Vertex Corrections Research Articles

The plain and simple parquet approximation: single-and multi-boson exchange in the two-dimensional Hubbard model

The parquet approach to vertex corrections is unbiased but computationally demanding. Most applications are therefore restricted to small cluster sizes or rely on various simplifying approximations. We have recently shown that the bosonization of the parquet diagrams provides interpretative and algorithmic advantages over the original purely fermionic formulation. Here, we present first results of the numerical implementation of this method by applying it to the half-filled Hubbard model on the square lattice at weak coupling. The improved algorithmic performance allows us to evaluate the parquet approximation for a 16times 16 lattice, retaining the full momentum and frequency structure of the various vertex functions. We discuss their symmetries and consider parametrizations of their momentum dependence using the truncated-unity approximation.Graphical abstract

Characterization of collective excitations in weakly coupled disordered superconductors

Isolated islands in two-dimensional strongly-disordered and strongly-coupled superconductors become optically active inducing sub-gap collective excitations in the ac conductivity. Here, we investigate the fate of these excitations as a function of the disorder strength in the experimentally relevant case of weak electron-phonon coupling. An explicit calculation of the ac conductivity, that includes vertex corrections to restore gauge symmetry, reveals the existence of collective sub-gap excitations, related to phase fluctuations and therefore identified as the Goldstone modes, for intermediate to strong disorder. As disorder increases, the shape of the sub-gap excitation transits from peaked close to the spectral gap to a broader distribution reaching much smaller frequencies. Phase-coherence still holds in part of this disorder regime. The requirement to observe sub-gap excitations is not the existence of isolated islands acting as nano-antennas but rather the combination of a sufficiently inhomogeneous order parameter with a phase fluctuation correlation length smaller than the system size. Our results indicate that, by tuning disorder, the Goldstone mode may be observed experimentally in metallic superconductors based for instance on Al, Sn, Pb or Nb.

Carbonaceous sulfur hydride system: The strong-coupled room-temperature superconductor with a low value of Ginzburg–Landau parameter

The superconducting state in a carbonaceous sulfur hydride (C–S–H) system is probably characterized by the record-high critical temperature of 288 K (p≈267 GPa). We determined the properties of the C–S–H superconducting phase within the scope of both classical Eliashberg equations and the Eliashberg equations with vertex corrections. We took into account the scenarios pertinent to either the intermediate or the high value of an electron–phonon coupling constant (λ≈0.75 or λ≈3.3, respectively). The scenario for the intermediate value, however, cannot be actually realized due to the anomalously high value of the logarithmic phonon frequency (ωln/kB=7150 K) it would require. On the other hand, we found it possible to reproduce correctly the value of TC and other thermodynamic quantities in the case of strong coupling, with all the reservations discussed in the presented paper. The vertex corrections lower the order parameter values within the range from ≈50 K to ≈275 K. For the upper critical field HC2≈27 T, the Ginzburg–Landau parameter κ is of the order of 1.7. The strong-coupling scenario for the C–S–H system is also suggested by the high values of λ estimated for H3S (λ≈2.1, κ≈1.5), LaH10 (λ≈2.8–3.9, κ≈1.6), and YH6 (λ≈1.7, κ≈1.3) compounds. In the case of the C–S–H system, we also anticipate the presence of the antiferromagnetic state above the superconducting state like in the dense CS2 superconductor. For p≈174 GPa and TC≈180 K, the magnetic ordering transition occurs at TN≈213 K.

Simplification of the Local Full Vertex in the Impurity Problem in DMFT and Its Applications for the Nonlocal Correlation

The two-particle vertex function is crucial for the diagrammatic extensions beyond DMFT for the nonlocal fluctuation. However, estimating the two-particle quantities is still a challenging task. In this study, we propose a simplification of the local two-particle full vertex and, using the simplified full vertex, we develop two methods to take into account the nonlocal fluctuation. We apply these methods to several models and confirm that our methods can capture important behaviors such as the pseudo gap in the DMFT + nonlocal calculation. In addition, the numerical costs are largely reduced compared to the conventional methods.

