Vertex Corrections Research Articles

Development and investigation of methods of graphic-functional modeling of distributed systems

During the research, the method of geometric modeling of distributed systems and related technological objects was developed. The method is based on the use of functional graphs. In the context of the research, the main differences between such graphs are: modeling of technological objects of distributed systems only by vertices without the use of edges to reproduce the mentioned objects; using the edges exclusively for the reproduction of the connections between the objects. Weighing of the vertices of the specified graphs is performed using the assigned functions or functionals with a complete absence of weights of the edges.In contrast to the closest analogues, the basis of the analytical interpretation of the formed graphic models in the proposed method is formed not by the incidence matrices, but by the parametric-topological adjacency matrices. In such conditions, the principles of assigning weight coefficients of graph elements change significantly: instead of the positional distribution of the weight sets elements between the matrix cells, the assignment of the specified elements as function arguments as a part of functional vertices is used. In the mentioned approach, a diagonal way of prescribing functions or functionals of vertices in the adjacency matrix is used. The assignment of the connections between the graph elements in the analytical interpretation is performed according to the introduced positional principle with the use of positive or negative logic. With this approach, the possibility of analytical formation of multiple connections between the vertices with an arbitrary number and direction, which was not used in the matrix of adjacency previously, is reached. Moreover, assigning functional dependencies to the graph elements allows the reproduction of not only static, but also the dynamic characteristics of the modeled objects in the geometric model.The practical value of the proposed method is increasing the universality and simplifying the automated configuration of software management systems. Achieving this result is possible by reducing the amount of input data and the possibility of introducing additional functions of control objects without the modification of the output code. In addition, the improvement of the formalized drawing up of technical tasks in the development of technical documentation and hardware distribution systems is provided. Furthermore, the integration of the method into existing CAE and CAD systems is possible, which provides opportunities for building up and creating fundamentally new similar systems.Further development of the proposed method consists in solving the problems related to the optimization of the distribution of arguments of vertex functions by cells of parametric-topological matrices.

Anomalously high value of Coulomb pseudopotential for the H5S2 superconductor

The H5S2 and H2S compounds are the two candidates for the low-temperature phase of compressed sulfur-hydrogen system. We have shown that the value of Coulomb pseudopotential (μ*) for H5S2 ([TC]exp = 36 K and p = 112 GPa) is anomalously high. The numerical results give the limitation from below to μ* that is equal to 0.402 (μ* = 0.589), if we consider the first order vertex corrections to the electron-phonon interaction). Presented data mean that the properties of superconducting phase in the H5S2 compound can be understood within the classical framework of Eliashberg formalism only at the phenomenological level (μ* is the parameter of matching the theory to the experimental data). On the other hand, in the case of H2S it is not necessary to take high value of Coulomb pseudopotential to reproduce the experimental critical temperature relatively well (μ* = 0.15). In our opinion, H2S is mainly responsible for the observed superconductivity state in the sulfur-hydrogen system at low temperature.

Counting Feynman diagrams via many-body relations.

We present an iterative algorithm to count Feynman diagrams via many-body relations. The algorithm allows us to count the number of diagrams of the exact solution for the general fermionic many-body problem at each order in the interaction. Further, we apply it to different parquet-type approximations and consider spin-resolved diagrams in the Hubbard model. Low-order results and asymptotics are explicitly discussed for various vertex functions and different two-particle channels. The algorithm can easily be implemented and generalized to many-body relations of different forms and levels of approximation.

Functional approach to the electronic and bosonic dynamics of many-body systems perturbed with an arbitrary strong electron-boson interaction

We present a formal derivation of the many-body perturbation theory for a system of electrons and bosons subject to a nonlinear electron-boson coupling. The interaction is treated at an arbitrary high order of bosons scattered. The considered Hamiltonian includes the well-known linear coupling as a special limit. This is the case, for example, of the Holstein and Fr\"{o}hlich Hamiltonians. Indeed, whereas linear coupling have been extensively studied, the scattering processes of electrons with multiple bosonic quasiparticles are largely unexplored. We focus here on a self-consistent theory in terms of dressed propagators and generalize the Hedin's equations using the Schwinger technique of functional derivatives. The method leads to an exact derivation of the electronic and bosonic self-energies, expressed in terms of a new family of vertex functions, high order correlators and bosonic and electronic mean-field potentials. In the electronic case we prove that the mean-field potential is the $n$th-order extension of the well-known Debye-Waller potential. We also introduce a bosonic mean-field potential entirely dictated by nonlinear electron-boson effects. The present scheme, treating electrons and bosons on an equal footing, demonstrates the full symmetry of the problem. The vertex functions are shown to have purely electronic and bosonic character as well as a mixed electron-boson one. These four vertex functions are shown to satisfy a generalized Bethe-Salpeter equation. Multi bosons response functions are also studied and explicit expressions for the two and the three bosons case are given.

