Vertex Corrections Research Articles

The influence of anisotropic Rashba spin–orbit coupling on current-induced spin polarization in graphene

We consider a disordered graphene layer with anisotropic Rashba spin–orbit coupling subjected to a longitudinal electric field. Using the linear response theory we calculate current-induced spin polarization including in-plane normal and parallel components with respect to the electric field direction. Unlike the case of isotropic Rashba spin–orbit where the normal component of spin polarization is linear in terms of Fermi energy around the Dirac point, anisotropic Rashba spin–orbit can result in non-linear dependence of this component at such energies within the Lifshitz points. Furthermore, we show that anisotropic Rashba interaction allows for tuning the direction of spin polarization from perpendicular direction to the parallel one such that for certain values of Rashba parameters the magnitudes of both components can also be quenched. The effect of carriers scattering on randomly distributed non-magnetic disorders is also taken into account by calculating vertex correction. This results in modification of spin polarization components depending on the relative strength of Rashba parameters.

Development of the self-consistent approximation and its application to the problem of magnetoelastic resonance in an inhomogeneous medium

Our previously proposed approximation involving both the first and second terms of the expansion of the vertex function is generalized to the system of two interacting wavefields of different physical nature. A system of self-consistent equations for the matrix Green’s function and matrix vertex function is derived. On the basis of this matrix generalization of the new self-consistent approximation, a theory of magnetoelastic resonance is developed for a ferromagnetic model, where the magnetoelastic coupling parameter e(x) is inhomogeneous. Equations for magnetoelastic resonance are analyzed for one-dimensional inhomogeneities of the coupling parameter. The diagonal and off-diagonal elements of the matrix Green’s function of the system of coupled spin and elastic waves are calculated with the change in the ratio between the average value e and rms fluctuation Δe of the coupling parameter between waves from the homogeneous case (e ≠ 0, Δe = 0) to the extremely randomized case (e = 0, Δe ≠ 0) at various correlation wavenumbers of inhomogeneities k c. For the limiting case of infinite correlation radius (k c = 0), in addition to approximate expressions, exact analytical expressions corresponding to the summation of all diagrams of elements of the matrix Green’s function are obtained. The results calculated for an arbitrary k c value in the new self-consistent approximation are compared to the results obtained in the standard self-consistent approximation, where only the first term of the expansion of the vertex function is taken into account. It is shown that the new approximation corrects disadvantages of the Green’s functions calculated in the standard approximation such as the dome shape of resonances and bends on the sides of resonance peaks. The appearance of a fine structure of the spectrum in the form of a narrow resonance on the Green’s function of spin waves and a narrow antiresonance on the Green’s function of elastic waves, which was previously predicted in the standard self-consistent approximation, is confirmed. With an increase in the parameter k c, the Green’s functions calculated in the standard and new approximations approach each other and almost coincide with each other at k c/k ≥ 0.5. At the same time, the results of this work indicate that the new self-consistent approximation has a certain advantage for studying the problems of stochastic radiophysics in media with long-wavelength inhomogeneities (small k c values), because it describes both the shape and width of peaks much better than the standard approximation.

Heavy-quark form factors in the large beta _0 limit

Heavy-quark form factors are calculated at beta _0 alpha _s sim 1 to all orders in alpha _s at the first order in 1/beta _0. Using the inversion relation generalized to vertex functions, we reduce the massive on-shell Feynman integral to the HQET one. This HQET vertex integral can be expressed via a {}_2F_1 function; the nth term of its varepsilon expansion is explicitly known. We confirm existing results for n_l^{L-1} alpha _s^L terms in the form factors (up to L=3), and we present results for higher L.

Connecting dark matter annihilation to the vertex functions of Standard Model fermions

We consider scenarios in which dark matter is a Majorana fermion which couples to Standard Model fermions through the exchange of charged mediating particles. The matrix elements for various dark matter annihilation processes are then related to one-loop corrections to the fermion-photon vertex, where dark matter and the charged mediators run in the loop. In particular, in the limit where Standard Model fermion helicity mixing is suppressed, the cross section for dark matter annihilation to various final states is related to corrections to the Standard Model fermion charge form factor. These corrections can be extracted in a gauge-invariant manner from collider cross sections. Although current measurements from colliders are not precise enough to provide useful constraints on dark matter annihilation, improved measurements at future experiments, such as the International Linear Collider, could improve these constraints by several orders of magnitude, allowing them to surpass the limits obtainable by direct observation.