Capped vertex with descendants for zero dimensional A∞ quiver varieties

In this paper, we study the capped vertex functions associated to certain zero-dimensional type-A Nakajima quiver varieties. The insertion of descendants into the vertex functions can be expressed by the Macdonald operators, which leads to explicit combinatorial formulas for the capped vertex functions.We determine the monodromy of the vertex functions and show that it coincides with the elliptic R-matrix of symplectic dual variety. We apply our results to give the vertex functions and the characters of the tautological bundles on the quiver varieties formed from arbitrary stability conditions.

Current noise and Keldysh vertex function of an Anderson impurity in the Fermi-liquid regime

We present a complete microscopic Fermi-liquid description for next-to-leading order transport through an Anderson impurity under a finite bias voltage $V$. It is applicable to multilevel quantum dots without particle-hole or time-reversal symmetry and is constructed based on the nonequilibrium Keldysh formalism, taking into account the current conservation between electrons in the impurity levels and the conduction bands. Specifically, we derive the formula for the current noise generated in the steady flow up to terms of order ${(eV)}^{3}$ at zero temperature $T=0$. To this end, we calculate the Keldysh vertex functions ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\sigma}{\ensuremath{\sigma}}^{\ensuremath{'}};{\ensuremath{\sigma}}^{\ensuremath{'}}\ensuremath{\sigma}}^{{\ensuremath{\nu}}_{1}{\ensuremath{\nu}}_{2};{\ensuremath{\nu}}_{3}{\ensuremath{\nu}}_{4}}(\ensuremath{\omega},{\ensuremath{\omega}}^{\ensuremath{'}};{\ensuremath{\omega}}^{\ensuremath{'}},\ensuremath{\omega})$, which depend on branches ${\ensuremath{\nu}}_{1},\phantom{\rule{4pt}{0ex}}{\ensuremath{\nu}}_{2},\phantom{\rule{4pt}{0ex}}{\ensuremath{\nu}}_{3}$, and ${\ensuremath{\nu}}_{4}$ of the time-loop contour and on spin degrees of freedom $\ensuremath{\sigma}$ and ${\ensuremath{\sigma}}^{\ensuremath{'}}$, up to linear order terms with respect to $eV$, $T$, and frequencies $\ensuremath{\omega}$ and ${\ensuremath{\omega}}^{\ensuremath{'}}$. The coefficients of these linear order terms are determined by a set of the parameters, defined with respect to the equilibrium ground state: the phase shift, static susceptibilities, and nonlinear three-body susceptibilities of the impurity electrons. The low-energy expressions of the vertex components are shown to satisfy the Ward identities with the Keldysh Green's functions expanded up to terms of order ${\ensuremath{\omega}}^{2}$, ${(eV)}^{2}$, and ${T}^{2}$. We also find that the imaginary part of the Ward identities can be described in terms of the $eV$-dependent collision integrals for a single-quasiparticle excitation and that for a single quasiparticle-quasihole pair excitation. These collision integrals ensure the current conservation of the next-to-leading order Fermi-liquid transport due to the quasiparticles with a finite damping rate.

Temperature and doping dependence of the singlet and triplet pair susceptibilities in the one-band Hubbard model based on the dynamical mean-field theory

The superconductivity in the one-band Hubbard model on a Bethe lattice is investigated on the basis of the dynamical mean-field theory (DMFT) in which the irreducible vertex function has no k-dependence and then only the s-wave superconductivities with the spin-singlet even-frequency and the spin-triplet odd-frequency pairings are possibly realized. We calculate the pair susceptibility by solving the Bethe-Salpeter equation with the use of the linearized DMFT and find that the spin-singlet pair susceptibility is suppressed by the on-site Coulomb interaction U especially for the region near the Mott transition. On the other hand, the spin-triplet pair susceptibility is apparently enhanced by U in the strong correlation regime at very low temperatures where the effective interaction for the triplet pairing is considered to be attractive and then the triplet superconductivity is expected to be realized.