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.

On the Relation between Equation-of-Motion Coupled-Cluster Theory and the GW Approximation.

We discuss the analytic and diagrammatic structure of ionization potential (IP) and electron affinity (EA) equation-of-motion coupled-cluster (EOM-CC) theory, in order to put it on equal footing with the prevalent GW approximation. The comparison is most straightforward for the time-ordered one-particle Green's function, and we show that the Green's function calculated by EOM-CC with single and double excitations (EOM-CCSD) includes fewer ring diagrams at higher order than does the GW approximation, due to the former's unbalanced treatment of time-ordering. However, the EOM-CCSD Green's function contains a large number of vertex corrections, including ladder diagrams, mixed ring-ladder diagrams, and exchange diagrams. By including triple excitations, the EOM-CCSDT Green's function includes all diagrams contained in the GW approximation, along with many high-order vertex corrections. In the same language, we discuss a number of common approximations to the EOM-CCSD equations, many of which can be classified as elimination of diagrams. Finally, we present numerical results by calculating the principal charged excitations energies of the molecules contained in the so-called GW100 test set [ J. Chem. Theory Comput. 2015 , 11 , 5665 - 5687 ]. We argue that (in molecules) exchange is as important as screening, advocating for a Hartree-Fock reference and second-order exchange in the self-energy.

Beyond the random phase approximation with a local exchange vertex

With the aim of constructing an electronic structure approach that systematically goes beyond the GW and random phase approximation (RPA) we introduce a vertex correction based on the exact-exchange (EXX) potential of time-dependent density functional theory. The EXX vertex function is constrained to be local but is expected to capture similar physics as the Hartree-Fock vertex. With the EXX vertex we then consistently unify different beyond-RPA approaches such as the various re-summations of RPA with exchange (RPAx) and the second order screened exchange (SOSEX) approximation. The theoretical analysis is supported by numerical studies on the hydrogen dimer and the electron gas, and we discuss the importance of including the vertex correction in both the screened interaction and the self energy. Finally, we give details on our implementation within the plane-wave pseudo potential framework and demonstrate the excellent performance of the different RPAx methods in describing the energetics of hydrogen and van der Waals bonds.

Equivalence of One Loop Diagrams in Covariant and Light Front QED Revisited

We revisit the proof of equivalence of one loop expressions for fermion self-energy and vertex correction in light-front QED and Covariant QED at the Feynman diagram level and generalize, to all components, the proof of equivalence for the one loop vertex correction diagram which was presented earlier by us only for the $+$ component of $\Lambda^\mu$. We demonstrate, in the general case also, that the equivalence cannot be established without the third term in the three-term photon propagator in light cone gauge.

Local vertex corrections from exchange-correlation kernels with a discontinuity

The fundamental gap of an interacting many-electron system is given by the sum of the single-particle Kohn-Sham gap and the derivative discontinuity. The latter can be generated by advanced approximations to the exchange-correlation (XC) energy and is the key quantity to capture strong correlation with density functional theory (DFT). In this work we derive an expression for the derivative discontinuity in terms of the XC kernel of time-dependent density functional theory and demonstrate the crucial role of a discontinuity in the XC kernel itself. By relating approximate XC kernels to approximate local vertex corrections we then generate beyond-GW self-energies that include a discontinuity in the local vertex function. The quantitative importance of this result is illustrated with a numerical study of the local exchange vertex on model systems.