Competing Unconventional Charge-Density-Wave States in Cuprate Superconductors: Spin-Fluctuation-Driven Mechanism

To understand the origin of unconventional charge-density-wave (CDW) states in cuprate superconductors, we establish the self-consistent CDW equation, and analyze the CDW instabilities based on the realistic Hubbard model, without assuming any $q$-dependence and the form factor. Many higher-order many-body processes, which are called the vertex corrections, are systematically generated by solving the CDW equation. When the spin fluctuations are strong, the uniform $q=0$ nematic CDW with $d$-form factor shows the leading instability. The axial nematic CDW instability at $q = Q_a = (\delta,0)$ ($\delta \approx \pi/2$) is the second strongest, and its strength increases under the static uniform CDW order. The present theory predicts that uniform CDW transition emerges at a high temperature, and it stabilize the axial $q = Q_a$ CDW at $T = T_{CDW}$. It is confirmed that the higher-order Aslamazov-Larkin processes cause the CDW orders at both $q = 0$ and $Q_a$.

Dynamical electron-phonon coupling, GW self-consistency, and vertex effect on the electronic band gap of ice and liquid water

We study the impact of dynamical electron-phonon (el-ph) effects on the electronic band gap of ice and liquid water by accounting for frequency-dependent Fan contributions in the el-ph mediated self-energy within the many-body perturbation theory (MBPT). We find that the dynamical el-ph coupling effects greatly reduce the static el-ph band-gap correction of the hydrogen-rich molecular ice crystal from$\ensuremath{-}2.46$ to $\ensuremath{-}0.23$ eV in great contrast to the result of Monserrat et al. [Phys. Rev. B 92, 140302 (2015)]. This is of particular importance as otherwise the static el-ph gap correction would considerably reduce the electronic band gap, leading to considerable underestimation of the intense peaks of optical absorption spectra of ice which would be in great disagreement to experimental references. By contrast, the static el-ph gap correction of liquid water is very moderate ($\ensuremath{-}0.32$ eV), and inclusion of dynamical effects slightly reduces the gap correction to $\ensuremath{-}0.19$ eV. Further, we determine the diverse sensitivity of ice and liquid water to the $GW$ self-consistency and show that the energy-only self-consistent approach (${G}_{n}{W}_{n}$) exhibits large implicit vertex character in comparison to the quasiparticle self-consistent approach, for which an explicit calculation of vertex corrections is necessary for good agreement with experiment.

Four-loop QCD propagators and vertices with one vanishing external momentum

We have computed the self-energies and a set of three-particle vertex functions for massless QCD at the four-loop level in the overline{mathrm{MS}} renormalization scheme. The vertex functions are evaluated at points where one of the momenta vanishes. Analytical results are obtained for a generic gauge group and with the full gauge dependence, which was made possible by extensive use of the Forcer program for massless four-loop propagator integrals. The bare results in dimensional regularization are provided in terms of master integrals and rational coefficients; the latter are exact in any space-time dimension. Our results can be used for further precision investigations of the perturbative behaviour of the theory in schemes other than overline{mathrm{MS}} . As an example, we derive the five-loop beta function in a relatively common alternative, the minimal momentum subtraction (MiniMOM) scheme.

Elastic scattering of electrons by nuclei of half-integer spin and discrete symmetries

Analytical expression is obtained for the right-left asymmetry A RL (3/2) for the process of elastic scattering of the longitudinally polarized electrons by nuclei with spin 3/2, described in the framework of Rarita-Schwinger formalism by invariant form factors of electromagnetic and weak vertex functions. It is shown, that this asymmetry directly depends on the nuclear anapole form factors G 1 (n), and structurally is equivalent to asymmetry A RL (1/2), which arises in electron scattering by proton.

