Self-energy Research Articles

Machine Learning Many-Body Green's Functions for Molecular Excitation Spectra.

We present a machine learning (ML) framework for predicting Green's functions of molecular systems, from which photoemission spectra and quasiparticle energies at quantum many-body level can be obtained. Kernel ridge regression is adopted to predict self-energy matrix elements on compact imaginary frequency grids from static and dynamical mean-field electronic features, which gives direct access to real-frequency many-body Green's functions through analytic continuation and Dyson's equation. Feature and self-energy matrices are represented in a symmetry-adapted intrinsic atomic orbital plus projected atomic orbital basis to enforce rotational invariance. We demonstrate good transferability and high data efficiency of the proposed ML method across molecular sizes and chemical species by showing accurate predictions of density of states (DOS) and quasiparticle energies at the level of many-body perturbation theory (GW) or full configuration interaction. For the ML model trained on 48 out of 1995 molecules randomly sampled from the QM7 and QM9 data sets, we report the mean absolute errors of ML-predicted highest occupied and lowest unoccupied molecular orbital energies to be 0.13 and 0.10 eV, respectively, compared to GW@PBE0. We further showcase the capability of this method by applying the same ML model to predict DOS for significantly larger organic molecules with up to 44 heavy atoms.

Clothed particle representation in quantum field theory: Fermion mass renormalization due to vector boson exchange

Clothed particle representation in quantum field theory: Fermion mass renormalization due to vector boson exchange

Completing Dirac’s work. The Dirac electron is a 2D hologram

The QED-physical (QED-P) theory [J. H. Wilson, Phys. Essays 35, 5 (2022)] is actually the theory Dirac sought in 1962 in his attempt to predict the muon as an “extensible model of the electron.” Recently, Lerche attempted to produce a classical solution to Dirac’s equation for the radial motion of an extensible, basically classical, model of the electron. Both Dirac and Lerche proceeded in the wrong, classical direction in this effort. The QED-P center of charge (CoC) position operator is derived directly from the Dirac equation (DE) CoC velocity operator, cα with no ad hoc assumptions. QED-P was integrated with QED into a single theory, and that integration is proved by the highly accurate estimates of QED that are dependent on the DE velocity operator cα. Both QED and QED-P are based directly on the same Dirac Equation (DE) four current c(α,I) that QED couples with an external electromagnetic field. QED uses covariant perturbation theory to produce highly accurate results, except for the electron selfenergy, which is infinite. The DE velocity operator, cα, is the spatial part of the free electron four current, and has highly unusual properties compared to classical velocity vectors. QED could not produce highly accurate answers without the 4 × 4 complex matrix cα as the electron CoC “velocity” operator. QED-P simply integrates the same Dirac equation four current used so successfully in QED, and produces the discrete internal spatial and time coordinate operators (ISaTCOs) to give the electron field’s internal structure a very specific, but highly, nonclassical geometric description, with no ad hoc assumptions. QED and QED-P are complementary theories, and both are proven to be true by the accurate results of QED. The physical interpretation of QED-P is discussed in this paper as a two-dimensional, rapidly vibrating “point” charge, that is always located on a 2D CoC sphere in the electron rest frame, oscillating rapidly through eight eigenvalues with an ISaTCO period of ∼10−22 s. The fact that the CoC’s eight ISaTCO spatial eigenvectors are always are located on a 2D shell encompassing the electron’s 3D “space” inside the CoC’s 2D shell is a direct consequence of the DE, and nothing else. In this paper, it is shown that the “discrete” ISaTCOs produce a one dimensional, discrete quantum harmonic oscillator with its ISaTCOs always located on a 2D CoC shell. The CoC shell is an “2D hologram” emerging from a 3D volume inside the 2D CoC shell with a vibrational electronic clock” producing an internal phase that is propagated throughout space/time. The QED-P point electron charge rotates 720° to complete one internal electron discrete period. The electron’s spin and magnetic moment [J. H. Wilson, Phys. Essays 29, 402 (2016); ibid. 31(1), 59‐67 (2018); ibid. 34, 17 (2021)] are generated by the CoC ISaTCO in QED-P, and there is no need for “intrinsic” properties. The QED-P electron properties described above d are far different than the standard model’s very small point particle with intrinsic properties of spin and magnetic moment.