Neutrino propagation in an electron background with an inhomogeneous magnetic field

We study the electromagnetic coupling of a neutrino that propagates in a two-stream electron background medium. Specifically, we calculate the electromagnetic vertex function for a medium that consists of a normal electron background plus another electron stream background that is moving with a velocity four-vector v^mu relative to the normal background. The results can be used as the basis for studying the neutrino electromagnetic properties and various processes in such a medium. As an application, we calculate the neutrino dispersion relation in the presence of an external magnetic field (mathbf {B}), focused in the case in which B is inhomogeneous, keeping only the terms of the lowest order in 1/m^2_W and linear in the B and its gradient. We show that the dispersion relation contains additional anisotropic terms involving the derivatives of mathbf {B}, such as the gradient of {hat{k}}cdot (mathbf {v}times mathbf {B}), which involve the stream background velocity, and a term of the form {hat{k}}cdot (nabla times mathbf {B}) that can be present in the absence of the stream background, in addition to a term of the form {hat{k}}cdot mathbf {v} and the well known term {hat{k}}cdot mathbf {B} that arises in the constant mathbf {B} case. The derivative-dependent terms are even under a CP transformation. As a result, in contrast to the latter two just mentioned, they depend on the sum of the particle and antiparticle densities and therefore can be non-zero in a CP-symmetric medium in which the particle and antiparticle densities are equal.

Divergences of the irreducible vertex functions in correlated metallic systems: Insights from the Anderson impurity model

In this work, we analyze in detail the occurrence of divergences in the irreducible vertex functions for one of the fundamental models of many-body physics: the Anderson impurity model (AIM). These divergences, a surprising hallmark of the breakdown of many-electron perturbation theory, have been recently observed in several contexts, including the dynamical mean-field solution of the Hubbard model. The numerical calculations for the AIM presented in this work, as well as their comparison with the corresponding results for the Hubbard model, allow us to clarify several open questions about the properties of vertex divergences in a particularly interesting context, the correlated metallic regime at low temperatures. Specifically, our analysis (i) rules out explicitly the transition to a Mott-insulating phase, but not the more general suppression of charge fluctuations (proposed in [O. Gunnarsson et al., Phys. Rev. B 93, 245102 (2016)]), as a necessary condition for the occurrence of vertex divergences, (ii) clarifies their relation with the underlying Kondo physics, and, eventually, (iii) individuates which divergences might also appear on the real-frequency axis in the limit of zero temperature, through the discovered scaling properties of the singular eigenvectors.

GW(Γ) method without the Bethe-Salpeter equation for photoabsorption energies of spin-polarized systems

The one-shot $GW$ method, beginning with the local density approximation (LDA), enables one to calculate photoemission and inverse photoemission spectra. In order to calculate photoabsorption spectra, one had to additionally solve the Bethe-Salpeter equation (BSE) for the two-particle (electron-hole) Green's function, which doubly induces evaluation errors. It has been recently reported that the $GW$+BSE method significantly underestimates the experimental photoabsorption energies (PAEs) of small molecules. In order to avoid these problems, we propose to apply the $GW(\mathrm{\ensuremath{\Gamma}})$ method not to the neutral ground state but to the cationic state to calculate PAEs without solving the BSE, which allows a rigorous one-to-one correspondence between the photoabsorption peak and the ``extended'' quasiparticle level. We applied the self-consistent linearized $GW\mathrm{\ensuremath{\Gamma}}$ method including the vertex correction $\mathrm{\ensuremath{\Gamma}}$ to our method, and found that this method gives the PAEs of B, ${\mathrm{Na}}_{3}$, and ${\mathrm{Li}}_{3}$ to within $0.1\phantom{\rule{0.28em}{0ex}}\mathrm{eV}$ accuracy.

Dynamically enhanced magnetic incommensurability: Effects of local dynamics on nonlocal spin correlations in a strongly correlated metal

We compute the spin susceptibility of the two-dimensional Hubbard model away from half-filling, and analyze the impact of frequency dependent vertex corrections as obtained from the dynamical mean field theory (DMFT). We find that the local dynamics captured by the DMFT vertex strongly affects non-local spin correlations, and thus the momentum dependence of the spin susceptibility. While the widely used random phase approximation yields commensurate N\'eel-type antiferromagnetism as the dominant instability over a wide doping range, the vertex corrections favor incommensurate ordering wave vectors away from $(\pi,\pi)$. Our results indicate that the connection between the magnetic ordering wave vector and the Fermi surface geometry, familiar for weakly interacting systems, can hold in a strongly correlated metal, too.

