields gauge independent order parameters for confinement and DCSB in qQED3.

Chemical Potential Dependence of Vertices

Based on the rainbow-ladder approximation of the Dyson–Schwinger equations and the assumption of the analyticity of the fermion-boson vertex in the neighborhood of zero chemical potential (μ = 0) and neglecting the μ-dependence of the dressed gluon propagator, we apply the method in [Phys. Rev. C 71 (2005) 015205] of studying the dressed quark propagator at finite chemical potential to prove that the general fermion-boson vertex at finite μ can also be obtained from the one at μ = 0 by a simple shift of variables. Using this result we extend the results of [Phys. Lett. B 420 (1998) 267] to the situation of finite chemical potential and show that under the approximations we have taken, the Gell–Mann–Oakes–Renner relation also holds at finite chemical potential.

Full fermion-boson vertex function derived in terms of symmetry relations in Abelian gauge theory

Nonperturbative studies such as confinement and dynamical chiral symmetry breaking need the nonperturbative interacting vertex functions. In this paper, an approach to determining the full fermion-boson vertex function in four-dimensional Abelian gauge theory is presented: this full vertex function is derived in terms of a set of normal (longitudinal) and transverse Ward-Takahashi relations for the fermion-boson (vector) and axial-vector vertices in the momentum space in the case of massless fermion. Such a derived fermion-boson vertex function should be satisfied both perturbatively and nonperturbatively. The fact that such a derived full fermion-boson vertex function to one-loop order holds indeed is proven and the nonperturbative form of this vertex is also under discussion.

Renormalization group flow for fermionic superfluids at zero temperature

We present a comprehensive analysis of quantum fluctuation effects in the superfluid ground state of an attractively interacting Fermi system, employing the attractive Hubbard model as a prototype. The superfluid order parameter and fluctuations thereof are implemented by a bosonic Hubbard-Stratonovich field, which splits into two components corresponding to longitudinal and transverse (Goldstone) fluctuations. Physical properties of the system are computed from a set of approximate flow equations obtained by truncating the exact functional renormalization group flow of the coupled fermion-boson action. The equations capture the influence of fluctuations on nonuniversal quantities such as the fermionic gap, as well as the universal infrared asymptotics present in every fermionic superfluid. We solve the flow equations numerically in two dimensions and compute the asymptotic behavior analytically in two and three dimensions. The fermionic gap $\ensuremath{\Delta}$ is reduced significantly compared to the mean-field gap, and the bosonic order parameter $\ensuremath{\alpha}$, which is equivalent to $\ensuremath{\Delta}$ in mean-field theory, is suppressed to values below $\ensuremath{\Delta}$ by fluctuations. The fermion-boson vertex is only slightly renormalized. In the infrared regime, transverse order-parameter fluctuations associated with the Goldstone mode lead to a strong renormalization of longitudinal fluctuations: the longitudinal mass and the bosonic self-interaction vanish linearly as a function of the scale in two dimensions, and logarithmically in three dimensions, in agreement with the exact behavior of an interacting Bose gas.

TRANSVERSE WARD–TAKAHASHI RELATION FOR THE FERMION–BOSON VERTEX FUNCTION IN FOUR-DIMENSIONAL ABELIAN GAUGE THEORY

In this paper, we give a general formula of the transverse Ward–Takahashi relation for the fermion–boson vertex function in momentum space in four-dimensional Abelian gauge theory, with the complete expression of the integral-term involving the nonlocal axial-vector vertex function, from which the corresponding one-loop expression is derived straightforwardly. Then we deduce carefully this transverse Ward–Takahashi relation to one-loop order in d dimensions, with d = 4 + ∊. The result shows that this relation holds in d dimensions if the combination ελμνργργ5 in d dimensions is replaced with -½{γλ, σμν} by the identity of γ-matrices, and there is no additional piece proportional to (d-4) to include for this relation to hold in four dimensions. This result is further confirmed by an explicit computation of terms in this transverse WT relation to one-loop order. We also discuss the momentum space representation of the nonlocal term in the transverse WT relation.

Fermion-Boson Vertex at Finite Chemical Potential

Based on the Ward–Takahashi identity at finite chemicalpotential and Lorentz structure analysis, we generalize theBall–Chiu vertex to the case of nonzero chemical potentialand obtain the general form of the fermion-boson vertex inQED at finite chemical potential.

Checking the transverse Ward–Takahashi relation at one-loop order in four dimensions

Some time ago Takahashi derived the so-called transverse relations relating Green's functions of different orders to complement the well-known Ward–Green–Takahashi identities of gauge theories by considering wedge rather than inner products. These transverse relations have the potential to determine the full fermion–boson vertex in terms of the renormalization functions of the fermion propagator. He and Yu have given an indicative proof at one-loop level in four dimensions. However, their construct involves the fourth-rank Levi–Civita tensor defined only unambiguously in four dimensions exactly where the loop integrals diverge. Consequently, here we explicitly check the proposed transverse Ward–Takahashi relation holds at one-loop order in d-dimensions, with d = 4 + ε.

TRANSVERSE WARD–TAKAHASHI RELATION FOR THE FERMION–BOSON VERTEX TO ONE-LOOP ORDER

In this paper, the transverse Ward–Takahashi relation for the fermion–boson vertex in momentum space is derived in four-dimensional Abelian gauge theory. We show that, by a formal derivation, the transverse Ward–Takahashi relation to one-loop order is satisfied. We also calculate the transverse Ward–Takahashi relation to one-loop order in an arbitrary covariant gauge in the case of massless fermions and find that the result is exactly the same as we obtain in terms of the one-loop fermion–boson vertex calculated in perturbation theory by using Feynman rules. This provides an approach to determine the transverse part of the vertex.

The nonperturbative propagator and vertex in massless quenched QEDd

It is well known how multiplicative renormalizability of the fermion propagator, through its Schwinger-Dyson equation, imposes restrictions on the 3-point fermion-boson vertex in massless quenched quantum electrodynamics in 4-dimensions (QED$_4$). Moreover, perturbation theory serves as an excellent guide for possible nonperturbative constructions of Green functions. We extend these ideas to arbitrary dimensions $d$. The constraint of multiplicative renormalizability of the fermion propagator is generalized to a Landau-Khalatnikov-Fradkin transformation law in $d$-dimensions and it naturally leads to a constraint on the fermion-boson vertex. We verify that this constraint is satisfied in perturbation theory at the one loop level in 3-dimensions. Based upon one loop perturbative calculation of the vertex, we find additional restrictions on its possible nonperturbative forms in arbitrary dimensions.