On the Anomalous Dimension in QCD

The anomalous dimension γm=1 in the infrared region near the conformal edge in the broken phase of the large Nf QCD has been shown by the ladder Schwinger–Dyson equation and also by the lattice simulation for Nf=8 and for Nc=3. Recently, Zwicky made another independent argument (without referring to explicit dynamics) for the same result, γm=1, by comparing the pion matrix element of the trace of the energy-momentum tensor π(p2)|(1+γm)·∑i=1Nfmfψ¯iψi|π(p1)=π(p2)|θμμ|π(p1)=2Mπ2 (up to trace anomaly) with the estimate of π(p2)|2·∑i=1Nfmfψ¯iψi|π(p1)=2Mπ2 through the Feynman–Hellmann theorem combined with an assumption Mπ2∼mf characteristic of the broken phase. We show that this is not justified by the explicit evaluation of each matrix element based on the dilaton chiral perturbation theory (dChPT): π(p2)|2·∑i=1Nfmfψ¯iψi|π(p1)=2Mπ2+[(1−γm)Mπ2·2/(1+γm)]=2Mπ2·2/(1+γm)≠2Mπ2 in contradiction with his estimate, which is compared with π(p2)|(1+γm)·∑i=1Nfmfψ¯iψi|π(p1)=(1+γm)Mπ2+[(1−γm)Mπ2]=2Mπ2 (both up to trace anomaly), where the terms in [] are from the σ (pseudo-dilaton) pole contribution. Thus, there is no constraint on γm when the σ pole contribution is treated consistently for both. We further show that the Feynman–Hellmann theorem is applied to the inside of the conformal window where dChPT is invalid and the σ pole contribution is absent, and with Mπ2∼mf2/(1+γm) instead of Mπ2∼mf, we have the same result as ours in the broken phase. A further comment related to dChPT is made on the decay width of f0(500) to ππ for Nf=2. It is shown to be consistent with the reality, when f0(500) is regarded as a pseudo-NG boson with the non-perturbative trace anomaly dominance.

Compton scattering in dielectric medium

We determine the Compton effect from the Volkov solution of the Dirac equation for a process in medium with the index of refraction n. Volkov solution involves the mass shift, or, the mass renormalization of an electron. We determine the modified Compton formula for the considered physical situation. The index of refraction causes that the wave lengths of the scattered photons are shorter for some angles than the wave lengths of the original photons. This is anomalous Compton effect.

Schwinger–Dyson Equation for Quarks in a QCD Inspired Model

Schwinger–Dyson Equation for Quarks in a QCD Inspired Model

Electron-phonon coupling from GW perturbation theory: Practical workflow combining BerkeleyGW, ABINIT, and EPW

We present a workflow of practical calculations of electron-phonon (e-ph) coupling with many-electron correlation effects included using the GW perturbation theory (GWPT). This workflow combines BerkeleyGW, ABINIT, and EPW software packages to enable accurate e-ph calculations at the GW self-energy level, going beyond standard calculations based on density functional theory (DFT) and density-functional perturbation theory (DFPT). This workflow begins with DFT and DFPT calculations (ABINIT) as starting point, followed by GW and GWPT calculations (BerkeleyGW) for the quasiparticle band structures and e-ph matrix elements on coarse electron k- and phonon q-grids, which are then interpolated to finer grids through Wannier interpolation (EPW) for computations of various e-ph coupling determined physical quantities such as the electron self-energies or solutions of anisotropic Eliashberg equations, among others. A gauge-recovering symmetry unfolding technique is developed to reduce the computational cost of GWPT (as well as DFPT) while fulfilling the gauge consistency requirement for Wannier interpolation.

Strong fluctuation theory of Tatarskii: quantum field theoretic approach – a critical assessment

In this paper, we carried out a critical investigation of the strong fluctuation theory using a quantum field theoretic approach [Tatarskii VI. The effects of the turbulent atmosphere on wave propagation. Jerusalem: IPST; 1971]. We employed the Dyson and Bethe-Salpeter equations to derive the radiative transfer equation (RTE) for the Green's function of the radiant intensity. We proceeded to obtain estimates of the domain of validity of the RTE. The results thus obtained satisfy both the optical theorem for waves in random media and energy conservation. We also derived the RTE for our problem by employing a recently developed scale-separated asymptotic theory [Bal G., Komorowski T, Ryzhik L. Kinetic limits of waves in random media. Kinet Relat Models. 2010;3:529–644]. Although the RTE obtained by the two approaches are identical, we explain that the applicability regimes, and hence validity conditions are different.