Flexible toggles and symmetric invertible asynchronous elementary cellular automata

A sequential dynamical system (SDS) consists of a graph G with vertices v1,…,vn, a state set A, a collection of “vertex functions” {fvi}i=1n, and a permutation π∈Sn that specifies how to compose these functions to yield the SDS map [G,{fvi}i=1n,π]:An→An. In this paper, we study symmetric invertible SDS defined over the cycle graph Cn using the set of states F2. These are, in other words, asynchronous elementary cellular automata (ECA) defined using ECA rules 150 and 105. Each of these SDS defines a group action on the set F2n of n-bit binary vectors. Because the SDS maps are products of involutions, this relates to generalized toggle groups, which Striker recently introduced. In this paper, we further generalize the notion of a generalized toggle group to that of a flexible toggle group; the SDS maps we consider are examples of Coxeter elements of flexible toggle groups.Our main result is the complete classification of the dynamics of symmetric invertible SDS defined over cycle graphs using the set of states F2 and the identity update order π=123⋯n. More precisely, if T denotes the SDS map of such an SDS, then we obtain an explicit formula for |Perr(T)|, the number of periodic points of T of period r, for every positive integer r. It turns out that if we fix r and vary n and T, then |Perr(T)| only takes at most three nonzero values.

Overcomplete compact representation of two-particle Green's functions

Two-particle Green's functions and the vertex functions play a critical role in theoretical frameworks for describing strongly correlated electron systems. However, numerical calculations at the two-particle level often suffer from large computation time and massive memory consumption. We derive a general expansion formula for the two-particle Green's functions in terms of an overcomplete representation based on the recently proposed ``intermediate representation'' basis. The expansion formula is obtained by decomposing the spectral representation of the two-particle Green's function. We demonstrate that the expansion coefficients decay exponentially, while all high-frequency and long-tail structures in the Matsubara-frequency domain are retained. This representation therefore enables efficient treatment of two-particle quantities and opens a route to the application of modern many-body theories to realistic strongly correlated electron systems.

Finite frequency current noise in the Holstein model

We investigate the effects of local vibrational excitations in the nonsymmetrized current noise $S(\omega)$ of a nanojunction. For this purpose, we analyze a simple model - the Holstein model - in which the junction is described by a single electronic level that is coupled to two metallic leads and to a single vibrational mode. Using the Keldysh Green's function technique, we calculate the nonsymmetrized current noise to the leading order in the charge-vibration interaction. For the noise associated to the latter, we identify distinct terms corresponding to the mean-field noise and the vertex correction. The mean-field result can be further divided into an elastic correction to the noise and in an inelastic correction, the second one being related to energy exchange with the vibration. To illustrate the general behavior of the noise induced by the charge-vibration interaction, we consider two limit cases. In the first case, we assume a strong coupling of the dot to the leads with an energy-independent transmission whereas in the second case we assume a weak tunneling coupling between the dot and the leads such that the transport occurs through a sharp resonant level. We find that the noise associated to the vibration-charge interaction shows a complex pattern as a function of the frequency $\omega$ and of the transmission function or of the dot's energy level. Several transitions from enhancement to suppression of the noise occurs in different regions, which are determined, in particular, by the vibrational frequency. Remarkably, in the regime of an energy-independent transmission, the zero order elastic noise vanishes at perfect transmission and at positive frequency whereas the noise related to the charge-vibration interaction remains finite enabling the analysis of the pure vibrational-induced current noise.

Probing ionization potential, electron affinity and self-energy effect on the spectral shape and exciton binding energy of quantum liquid water with self-consistent many-body perturbation theory and the Bethe–Salpeter equation

An accurate theoretical prediction of ionization potential (IP) and electron affinity (EA) is key in understanding complex photochemical processes in aqueous environments. There have been numerous efforts in literature to accurately predict IP and EA of liquid water, however with often conflicting results depending on the level of theory and the underlying water structures. In a recent study based on hybrid-non-self-consistent many-body perturbation theory (MBPT) Gaiduk et al (2018 Nat. Commun. 9 247) predicted an IP of 10.2 eV and EA of 0.2 eV, resulting in an electronic band gap (i.e. electronic gap (IP-EA) as measured by photoelectron spectroscopy) of about 10 eV, redefining the widely cited experimental gap of 8.7 eV in literature. In the present work, we show that GW self-consistency and an implicit vertex correction in MBPT considerably affect recently reported EA values by Gaiduk et al (2018 Nat. Commun. 9 247) by about 1 eV. Furthermore, the choice of pseudo-potential is critical for an accurate determination of the absolute band positions. Consequently, the self-consistent GW approach with an implicit vertex correction based on projector augmented wave (PAW) method on top of quantum water structures predicts an IP of 10.2, an EA of 1.1, a fundamental gap of 9.1 eV and an exciton binding (Eb) energy of 0.9 eV for the first absorption band of liquid water via the Bethe–Salpeter equation (BSE). Only within such a self-consistent approach a simultanously accurate prediction of IP, EA, Eg, Eb is possible.

