Lowest-order Vertex Corrections Research Articles

Pion-mediated Cooper pairing of neutrons: beyond the bare vertex approximation

In some quantum many particle systems, the fermions could form Cooper pairs by exchanging intermediate bosons. This then drives a superconducting phase transition or a superfluid transition. Such transitions should be theoretically investigated by using proper non-perturbative methods. Here we take the neutron superfluid transition as an example and study the Cooper pairing of neutrons mediated by neutral π-mesons in the low-density region of a neutron matter. We perform a non-perturbative analysis of the neutron–meson coupling and compute the pairing gap Δ, the critical density ρ c , and the critical temperature T c by solving the Dyson–Schwinger equation of the neutron propagator. We first carry out calculations under the widely used bare vertex approximation and then incorporate the contribution of the lowest-order vertex correction. This vertex correction is not negligible even at low densities and its importance is further enhanced as the density increases. The transition critical line on density-temperature plane obtained under the bare vertex approximation is substantially changed after including the vertex correction. These results indicate that the vertex corrections play a significant role and need to be seriously taken into account.

Nonadiabatic superconductivity in a Li-intercalated hexagonal boron nitride bilayer.

When considering a Li-intercalated hexagonal boron nitride bilayer (Li-hBN), the vertex corrections of electron–phonon interaction cannot be omitted. This is evidenced by the very high value of the ratio λωD/εF ≈ 0.46, where λ is the electron–phonon coupling constant, ωD is the Debye frequency, and εF represents the Fermi energy. Due to nonadiabatic effects, the phonon–induced superconducting state in Li-hBN is characterized by much lower values of the critical temperature (TLOVCC ∈ {19.1, 15.5, 11.8} K, for μ* ∈ {0.1, 0.14, 0.2}, respectively) than would result from calculations not taking this effect into account (TMEC∈ {31.9, 26.9, 21} K). From the technological point of view, the low value of TC limits the possible applications of Li-hBN. The calculations were carried out under the classic Migdal–Eliashberg formalism (ME) and the Eliashberg theory with lowest-order vertex corrections (LOVC). We show that the vertex corrections of higher order (λ3) lower the value of TLOVCC by a few percent.

Stochastic lists: Sampling multivariable functions with population methods

We introduce the method of stochastic lists to deal with a multi-variable positive function, defined by a self-consistent equation, typical for certain problems in physics and mathematics. In this approach, the function's properties are represented statistically by lists containing a large collection of sets of coordinates (or "walkers") that are distributed according to the function's value. The coordinates are generated stochastically by the Metropolis algorithm and may replace older entries according to some protocol. While stochastic lists offer a solution to the impossibility of efficiently computing and storing multi-variable functions without a systematic bias, extrapolation in the inverse of the number of walkers is usually difficult, even though in practice very good results are found already for short lists. This situation is reminiscent of diffusion Monte Carlo, and is hence generic for all population-based methods. We illustrate the method by computing the lowest-order vertex corrections in Hedin's scheme for the Fr\"ohlich polaron and the ground state energy and wavefunction of the Heisenberg model in two dimensions.

Quantitative analysis of nonadiabatic effects in dense H3S and PH3 superconductors.

The comparison study of high pressure superconducting state of recently synthesized H3S and PH3 compounds are conducted within the framework of the strong-coupling theory. By generalization of the standard Eliashberg equations to include the lowest-order vertex correction, we have investigated the influence of the nonadiabatic effects on the Coulomb pseudopotential, electron effective mass, energy gap function and on the 2Δ(0)/TC ratio. We found that, for a fixed value of critical temperature (178 K for H3S and 81 K for PH3), the nonadiabatic corrections reduce the Coulomb pseudopotential for H3S from 0.204 to 0.185 and for PH3 from 0.088 to 0.083, however, the electron effective mass and ratio 2Δ(0)/TC remain unaffected. Independently of the assumed method of analysis, the thermodynamic parameters of superconducting H3S and PH3 strongly deviate from the prediction of BCS theory due to the strong-coupling and retardation effects.

Diagrammatic expansion for positive spectral functions beyondGW: Application to vertex corrections in the electron gas

We present a diagrammatic approach to construct self-energy approximations within many-body perturbation theory with positive spectral properties. The method cures the problem of negative spectral functions which arises from a straightforward inclusion of vertex diagrams beyond the GW approximation. Our approach consists of a two-steps procedure: we first express the approximate many-body self-energy as a product of half-diagrams and then identify the minimal number of half-diagrams to add in order to form a perfect square. The resulting self-energy is an unconventional sum of self-energy diagrams in which the internal lines of half a diagram are time-ordered Green's functions whereas those of the other half are anti-time-ordered Green's functions, and the lines joining the two halves are either lesser or greater Green's functions. The theory is developed using noninteracting Green's functions and subsequently extended to self-consistent Green's functions. Issues related to the conserving properties of diagrammatic approximations with positive spectral functions are also addressed. As a major application of the formalism we derive the minimal set of additional diagrams to make positive the spectral function of the GW approximation with lowest-order vertex corrections and screened interactions. The method is then applied to vertex corrections in the three-dimensional homogeneous electron gas by using a combination of analytical frequency integrations and numerical Monte-Carlo momentum integrations to evaluate the diagrams.