Self-consistent solution of Hedin's equations: Semiconductors and insulators

The band gaps of a few selected semiconductors/insulators are obtained from the self-consistent solution of the Hedin's equations. Two different schemes to include the vertex corrections are studied: (i) the vertex function of the first-order (in the screened interaction $W$) is applied in both the polarizability $P$ and the self-energy $\Sigma$, and (ii) the vertex function obtained from the Bethe-Salpeter equation is used in $P$ whereas the vertex of the first-order is used in $\Sigma$. Both schemes show considerable improvement in the accuracy of the calculated band gaps as compared to the self-consistent $GW$ approach (sc$GW$) and to the self-consistent quasi-particle $GW$ approach (QS$GW$). To further distinguish between the performances of two vertex-corrected schemes one has to properly take into account the effect of the electron-phonon interaction on the calculated band gaps which appears to be of the same magnitude as the difference between schemes i) and ii).

Calculation of two-particle quantities in the typical medium dynamical cluster approximation

The mean-field theory for disordered electron systems without interaction is widely and successfully used to describe equilibrium properties of materials over the whole range of disorder strengths. However, it fails to take into account the effects of quantum coherence and information of localization. Vertex corrections due to multiple back-scatterings may drive the electrical conductivity to zero and make expansions around the mean field in strong disorder problematic. Here, we present a method for the calculation of two-particle quantities which enable us to characterize the metal-insulator transitions (MIT) in disordered electron systems by using the Typical Medium Dynamical Cluster Approximation (TMDCA). We show how to include vertex corrections and the information about the mobility edge to the typical mean-field theory. We successfully demonstrate the application of the developed method by showing that the conductivity formulated in this way properly characterizes the MIT in disordered systems.

Local correlation functions of arbitrary order for the Falicov-Kimball model

Local n-particle vertex functions represent the crucial ingredient for all diagrammatic extensions of dynamical mean field theory (DMFT). Hitherto their application has been restricted -with a few exceptions- to the n=2-particle vertex while higher-order vertices have been neglected. In this paper we derive a general analytical expression for the n-particle (one-particle reducible) vertex of the Falicov-Kimball model (FKM). We observe that the magnitude of such vertex functions itself strongly increases with the particle-number n. On the other hand, their effect on generic Feynman diagrams remains rather moderate due to the damping effect of the Green's functions present in such diagrams. Nevertheless, they yield important contributions to the self-energy corrections calculated in diagrammatic extensions of DMFT as we explicitly demonstrate using the example of dual fermion (DF) calculations for the two-dimensional FKM at quarter filling of stationary f-electrons where corrections to the self-energy obtained from the three-particle vertex are indeed comparable in magnitude to corresponding corrections stemming from the two-particle vertex.

Ladder approximation for the AC conductivity in the generalized two-dimensional Hubbard model

We calculate the optical conductivity of the generalized two-dimensional Hubbard model including vertex corrections, by using a ladder approximation in the diagrammatic expansion. We have obtained a superconductor behavior for this system at low temperature. Employing the ladder approach, we have included the influence of the electron-electron interaction on a previous result obtained in the mean field approximation. We have obtained the behavior of the optical conductivity, employing the ladder approximation, tending to zero in the DC limit. Since this approximation is better than the mean field approximation (it include information about the electron-electron interaction) then we have a better description to the behavior of the AC optical conductivity for the two-dimensional Hubbard model.

Standard coupling unification in SO(10), hybrid seesaw neutrino mass and leptogenesis, dark matter, and proton lifetime predictions

We discuss gauge coupling unification of SU(3)C × SU(2)L × U(1)Y descending directly from non-supersymmetric SO(10) while providing solutions to the three out-standing problems of the standard model: neutrino masses, dark matter, and the baryon asymmetry of the universe. Conservation of matter parity as gauged discrete symmetry for the stability and identification of dark matter in the model calls for high-scale spontaneous symmetry breaking through 126H Higgs representation. This naturally leads to the hybrid seesaw formula for neutrino masses mediated by heavy scalar triplet and right-handed neutrinos. Being quadratic in the Majorana coupling, the seesaw formula predicts two distinct patterns of right-handed neutrino masses, one hierarchical and another not so hierarchical (or compact), when fitted with the neutrino oscillation data. Predictions of the baryon asymmetry via leptogenesis are investigated through the decays of both the patterns of RHν masses. A complete flavor analysis has been carried out to compute CP-asymmetries including washouts and solutions to Boltzmann equations have been utilised to predict the baryon asymmetry. The additional contribution to vertex correction mediated by the heavy left-handed triplet scalar is noted to contribute as dominantly as other Feynman diagrams. We have found successful predictions of the baryon asymmetry for both the patterns of right-handed neutrino masses. The SU(2)L triplet fermionic dark matter at the TeV scale carrying even matter parity is naturally embedded into the non-standard fermionic representation 45F of SO(10). In addition to the triplet scalar and the triplet fermion, the model needs a nonstandard color octet fermion of mass ∼ 5 × 107 GeV to achieve precision gauge coupling unification at the GUT mass scale MU0 = 1015.56 GeV. Threshold corrections due to superheavy components of 126H and other representations are estimated and found to be substantial. It is noted that the proton life time predicted by the model is accessible to the ongoing and planned experiments over a wide range of parameter space.