Dimerization tendencies of the pyrochlore Heisenberg antiferromagnet: A functional renormalization group perspective

We investigate the ground state properties of the spin-$1/2$ pyrochlore Heisenberg antiferromagnet using pseudofermion functional renormalization group techniques. The first part of our analysis is based on an enhanced parton mean-field approach which takes into account fluctuation effects from renormalized vertex functions. Our implementation of this technique extends earlier approaches and resolves technical difficulties associated with a diagrammatic overcounting. Using various parton ans\"atze for quantum spin liquids, dimerized and nematic states our results indicate a tendency for lattice symmetry breaking in the ground state. While overall quantum spin liquids seem unfavorable in this system, the recently proposed monopole state still shows the strongest support among all spin liquid ans\"atze that we have tested, which is further confirmed by our complementary variational Monte Carlo calculations. In the second part of our investigation, we probe lattice symmetry breaking more directly by applying the pseudofermion functional renormalization group to perturbed systems. Our results from this technique confirm that the system's ground state either exhibits broken $C_3$ rotation symmetry, or a combination of inversion and $C_3$ symmetry breaking.

Hadronic three-body D decays mediated by scalar resonances

We study the quasi-two-body $D\ensuremath{\rightarrow}SP$ decays and the three-body $D$ decays proceeding through intermediate scalar resonances, where $S$ and $P$ denote scalar and pseudoscalar mesons, respectively. Our main results are the following: (i) Certain external and internal $W$-emission diagrams with the emitted meson being a scalar meson are na\"{\i}vely expected to vanish, but they actually receive contributions from vertex and hard spectator-scattering corrections beyond the factorization approximation. (ii) For light scalars with masses below or close to 1 GeV, it is more sensible to study three-body decays directly and compare with experiment as the two-body branching fractions are either unavailable or subject to large finite-width effects of the scalar meson. (iii) We consider the two-quark (scheme I) and four-quark (scheme II) descriptions of the light scalar mesons, and find the latter generally in better agreement with experiment. This is in line with recent BESIII measurements of semileptonic charm decays that prefer the tetraquark description of light scalars produced in charmed meson decays. (iv) The topological amplitude approach fails here as the $D\ensuremath{\rightarrow}SP$ decay branching fractions cannot be reliably inferred from the measurements of three-body decays, mainly because the decay rates cannot be factorized into the topological amplitude squared and the phase space factor. (v) The predicted rates for ${D}^{0}\ensuremath{\rightarrow}{f}_{0}P,{a}_{0}P$ are generally smaller than experimental data by one order of magnitude, presumably implying the significance of $W$-exchange amplitudes. (vi) The $W$-annihilation amplitude is found to be very sizable in the $SP$ sector with $|A/T{|}_{SP}\ensuremath{\sim}1/2$, contrary to its suppression in the $PP$ sector with $|A/T{|}_{PP}\ensuremath{\sim}0.18$. (vii) Finite-width effects are very important for the very broad $\ensuremath{\sigma}/{f}_{0}(500)$ and $\ensuremath{\kappa}/{K}_{0}^{*}(700)$ mesons. The experimental branching fractions $\mathcal{B}({D}^{+}\ensuremath{\rightarrow}\ensuremath{\sigma}{\ensuremath{\pi}}^{+})$ and $\mathcal{B}({D}^{+}\ensuremath{\rightarrow}{\overline{\ensuremath{\kappa}}}^{0}{\ensuremath{\pi}}^{+})$ are thus corrected to be $(3.8\ifmmode\pm\else\textpm\fi{}0.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ and $(6.{7}_{\ensuremath{-}4.5}^{+5.6})%$, respectively.