Electron-electron interactions in the vacuum polarization of graphene

We discuss the effect of electron-electron interactions on the static polarization properties of graphene beyond RPA. Divergent self-energy corrections are naturally absorbed into the renormalized coupling constant $\alpha$. We find that the lowest order vertex correction, which is the first non-trivial correlation contribution, is finite, and about 30% of the RPA result at strong coupling $\alpha \sim 1$. The vertex correction leads to further reduction of the effective charge. Finite contributions to dielectric screening are expected in all orders of perturbation theory.

Breakdown of Migdal–Eliashberg Theory via Catastrophic Vertex Divergence at Low Phonon Frequency

We investigate the applicability of Migdal–Eliashberg (ME) theory by revisiting Migdal’s analysis within the dynamical mean-field theory framework. First, we compute spectral functions, the quasi-particle weight, the self energy, renormalised phonon frequency and resistivity curves of the half-filled Holstein model. We demonstrate how ME theory has a phase-transition-like instability at intermediate coupling, and how the Engelsberg–Schrieffer (ES) picture is complicated by low-energy excitations from higher order diagrams (demonstrating that ES theory is a very weak coupling approach). Through consideration of the lowest-order vertex correction, we analyse the applicability of ME theory close to this transition. We find a breakdown of the theory in the intermediate coupling adiabatic limit due to a divergence in the vertex function. The region of applicability is mapped out, and it is found that ME theory is only reliable in the weak coupling adiabatic limit, raising questions about the accuracy of recent analyses of cuprate superconductors which do not include vertex corrections.

Electron and phonon dispersions of the two-dimensional Holstein model: effects ofvertex and non-local corrections

I apply the newly developed dynamical cluster approximation (DCA) to thecalculation of the electron and phonon dispersions in the two-dimensional Holsteinmodel. In contrast to previous work, the DCA enables the effects of spatialfluctuations (non-local corrections) to be examined. Approximations neglectingand incorporating lowest-order vertex corrections are investigated. I calculate thephonon density of states, the renormalized phonon dispersion, the electrondispersion and electron spectral functions. I demonstrate how vertex correctionsstabilize the solution, stopping a catastrophic softening of the (π, π) phononmode. A kink in the electron dispersion is found in the normal state along the (ζ, ζ)symmetry direction in both the vertex- and non-vertex-corrected theories for low phononfrequencies, corresponding directly to the renormalized phonon frequency at the (π, 0)point. This kink is accompanied by a sudden drop in the quasi-particle lifetime.Vertex and non-local corrections enhance the effects at large bare phononfrequencies.

Migdal's theorem in heavy fermion systems

The problem of non-adiabaticity in strongly correlated electron systems has been a subject of considerable attention. We concentrate on the role of vertex corrections to electron–phonon interaction for heavy fermion systems in the normal state. The range of energy transfer important for heavy fermion systems is not set by the energy characteristic of phonons (Debye energy), as in the case of normal metals, but is determined by the characteristic energy scale T ∗ (spin fluctuation temperature). To preserve characteristic features of heavy fermion systems we introduce a very simple model assuming a huge density of states near the Fermi level and a very large electron effective mass. We present an estimate of the lowest-order vertex correction to electron–phonon interaction and show that the Migdal's theorem is not valid for heavy fermion systems.

Vertex-corrected perturbation theory for the electron-phonon problem with nonconstant density of states

A series of weak-coupling perturbation theories which include the lowest-order vertex corrections are applied to the attractive Holstein model in infinite dimensions. The approximations are chosen to reproduce the iterated perturbation theory in the limit of half-filling and large phonon frequency (where the Holstein model maps onto the Hubbard model). Comparison is made with quantum Monte Carlo solutions to test the accuracy of different approximation schemes.

Vertex-corrected tunneling inversion in electron-phonon mediated superconductors: Pb

The McMillan-Rowell tunneling inversion program, which extracts the electron-phonon spectral function ${\mathrm{\ensuremath{\alpha}}}^{2}$F(\ensuremath{\Omega}) and the Coulomb pseudopotential ${\mathrm{\ensuremath{\mu}}}^{\mathrm{*}}$ from experimental tunneling data, is generalized to include the lowest-order vertex correction. We neglect the momentum dependence of the electron-phonon matrix elements, which is formally equivalent to using a local approximation. The perturbation theory is performed on the imaginary axis and then an exact analytic continuation is employed to produce the tunneling density of states on the real axis. Comparison is made with the experimental data for Pb. We find \ensuremath{\lambda} is increased by approximately 1% due to the vertex corrections.

