Target-oriented inversion using the Patched Green′s function method

We have designed a target-oriented methodology to perform Full Waveform Inversion using a frequency-domain wave propagator based on the so-called Patched Green’s Function (PGF) technique. Originally developed in condensed matter physics to describe electronic waves in materials, the PGF technique is easily adaptable to the case of wave propagation in a spatially variable media in general. By dividing the entire computational domain into two sections, namely the target area and the outside target area, we calculate the Green Functions related to each section separately. The calculations related to the section outside the target are performed only once at the beginning of inversion, whereas the calculations in the target area are performed repeatedly for each iteration of the inversion process. With the Green Functions of the separate areas, we calculate the Green Functions of the two systems patched together through the application of a Recursive Dyson equation. By performing 2D and time-lapse experiments on the Marmousi model and a Brazilian Pre-salt velocity model, we demonstrate that the target-oriented PGF reduces the computational time of the inversion without compromising accuracy. In fact, when compared with conventional FWI results, the PGF-based calculations are identical but done in a fraction of the time.

Low rank compression in the numerical solution of the nonequilibrium Dyson equation

We propose a method to improve the computational and memory efficiency of numerical solvers for the nonequilibrium Dyson equation in the Keldysh formalism. It is based on the empirical observation that the nonequilibrium Green's functions and self energies arising in many problems of physical interest, discretized as matrices, have low rank off-diagonal blocks, and can therefore be compressed using a hierarchical low rank data structure. We describe an efficient algorithm to build this compressed representation on the fly during the course of time stepping, and use the representation to reduce the cost of computing history integrals, which is the main computational bottleneck. For systems with the hierarchical low rank property, our method reduces the computational complexity of solving the nonequilibrium Dyson equation from cubic to near quadratic, and the memory complexity from quadratic to near linear. We demonstrate the full solver for the Falicov-Kimball model exposed to a rapid ramp and Floquet driving of system parameters, and are able to increase feasible propagation times substantially. We present examples with 262 144 time steps, which would require approximately five months of computing time and 2.2 TB of memory using the direct time stepping method, but can be completed in just over a day on a laptop with less than 4 GB of memory using our method. We also confirm the hierarchical low rank property for the driven Hubbard model in the weak coupling regime within the GW approximation, and in the strong coupling regime within dynamical mean-field theory.

Electroweak SU(2)L × U(1)Y model with strong spontaneously fermion-mass-generating gauge dynamics

Higgs sector of the Standard model (SM) is replaced by quantum flavor dynamics (QFD), the gauged flavor SU(3)f symmetry with scale Λ. Anomaly freedom requires addition of three νR. The approximate QFD Schwinger-Dyson equation for the Euclidean infrared fermion self-energies Σf(p2) has the spontaneous-chiral-symmetry-breaking solutions ideal for seesaw: (1) Σf(p2) = $$ {M}_{fR}^2/p $$ M fR 2 / p where three Majorana masses MfR of νfR are of order Λ. (2) Σf(p2) = $$ {m}_f^2/p $$ m f 2 / p where three Dirac masses mf = m(0)1 + m(3)λ3 + m(8)λ8 of SM fermions are exponentially suppressed w.r.t. Λ, and degenerate for all SM fermions in f. (1) MfR break SU(3)f symmetry completely; m(3), m(8) superimpose the tiny breaking to U(1) × U(1). All flavor gluons thus acquire self-consistently the masses ∼ Λ. (2) All mf break the electroweak SU(2)L × U(1)Y to U(1)em. Symmetry partners of the composite Nambu-Goldstone bosons are the genuine Higgs particles: (1) three νR-composed Higgses χi with masses ∼ Λ. (2) Two new SM-fermion-composed Higgses h3, h8 with masses ∼ m(3), m(8), respectively. (3) The SM-like SM-fermion-composed Higgs h with mass ∼ m(0), the effective Fermi scale. Σf(p2)-dependent vertices in the electroweak Ward-Takahashi identities imply: the axial-vector ones give rise to the W and Z masses at Fermi scale. The polar-vector ones give rise to the fermion mass splitting in f. At the present exploratory stage the splitting comes out unrealistic.

Gluon dynamics from an ordinary differential equation

We present a novel method for computing the nonperturbative kinetic term of the gluon propagator from an ordinary differential equation, whose derivation hinges on the central hypothesis that the regular part of the three-gluon vertex and the aforementioned kinetic term are related by a partial Slavnov–Taylor identity. The main ingredients entering in the solution are projection of the three-gluon vertex and a particular derivative of the ghost-gluon kernel, whose approximate form is derived from a Schwinger–Dyson equation. Crucially, the requirement of a pole-free answer determines the initial condition, whose value is calculated from an integral containing the same ingredients as the solution itself. This feature fixes uniquely, at least in principle, the form of the kinetic term, once the ingredients have been accurately evaluated. In practice, however, due to substantial uncertainties in the computation of the necessary inputs, certain crucial components need be adjusted by hand, in order to obtain self-consistent results. Furthermore, if the gluon propagator has been independently accessed from the lattice, the solution for the kinetic term facilitates the extraction of the momentum-dependent effective gluon mass. The practical implementation of this method is carried out in detail, and the required approximations and theoretical assumptions are duly highlighted.

Quark mass function at finite density in real-time formalism

Chiral symmetry restoration of quarks is investigated at finite density in quantum chromodynamics. The critical chemical potentials at zero temperature, in which the chiral symmetry is restored, are calculated with the Schwinger-Dyson equation in the real-time formalism without the instantaneous exchange approximation. We present some properties of the quark mass functions and the quark propagators.

Many-body Dyson equation approach to the seniority model of pairing

As is well known, the single level seniority model of pairing has been solved exactly, since long using angular momentum algebra. It is shown that it can also be solved using the Dyson equation of standard many-body theory. The formalism shows some interesting many-body aspects.