Generic anisotropic Lifshitz scalar field theory: New massive minimal subtraction

We formulate the simplest minimal subtraction version for massive scalar fields with O(N) symmetry for generic anisotropic Lifshitz spacetimes. We introduce the simplest geometric concepts to define it as a special manifold. Then we restrict ourselves to flat Euclidean spaces. An appropriate partial-p operation is applied in the bare two-point vertex function diagrams, which separates the original diagram into a sum of two different integrals which are the coefficients of the corresponding polynomials in the mass and external momentum. Within the proposed method, the coefficient of the mass terms can be discarded and we obtain a minimal subtraction method almost identical to the same scheme in the massless theory in every external momentum/mass subspace. We restrict our demonstration of the method up to three-loop order in the two-point vertex part. We verify its consistency through a diagrammatic computation of static critical exponents, which validates the universality hypothesis.

Rise and fall of plaquette order in the Shastry-Sutherland magnet revealed by pseudofermion functional renormalization group

The Shastry-Sutherland (SS) model as a canonical example of frustrated magnetism has been extensively studied. The conventional wisdom has been that the transition from the plaquette valence bond order to the Neel order is direct and potentially realizes a deconfined quantum critical point beyond the Ginzburg-Landau paradigm. This scenario, however, was challenged recently by improved numerics from density matrix renormalization group which offers evidence for a narrow gapless spin liquid between the two phases. Prompted by this controversy and to shed light on this intricate parameter regime from a fresh perspective, we report high-resolution functional renormalization group analysis of the generalized SS model. The flows of over 50 million running couplings provide a detailed picture for the evolution of spin correlations as the frequency/energy scale is dialed from the ultraviolet to the infrared to yield the zero-temperature phase diagram. The singlet dimer phase emerges as a fixed point, the Neel order is characterized by divergence in the vertex function, while the transition into and out of the plaquette order is accompanied by pronounced peaks in the plaquette susceptibility. The plaquette order is suppressed before the onset of the Neel order, lending evidence for a finite spin liquid region for ${J}_{1}/{J}_{2}\ensuremath{\in}(0.77,0.82)$, where the flow is continuous without any indication of divergence.

Running coupling constant in thermal ϕ4 theory up to two loop order

Using the imaginary time formalism in thermal field theory, we derive the running coupling constant and running mass in two loop order. In the process, we express the imaginary time formalism of Feynman diagrams as the summation of nonthermal quantum field theory (QFT) Feynman diagrams with coefficients that depend on temperature and mass. Renormalization constants for thermal ${\ensuremath{\phi}}^{4}$ theory were derived using simple diagrammatic analysis. Our model links the nonthermal QFT and the imaginary time formalism by assuming both have the same mass scale $\ensuremath{\mu}$ and coupling constant $g$. When these results are combined with the renormalization group equations and applied simultaneously to thermal and nonthermal proper vertex functions, the coupling constant and running mass with implicit temperature dependence are obtained. We evaluated pressure for scalar particles in two loop orders at the zero external momentum limit by substituting the running mass result in the quasiparticle model.

Dynamical exponent of a quantum critical itinerant ferromagnet: A Monte Carlo study

We consider the effect of the coupling between 2D quantum rotors near an XY ferromagnetic quantum critical point and spins of itinerant fermions. We analyze how this coupling affects the dynamics of rotors and the self-energy of fermions.A common belief is that near a $q=0$ ferromagnetic transition, fermions induce an $\Omega/q$ Landau damping of rotors (i.e., the dynamical critical exponent is $z=3$) and Landau overdamped rotors give rise to non-Fermi liquid fermionic self-energy $\Sigma\propto \omega^{2/3}$. This behavior has been confirmed in previous quantum Monte Carlo (QMC) studies.Here we show that for the XY case the behavior is different.We report the results of large scale quantum Monte Carlo simulations,which show that at small frequencies $z=2$ and $\Sigma\propto \omega^{1/2}$. We argue that the new behavior is associated with the fact that a fermionic spin is by itself not a conserved quantity due to spin-spin coupling to rotors, and a combination of self-energy and vertex corrections replaces $1/q$ in the Landau damping by a constant. We discuss the implication of these results to experiments.