Lowest-order corrections to the RPA polarizability and GW self-energy of a semiconducting wire.

We present the results of the addition of lowest-order vertex and self-consistency corrections to the RPA polarizability and the GW self-energy for a semiconducting wire. It is found that, when starting from a local density approximation zeroth-order Green function and systematically including these corrections in both the polarizability and the self-energy, the correction to the non-self-consistent RPA-GW band gap is small. Partial inclusion of these corrections leads to very different band gaps. This sheds new light on the puzzling question why non-self-consistent RPA-GW calculations of band gaps have been so very successful.

Vertex corrections to the theory of superconductivity

With the eventual goal of understanding vertex corrections in the doped fullerenes and the bismuthates, assuming that they may be electron-phonon superconductors, we have constructed a weak coupling perturbation theory that includes the lowest-order vertex corrections (and reduces to Migdal-Eliashberg theory in the absence of the vertex corrections). Vertex corrections are studied in the local approximation which neglects the momentum dependence of the self energy. This approximation is exact in the limit of infinite dimensions and appears to be accurate in three dimensions. We present results for the superconducting transition temperature as a function of ω ϱ/E ƒ and λ and make comparison with similar results calculated in the infinite-dimensional Holstein model in the weak coupling limit. We find that vertex corrections reduce T c and the energy gap Δ o; however, the ratio 2 Δ 0/ k B T c remains unaffected. We also find that the isotope effect is reduced in the presence of vertex corrections.

Calculations of the superconducting critical temperature with vertex corrections.

The calculations of the superconducting transition temperature, including the lowest-order vertex corrections, are carried out for three models: the BCS-type instantaneous interaction, a retarded isotropic interaction described by the Eliashberg spectral function ${\mathrm{\ensuremath{\alpha}}}^{2}$F(\ensuremath{\Omega}) and a spin-fluctuation interaction with four \ensuremath{\delta}-function peaks in the corners of a square Brillouin zone. In the isotropic case the effect of higher-order self-energy diagrams is to suppress the critical temperature calculated within the mean-field approximation when the characteristic energy of the boson responsible for the pairing is not small compared to the Fermi energy. In the case of the model with sharp peaks in the momentum space the vertex correction can lead to an increase of ${\mathit{T}}_{\mathit{c}}$.

Lowest-order vertex-correction contribution to the direct gap of silicon.

We have calculated the contribution of the lowest-order vertex-correction diagram to the direct gap of silicon at the \ensuremath{\Gamma} point, taking into account the dynamic screening of the electron-electron interaction. Our best calculation yields a contribution of 0.12 eV. This result supports the assumption of the GW approximation that vertex corrections can be neglected. We do not find a significant shift of the absolute energies.

Lowest-order vertex corrections to the energy gap in covalent semiconductors.

The accuracy of the GW approximation is investigated by evaluating the lowest-order vertex contributions to the energy gap of covalent semiconductors. We compare for silicon the energy shifts of a plane-wave calculation with the results of a variational ansatz in local space. The gap corrections at the \ensuremath{\Gamma} point are found to be less than 0.1 eV. An analysis of the different diagrammatic contributions shows that there are two counteracting physical processes which lead to partial cancellations.

Hartree-Fock and lowest-order vertex-correction contribution to the direct gap of the semiconductor silicon.

We have calculated the contribution of the second-order vertex-correction diagram to the direct gap of silicon at the \ensuremath{\Gamma} point and have compared it with the Hartree-Fock contribution. Both contributions have been calculated by using the Monte Carlo method for the involved three- and six-dimensional integrations. Our results show that the second-order contribution is much smaller than the first-order Hartree-Fock contribution. Although we have calculated the vertex correction diagram using the bare instead of the screened Coulomb interaction, our results give a first indication that vertex corrections can indeed be neglected, an assumption which is inherent in the GW approximation to the electron self-energy in a semiconductor.

Lowest-Order Correction to Many-Fermion Vertex Function

The lowest-order vertex correction, the exchange counterpart of the polarization correction, is evaluated for all values of the energy and momentum transfer variables and for an arbitrary rotationally invariant two-body interaction. The method employs a dispersion theory technique and an angular expansion of the interaction in spherical harmonics. For any partial wave, the result is expressed as a one-dimensional integral over a closed interval that can easily be evaluated numerically once the form of the interaction is specified. Explicit expressions can be obtained for the long wavelength limit, and for an expansion in powers of the momentum transfer variable. In this case the result depends on the value of the interaction and its momentum derivatives when one of the momentum variables lies on the Fermi surface.