Functional equation for pure Yang–Mills Wilson-loop two-point correlator from Schwinger–Dyson equations

The physics of pure Yang–Mills can be described gauge-invariantly using n-point functions of Wilson loops. These n-point functions must obey functional Schwinger–Dyson equations, which highly simplify in the limit of large number of colors N. These equations have remained unsolved for several decades (except for trivial spacetimes).In this paper, I study the Schwinger–Dyson equations for the Wilson-loop two-point function in large-N U(N) pure Yang–Mills theory with a lattice regulator. I recast the equation in a form that makes it suitable for extraction of the Wilson-loop propagator (and thus the large-N glueball spectrum) using path integral techniques.

Many-body GW calculations with very large scale polarizable environments made affordable: A fully abinitio QM/QM approach.

We present a many-body GW formalism for quantum subsystems embedded in discrete polarizable environments containing up to several hundred thousand atoms described at a fully abinitio random phase approximation level. Our approach is based on a fragment approximation in the construction of the Green's function and independent-electron susceptibilities. Further, the environing fragments susceptibility matrices are reduced to a minimal but accurate representation preserving low order polarizability tensors through a constrained minimization scheme. This approach dramatically reduces the cost associated with inverting the Dyson equation for the screened Coulomb potential W, while preserving the description of short to long-range screening effects. The efficiency and accuracy of the present scheme is exemplified in the paradigmatic cases of fullerene bulk, surface, subsurface, and slabs with varying number of layers.

Theoretical study of electron compressibility in semiconductor quantum wires

The compressibility was calculated of electrons as a function of external magnetic, temperature and carrier density in the parabolic lateral barrier quantum wires (QWRs). The compressibility found by using Seitz's theorem in which electron self-energy is calculated within the random-phase approximation. All results for GaAs QWRs show that a large difference between the compressibility in low- and high-temperature region at external magnetic fields. These results can help clarify how the Coulomb interaction and exchange, correlation energy affect the compressibility of the semiconductor across a wide range of density, temperature, and external magnetic.

Schwinger poles of the three-gluon vertex: symmetry and dynamics

The implementation of the Schwinger mechanism endows gluons with a nonperturbative mass through the formation of special massless poles in the fundamental QCD vertices; due to their longitudinal character, these poles do not cause divergences in on-shell amplitudes, but induce detectable effects in the Green’s functions of the theory. Particularly important in this theoretical setup is the three-gluon vertex, whose pole content extends beyond the minimal structure required for the generation of a gluon mass. In the present work we analyze these additional pole patterns by means of two distinct, but ultimately equivalent, methods: the Slavnov–Taylor identity satisfied by the three-gluon vertex, and the nonlinear Schwinger–Dyson equation that governs the dynamical evolution of this vertex. Our analysis reveals that the Slavnov–Taylor identity imposes strict model-independent constraints on the associated residues, preventing them from vanishing. Approximate versions of these constraints are subsequently recovered from the Schwinger–Dyson equation, once the elements responsible for the activation of the Schwinger mechanism have been duly incorporated. The excellent coincidence between the two approaches exposes a profound connection between symmetry and dynamics, and serves as a nontrivial self-consistency test of this particular mass generating scenario.

Phase transitions in three-dimensional Dirac semi-metals using Schwinger–Dyson equations

Phase transitions in three-dimensional Dirac semi-metals using Schwinger–Dyson equations

Higher-order scattering of high-frequency ultrasound in strongly heterogeneous microstructures