Strong-Coupling Effects and Shear Viscosity in an Ultracold Fermi Gas

We theoretically investigate the shear viscosity $$\eta $$ , as well as the entropy density s, in the normal state of an ultracold Fermi gas. Including pairing fluctuations within the framework of a T-matrix approximation, we calculate these quantities in the Bardeen–Cooper–Schrieffer (BCS)–Bose–Einstein condensation (BEC) crossover region. We also evaluate $$\eta / s$$ , to compare it with the lower bound of this ratio, conjectured by Kovtun, Son, and Starinets (KSS bound). In the weak-coupling BCS side, we show that the shear viscosity $$\eta $$ is remarkably suppressed near the superfluid phase transition temperature $$T_{\mathrm{c}}$$ , due to the so-called pseudogap phenomenon. In the strong-coupling BEC side, we find that, within the neglect of the vertex corrections, one cannot correctly describe $$\eta $$ . We also show that $$\eta / s$$ decreases with increasing the interaction strength, to become very close to the KSS bound, $$\hbar /4\pi k_{\mathrm{B}}$$ , on the BEC side.

First-principles quantum transport method for disordered nanoelectronics: Disorder-averaged transmission, shot noise, and device-to-device variability

Because disorders are inevitable in realistic nanodevices, the capability to quantitatively simulate the disorder effects on electron transport is indispensable for quantum transport theory. Here, we report a unified and effective first-principles quantum transport method for analyzing effects of chemical or substitutional disorder on transport properties of nanoelectronics, including averaged transmission coefficient, shot noise, and disorder-induced device-to-device variability. All our theoretical formulations and numerical implementations are worked out within the framework of the tight-binding linear muffin tin orbital method. In this method, we carry out the electronic structure calculation with the density functional theory, treat the nonequilibrium statistics by the nonequilbrium Green's function method, and include the effects of multiple impurity scattering with the generalized nonequilibrium vertex correction (NVC) method in coherent potential approximation (CPA). The generalized NVC equations are solved from first principles to obtain various disorder-averaged two-Green's-function correlators. This method provides a unified way to obtain different disorder-averaged transport properties of disordered nanoelectronics from first principles. To test our implementation, we apply the method to investigate the shot noise in the disordered copper conductor, and find all our results for different disorder concentrations approach a universal Fano factor $1/3$. As the second test, we calculate the device-to-device variability in the spin-dependent transport through the disordered Cu/Co interface and find the conductance fluctuation is very large in the minority spin channel and negligible in the majority spin channel. Our results agree well with experimental measurements and other theories. In both applications, we show the generalized nonequilibrium vertex corrections play a determinant role in electron transport simulation. Our results demonstrate the effectiveness of the first-principles generalized CPA-NVC for atomistic analysis of disordered nanoelectronics, extending the capability of quantum transport simulation.

Two‐particle self‐consistent analysis for the electron‐hole asymmetry of superconductivity in cuprate superconductors

In the hole‐doped type cuprate superconductors, it is well‐known that the superconducting transition temperature Tc exhibits a dome‐like structure against doping. On the other hand, recent experiments unveil that Tc in the electron‐doped compounds shows a monotonic increase with decreasing the doping, at least down to a very small doping rate. Our recent study for the three‐band d–p model has unveiled that this asymmetric behavior can be explained as a combined effect of the intrinsic electron‐hole asymmetry in systems comprising Cu3 d and O2 p orbitals and the band‐filling‐dependent vertex correction . In the present study, we study another compound Tl2 Ba2 CuO6 to show that this explanation can be applied to other cuprate superconductors with the small orbital mixture. By varying the d–p offset, we also study how the strength of the d–p hybridization controls the spin fluctuation and hence the pairing interaction.