Absence of Long-Range Order in a Triangular Spin System with Dipolar Interactions.

The antiferromagnetic Heisenberg model on the triangular lattice is perhaps the best known example of frustrated magnets, but it orders at low temperatures. Recent density matrix renormalization group (DMRG) calculations find that the next nearest neighbor interaction J_{2} enhances the frustration, and it leads to a spin liquid for J_{2}/J_{1}∈(0.08,0.15). In addition, a DMRG study of a dipolar Heisenberg model with longer range interactions gives evidence for a spin liquid at a small dipole tilting angle θ∈[0,10°). In both cases, the putative spin liquid region appears to be small. Here, we show that for the triangular lattice dipolar Heisenberg model, a robust quantum paramagnetic phase exists in a surprisingly wide region, θ∈[0,54°), for dipoles tilted along the lattice diagonal direction. We obtain the phase diagram of the model by functional renormalization group (RG), which treats all magnetic instabilities on equal footing. The quantum paramagnetic phase is characterized by a smooth continuous flow of vertex functions and spin susceptibility down to the lowest RG scale, in contrast to the apparent breakdown of RG flow in phases with stripe or spiral order. Our finding points to a promising direction to search for quantum spin liquids in ultracold dipolar molecules.

Conditions where random phase approximation becomes exact in the high-density limit

It is shown that, in $d$-dimensional systems, the vertex corrections beyond the random phase approximation (RPA) or $GW$ approximation scales with the power $d\ensuremath{-}\ensuremath{\beta}\ensuremath{-}\ensuremath{\alpha}$ of the Fermi momentum if the relation between Fermi energy and Fermi momentum is ${\ensuremath{\epsilon}}_{\mathrm{f}}\ensuremath{\sim}{p}_{\mathrm{f}}^{\ensuremath{\beta}}$ and the interacting potential possesses a momentum power law of $\ensuremath{\sim}{p}^{\ensuremath{-}\ensuremath{\alpha}}$. The condition $d\ensuremath{-}\ensuremath{\beta}\ensuremath{-}\ensuremath{\alpha}l0$ specifies systems where RPA is exact in the high-density limit. The one-dimensional structure factor is found to be the interaction-free one in the high-density limit for contact interaction. A cancellation of RPA and vertex corrections render this result valid up to second order in contact interaction. For finite-range potentials of cylindrical wires a large-scale cancellation appears and is found to be independent of the width parameter of the wire. The proposed high-density expansion agrees with the quantum Monte Carlo simulations.

Conditions for l=1 Pomeranchuk instability in a Fermi liquid

We perform a microscropic analysis of how the constraints imposed by conservation laws affect $q=0$ Pomeranchuk instabilities in a Fermi liquid. The conventional view is that these instabilities are determined by the static interaction between low-energy quasiparticles near the Fermi surface, in the limit of vanishing momentum transfer $q$. The condition for a Pomeranchuk instability is set by $F^{c(s)}_l =-1$, where $F^{c(s)}_l$ (a Landau parameter) is a properly normalized partial component of the anti-symmetrized static interaction $F(k,k+q; p,p-q)$ in a charge (c) or spin (s) sub-channel with angular momentum $l$. However, it is known that conservation laws for total spin and charge prevent Pomeranchuk instabilities for $l=1$ spin- and charge- current order parameters. Our study aims to understand whether this holds only for these special forms of $l=1$ order parameters, or is a more generic result. To this end we perform a diagrammatic analysis of spin and charge susceptibilities for charge and spin density order parameters, as well as perturbative calculations to second order in the Hubbard $U$. We argue that for $l=1$ spin-current and charge-current order parameters, certain vertex functions, which are determined by high-energy fermions, vanish at $F^{c(s)}_{l=1}=-1$, preventing a Pomeranchuk instability from taking place. For an order parameter with a generic $l=1$ form-factor, the vertex function is not expressed in terms of $F^{c(s)}_{l=1}$, and a Pomeranchuk instability does occur when $F^{c(s)}_1=-1$. We argue that for other values of $l$, a Pomeranchuk instability occurs at $F^{c(s)}_{l} =-1$ for an order parameter with any form-factor