Ultrasonic waves get scattered from grain boundaries while propagating through polycrystals, thereby experiencing attenuation and a change in wavespeed. These grain boundaries act as the sole scattering sites for untextured aggregates of randomly oriented crystallites. In addition, porosity, texture, defects, precipitates, or inclusions also serve as scatterers when present in a microstructure. Existing analytical models rely on the first-order smoothing approximation (FOSA)-based homogenization to the mass operator series in the governing Dyson equation. They can accurately predict ultrasonic dispersion in some weakly inhomogeneous metals like Aluminum. However, while comparing against more realistic Finite Element (FE) predictions, serious discrepancies, as high as 70%, are found in the attenuation estimates for metals, like Lithium, that possess strong elastic fluctuations in their microstructures. Thus, the current model analytically investigates higher-order scattering effects corresponding to the third-order smoothing approximation (TOSA) for the first time. Current results reveal no higher-order scattering effects for Aluminum at any frequencies. However, the longitudinal attenuation estimates for Lithium are found to improve by 1.27% and 0.37% in the “stochastic” (high-frequency) and “Rayleigh” (low-frequency) regimes, respectively. The presentation will highlight the effects of including TOSA on analytical estimates of high-frequencyultrasonic dispersion in different strongly heterogeneous microstructures.

A Comment on the Classical Electron Self-Energy

A Comment on the Classical Electron Self-Energy

Gravitational form factors of a kink in 1 + 1 dimensional ϕ4 model

We calculate the one-loop correction to the distribution of energy-momentum tensor around a kink in 1 + 1 dimensional ϕ4 model. We employ the collective coordinate method to eliminate the zero mode that gives rise to infrared divergence. The ultraviolet divergences are removed by vacuum subtraction and mass renormalization. We obtain an analytic result that is finite and satisfies the momentum conservation. The total energy of the kink obtained from the spatial integral of energy density reproduces the known result. Our result obtained on a finite space has a spatially-uniform term that is inversely proportional to the spatial length.

A fast time domain solver for the equilibrium Dyson equation

We consider the numerical solution of the real-time equilibrium Dyson equation, which is used in calculations of the dynamical properties of quantum many-body systems. We show that this equation can be written as a system of coupled, nonlinear, convolutional Volterra integro-differential equations, for which the kernel depends self-consistently on the solution. As is typical in the numerical solution of Volterra-type equations, the computational bottleneck is the quadratic-scaling cost of history integration. However, the structure of the nonlinear Volterra integral operator precludes the use of standard fast algorithms. We propose a quasilinear-scaling FFT-based algorithm which respects the structure of the nonlinear integral operator. The resulting method can reach large propagation times and is thus well-suited to explore quantum many-body phenomena at low energy scales. We demonstrate the solver with two standard model systems: the Bethe graph and the Sachdev-Ye-Kitaev model.

On-shell renormalization of fermion masses, fields, and mixing matrices at 1-loop

On-shell renormalization of fermion masses, fields, and mixing matrices at 1-loop

Leading power accuracy in lattice calculations of parton distributions

In lattice-QCD calculations of parton distribution functions (PDFs) via large-momentum effective theory, the leading power (twist-three) correction appears as O(ΛQCD/Pz) due to the linear-divergent self-energy of Wilson line in quasi-PDF operators. For lattice data with hadron momentum Pz of a few GeV, this correction is dominant in matching, as large as 30% or more. We show how to eliminate this uncertainty through choosing the mass renormalization parameter consistently with the resummation scheme of the infrared-renormalon series in perturbative matching coefficients. An example on the lattice pion PDF data at Pz=1.9 GeV shows an improvement of matching accuracy by a factor of more than 3∼5 in the expansion region x=0.2∼0.5.

Planar degeneracy of the three-gluon vertex

We present a detailed exploration of certain outstanding features of the transversely-projected three-gluon vertex, using the corresponding Schwinger–Dyson equation in conjunction with key results obtained from quenched lattice simulations. The main goal of this study is the scrutiny of the approximate property denominated “planar degeneracy”, unveiled when the Bose symmetry of the vertex is properly exploited. The planar degeneracy leads to a particularly simple parametrization of the vertex, reducing its kinematic dependence to essentially a single variable. Our analysis, carried out in the absence of dynamical quarks, reveals that the planar degeneracy is particularly accurate for the description of the form factor associated with the classical tensor, for a wide array of arbitrary kinematic configurations. Instead, the remaining three form factors display considerable violations of this property. In addition, and in close connection with the previous point, we demonstrate the numerical dominance of the classical form factor over all others, except in the vicinity of the soft-gluon kinematics. The final upshot of these considerations is the emergence of a very compact description for the three-gluon vertex in general kinematics, which may simplify significantly nonperturbative applications involving this vertex.