Maximum entropy analytic continuation for frequency-dependent transport coefficients with nonpositive spectral weight

The computation of transport coefficients, even in linear response, is a major challenge for theoretical methods that rely on analytic continuation of correlations functions obtained numerically in Matsubara space. While maximum entropy methods can be used for certain correlation functions, this is not possible in general, important examples being the Seebeck, Hall, Nernst and Reggi-Leduc coefficients. Indeed, positivity of the spectral weight on the positive real-frequency axis is not guaranteed in these cases. The spectral weight can even be complex in the presence of broken time-reversal symmetry. Various workarounds, such as the neglect of vertex corrections or the study of the infinite frequency or Kelvin limits have been proposed. Here, we show that one can define auxiliary response functions that allow to extract the desired real-frequency susceptibilities from maximum entropy methods in the most general multiorbital cases with no particular symmetry. As a benchmark case, we study the longitudinal thermoelectric response and corresponding Onsager coefficient in the single-band two-dimensional Hubbard model treated with dynamical mean-field theory (DMFT) and continuous-time quantum Monte Carlo (CTQMC). We thereby extend to transport coefficients the maximum entropy analytic continuation with auxiliary functions (MaxEntAux method), developed for the study of the superconducting pairing dynamics of correlated materials.

Keldysh functional renormalization group for electronic properties of graphene

We construct a nonperturbative nonequilibrium theory for graphene electrons interacting via the instantaneous Coulomb interaction by combining the functional renormalization group method with the nonequilibrium Keldysh formalism. The Coulomb interaction is partially bosonized in the forward scattering channel resulting in a coupled Fermi-Bose theory. Quantum kinetic equations for the Dirac fermions and the Hubbard-Stratonovich boson are derived in Keldysh basis, together with the exact flow equation for the effective action and the hierarchy of one-particle irreducible vertex functions, taking into account a possible non-zero expectation value of the bosonic field. Eventually, the system of equations is solved approximately under thermal equilibrium conditions at finite temperature, providing results for the renormalized Fermi velocity and the static dielectric function, which extends the zero-temperature results of Bauer~\textit{et~al.}, Phys.\ Rev.\ B {\bf 92}, 121409 (2015).

Ab initio dynamical vertex approximation

Diagrammatic extensions of dynamical mean field theory (DMFT) such as the dynamical vertex approximation (D$\Gamma$A) allow us to include non-local correlations beyond DMFT on all length scales and proved their worth for model calculations. Here, we develop and implement an AbinitioD$\Gamma$A approach for electronic structure calculations of materials. Starting point is the two-particle irreducible vertex in the two particle-hole channels which is approximated by the bare non-local Coulomb interaction and all local vertex corrections. From this we calculate the full non-local vertex and the non-local self-energy through the Bethe-Salpeter equation. The AbinitioD$\Gamma$A approach naturally generates all local DMFT correlations and all non-local $GW$ contributions, but also further non-local correlations beyond: mixed terms of the former two and non-local spin fluctuations. We apply this new methodology to the prototypical correlated metal SrVO$_3$.

The S-wave resonance contributions in the B^{0}_{s} decays into psi (2S,3S) plus pion pair

The three-body decays B^0_s rightarrow psi (2S,3S) pi ^+ pi ^- are studied based on the perturbative QCD approach. With the help of the nonperturbative two-pion distribution amplitudes, the analysis is simplified into the quasi-two-body processes. Besides the traditional factorizable and nonfactorizable diagrams at the leading order, the next-to-leading order vertex corrections are also included to cancel the scale dependence. The f_0(980), f_0(1500) resonance contributions as well as the nonresonant contributions are taken into account using the presently known pi pi time-like scalar form factor for the sbar{s} component. It is found that the predicted B^0_s rightarrow psi (2S) pi ^+ pi ^- decay spectra in the pion pair invariant mass shows a similar behavior as the experiment. The calculated S-wave contributions to the branching ratio of B^0_s rightarrow psi (2S) pi ^+ pi ^- is 6.0times 10^{-5}, which is in agreement with the LHCb data mathcal {B}(B^0_s rightarrow psi (2S) pi ^+ pi ^-)=(7.2pm 1.2)times 10^{-5} within errors. The estimate of mathcal {B}(B^0_s rightarrow psi (3S) pi ^+ pi ^-) can reach the order of 10^{-5}, pending the corresponding measurements.