Full versus quasiparticle self-consistency in vertex-corrected GW approaches

Using seven semiconductors and insulators with band gaps covering the range from 1 to 10 eV we systematically explore the performance of two different variants of self-consistency associated with the famous Hedin system of equations: the full self-consistency and the so-called quasiparticle approximation to it. The pros and cons of these two variants of self-consistency are sufficiently well documented in the literature for the simplest GW approximation to the Hedin equations. Our study, therefore, aims primarily at the level of theory beyond GW approximation, i.e., at the level of theory which includes vertex corrections. Whereas quasiparticle self-consistency has certain advantages at the GW level (a well-known fact), the situation becomes quite different when vertex corrections are included. In the variant with full self-consistency, vertex corrections (both for polarizability and for self-energy) systematically reduce the calculated band gaps making them closer to the experimental values. In the variant with quasiparticle self-consistency, however, an inclusion of the same diagrams has a considerably larger effect and calculated band gaps become severely underestimated. Different effects of vertex corrections in two variants of self-consistency can be related to the $Z$-factor cancellation which plays a positive role in quasiparticle self-consistency at the GW level of theory but appears to be destructive for the quasiparticle approximation when higher-order diagrams are included. The second result of our study is that we were able to reproduce the results obtained with the questaal code using our flapwmbpt code when the same variant of self-consistency (quasiparticle) and the same level of vertex corrections (for polarizability only, static approximation for screened interaction, and Tamm-Dancoff approximation for the Bethe-Salpeter equation) are used.

Single-boson exchange representation of the functional renormalization group for strongly interacting many-electron systems

We present a reformulation of the functional renormalization group (fRG) for many-electron systems, which relies on the recently introduced single boson exchange (SBE) representation of the parquet equations [Phys. Rev. B 100, 155149 (2019)]. The latter exploits a diagrammatic decomposition, which classifies the contributions to the full scattering amplitude in terms of their reducibility with respect to cutting one interaction line, naturally distinguishing the processes mediated by the exchange of a single boson in the different channels. We apply this idea to the fRG by splitting the one-loop fRG flow equations for the vertex function into SBE contributions and a residual four-point fermionic vertex. Similarly as in the case of parquet solvers, recasting the fRG algorithm in the SBE representation offers both computational and interpretative advantages: the SBE decomposition not only significantly reduces the numerical effort of treating the high-frequency asymptotics of the flowing vertices, but it also allows for a clear physical identification of the collective degrees of freedom at play. We illustrate the advantages of an SBE formulation of fRG-based schemes, by computing through the merger of dynamical mean-field theory and fRG the susceptibilities and the Yukawa couplings of the two-dimensional Hubbard model from weak to strong coupling, for which we also present an intuitive physical explanation of the results. The SBE formulation of the one-loop flow equations paves a promising route for future multiboson and multiloop extensions of fRG-based algorithms.

Semiclassical response of disordered conductors: Extrinsic carrier velocity and spin and field-corrected collision integral

The semiclassical equations of motion are widely used to describe carrier transport in conducting materials. Nevertheless, the substantial challenge of incorporating disorder systematically into the semiclassical model persists, leading to quantitative inaccuracies and occasionally erroneous predictions for the expectation values of physical observables. In the present work we provide a general prescription for reformulating the semiclassical equations of motion for carriers in disordered conductors by taking the quantum mechanical density matrix as the starting point. We focus on external electric fields, without magnetic fields, and spin-independent disorder. The density matrix approach allows averaging over impurity configurations, and the trace of the velocity operator with the disorder-averaged density matrix can be reinterpreted as the semiclassical velocity weighted by the Boltzmann distribution function. Through this rationale the well-known intrinsic group and anomalous velocities are trivially recovered, while we demonstrate the existence of an extrinsic interband velocity, namely a disorder correction to the semiclassical velocity of Bloch electrons, mediated by the interband matrix elements of the Berry connection. A similar correction is present in the non-equilibrium expectation value of the spin operator, contributing to spin-orbit torques. To obtain agreement with diagrammatic approaches the scattering term in the Boltzmann equation is corrected to first order in the electric field, and the Boltzmann equation is solved up to sub-leading order in the disorder potential. Our prescription ensures all vertex corrections present in diagrammatic treatments are taken into account, and to illustrate this we discuss model cases in topological insulators, including the anomalous Hall effect as well as spin-orbit torques.

Large enhancement of Edelstein effect in Weyl semimetals from Fermi-arc surface states

One remarkable feature of Weyl semimetals is the manifestation of their topological nature in the form of the Fermi-arc surface states. In a recent calculation by \cite{Johansson2018}, the current-induced spin polarization or Edelstein effect has been predicted, within the semiclassical Boltzmann theory, to be strongly amplified in a Weyl semimetal TaAs due to the existence of the Fermi arcs. Motivated by this result, we calculate the Edelstein response of an effective model for an inversion-symmetry-breaking Weyl semimetal in the presence of an interface using linear response theory. The scatterings from scalar impurities are included and the vertex corrections are computed within the self-consistent ladder approximation. At chemical potentials close to the Weyl points, we find the surface states have a much stronger response near the interface than the bulk states by about one to two orders of magnitude. At higher chemical potentials, the surface states' response near the interface decreases to be about the same order of magnitude as the bulk states' response. We attribute this phenomenon to the decoupling between the Fermi arc states and bulk states at energies close to the Weyl points. The surface states which are effectively dispersing like a one-dimensional chiral fermion become nearly nondissipative. This leads to a large surface vertex correction and, hence, a strong enhancement of the surface states' Edelstein response.

Electromagnetic finite-size effects beyond the point-like approximation

We present a model-independent and relativistic approach to analytically derive electromagnetic finite-size effects beyond the point-like approximation. The key element is the use of electromagnetic Ward identities to constrain vertex functions, and structure-dependence appears via physical form-factors and their derivatives. We apply our general method to study the leading finitesize structure-dependence in the pseudoscalar mass (at order 1/L3) as well as in the leptonic decay amplitudes of pions and kaons (at order 1/L2). Knowledge of the latter is essential for Standard Model precision tests in the flavour physics sector from lattice simulations.

Two-loop rational terms for spontaneously broken theories

Rational counterterms are a key ingredient for the automation of loop calculations through numerical methods. Building on the recently established properties of rational terms of UV origin at two loops, in this paper we present a systematic method for the determination of rational counterterms within spontaneously broken theories. In particular we introduce a generalised vev-expansion approach that makes it possible to obtain the rational counterterms of UV origin for a spontaneously broken theory by means of calculations in the unbroken phase. The drastic simplifications that result from the underlying symmetry open the door to the efficient determination of rational counterterms for the full Standard Model at two loops. The renormalisation-scheme dependence is analysed in detail, and we show that rational counterterms need to be determined only once and for all in a generic renormalisation scheme for the symmetric phase and, a posteriori, they can be easily adapted to a wide range of physical renormalisation schemes for the spontaneously broken phase. As a first application we determine the full set of mathcal{O}left({alpha}_{mathrm{S}}^2right) rational counterterms of UV origin for the full Standard Model, i.e. for all superficially UV-divergent two-loop vertex functions involving combinations of gluons, quarks, electroweak vector bosons and scalar bosons.

Equivalence of light-front and covariant quantum electrodynamics at one-loop level and the form of the gauge boson propagator

We consider the three fundamental one loop Feynman diagrams of QED viz. vertex correction, fermion self-energy and vacuum polarization in the light-front gauge and discuss the equivalence of their standard covariant expressions with the light-front expressions obtained using light-cone time-ordered Hamiltonian perturbation theory. Although this issue has been considered by us and others previously, our emphasis in this article is on addressing the ambiguity regarding the correct form of the gauge boson propagator to be used in the light-front gauge. We generalize our earlier results and show, using an alternative method called the Asymptotic Method, how integrating over the light-front energy consistently in the covariant expression of each of the three one loop corrections leads to the propagating as well as the instantaneous diagrams of the light-front theory. In doing so, we re-establish the necessity of using the correct form of gauge boson propagator.