Schwinger Equation Research Articles

FFT-based multiscale scheme for homogenisation of heterogeneous plates including damage and failure

This paper introduces a new approach that combines FFT-based multiscale homogenisation with the Lippman–Schwinger equation to efficiently and accurately analyse plate structures with periodic micro-structures. For the first time, the plate periodic Lippman–Schwinger equation is derived to address the auxiliary cell problem. The proposed method comprises two key components: (i) a 3D plate formulation using a novel finite prism approach, and (ii) solving the Lippman–Schwinger equations by means of Green’s functions. Solutions for both the classical and first-order plate theories are derived and implemented for linear and nonlinear problems, completed with an open-source code provided.The efficiency and accuracy of the proposed method are assessed through several case studies, including complex woven composites. It is demonstrated that proposed method achieves a comparable level of accuracy to 3D solid problems while significantly reducing the required computing effort (i.e., milliseconds in computing time as opposed to hours required for 3D solid models). Moreover, nonlinear progressive damage problem involving an actual plate woven composite model is investigated. The results obtained show good agreement with experimental measurements for the progressive damage in plain woven composite, further highlighting the effectiveness of the proposed method.

Looking at QED with Dyson–Schwinger Equations: Basic Equations, Ward–Takahashi Identities and the Two-Photon-Two-Fermion Irreducible Vertex

A minimal truncated set of the integral Dyson–Schwinger equations, in Minkowski spacetime, that allows to explore QED beyond its perturbative solution is derived for general linear covariant gauges. The minimal set includes the equations for the fermion and photon propagators, the photon-fermion vertex, and the two-photon-two-fermion one-particle-irreducible diagram. If the first three equations are exact, to build a closed set of equations, the two-photon-two-fermion equation is truncated ignoring the contribution of Green functions with large number of external legs. It is shown that the truncated equation for the two-photon-two-fermion vertex reproduces the lowest-order perturbative result in the limit of the small coupling constant. Furthermore, this equation allows to define an iterative procedure to compute higher order corrections in the coupling constant. The Ward–Takahashi identity for the two-photon-two-fermion irreducible vertex is derived and solved in the soft photon limit, where one of the photon momenta vanish, in the low photon momenta limit and for general kinematics. The solution of the Ward–Takahashi identity determines the longitudinal component of the two-photon-two-fermion irreducible vertex, while it is proposed to use the Dyson–Schwinger equation to determine the transverse part of this irreducible diagram. The two-photon-two-fermion DSE is solved in heavy fermion limit, considering a simplified version of the QED vertices. The contribution of this irreducible vertex to a low-energy effective photon-fermion vertex is discussed and the fermionic operators that are generated are computed in terms of the fermion propagator functions.

Pion-mediated Cooper pairing of neutrons: beyond the bare vertex approximation

In some quantum many particle systems, the fermions could form Cooper pairs by exchanging intermediate bosons. This then drives a superconducting phase transition or a superfluid transition. Such transitions should be theoretically investigated by using proper non-perturbative methods. Here we take the neutron superfluid transition as an example and study the Cooper pairing of neutrons mediated by neutral π-mesons in the low-density region of a neutron matter. We perform a non-perturbative analysis of the neutron–meson coupling and compute the pairing gap Δ, the critical density ρ c , and the critical temperature T c by solving the Dyson–Schwinger equation of the neutron propagator. We first carry out calculations under the widely used bare vertex approximation and then incorporate the contribution of the lowest-order vertex correction. This vertex correction is not negligible even at low densities and its importance is further enhanced as the density increases. The transition critical line on density-temperature plane obtained under the bare vertex approximation is substantially changed after including the vertex correction. These results indicate that the vertex corrections play a significant role and need to be seriously taken into account.

An extended full field self-consistent cluster analysis framework for woven composite

In this work, the self-consistent clustering analysis (SCA) framework is extended to include homogenization and full field analysis of 3D anisotropic woven composite Representative Unit Cell (RUC). The developed extended framework has two new features, namely, (i), to reconstruct the local field variables, a strain refinement stage is presented by solving full field Lippmann–Schwinger equations within 3D anisotropic woven composite RUC following the online predictive stage in SCA, and (ii), discrete Green’s operator based on finite difference is adopted to improve the accuracy of refined point-wise physical field variables. To demonstrate the accuracy and efficiency of the proposed method, benchmark problems are analyzed, and results are compared to directly numerical simulation (DNS). For the reproducibility of presented results, the developed code can be freely downloaded from https://github.com/Tong-RuiLiu/Extended-SCA-and-FFT-based-strain-refinement-method-.

Investigation of the confinement in massive QED3

By the application of the Schwinger function, the deconfinement in massive quantum electrodynamics in [Formula: see text]-dimensional (QED3) is investigated. It is shown that oscillating behavior of the Schwinger function gradually disappears with fermion mass increasing based on coupled Dyson–Schwinger equations in massive QED3. On the other hand, we have shown that the confined fermion transfers gradually into the phase of deconfinement with the increasing boson mass, which is consistent with the result in the case of massless QED3. Finally, we emphasized that dynamical chiral symmetry breaking and confinement always appear simultaneously in massive QED3.

Influence of gauge boson mass on dynamical mass generation and chiral phase transition in thermal QED3

In this work, we quantitatively explore how a finite gauge boson mass affects dynamical mass generation and the chiral phase transition in thermal QED3. To this end, we numerically solve the Dyson–Schwinger equations by considering the contributions coming from the wave function renormalization and the spatial part of the boson propagator at first and then compute the chiral condensate and some of thermodynamic quantities. It is found that the wave function renormalization and the spatial part of the boson propagator are crucial for the numerical solutions of the dressing functions and the value of critical temperature. The behaviors of the thermodynamic quantities and the chiral condensate as a function of temperature consistently imply that the chiral phase transition is none other than a second order one. That critical temperature decreases with increasing gauge boson mass verifies that the gauge boson mass will weaken the interactions between fermions and so suppress dynamical mass generation.

Hawking radiation of rotating BTZ black hole based on modified dispersion relation and Rarita–Schwinger equation

In this paper, tunneling of fermions from rotating BTZ black hole is investigated using modified dispersion relation (MDR) and Rarita–Schwinger equation. The effect of MDR on the tunneling of fermions raises the Hawking temperature of rotating BTZ black hole. It is observed that the modified Hawking temperature of the black hole depends not only on the radial parameters of the black hole but also on the angular parameters of the black hole and the coupling constant [Formula: see text]. Further, the entropy and the heat capacity of the black hole are also studied.

Greybody factors emitted by a regular black hole in a non-minimally coupled Einstein–Yang–Mills theory

In this paper, we study the greybody factors (GFs) for fermions with different spins and bosons in the regular black hole (BH) predicted by a non-minimal Einstein–Yang–Mills (EYM) theory. We investigate the effect of magnetic charge on effective potentials and GFs. For this purpose, we consider the Dirac and Rarita–Schwinger, as well as Klein–Gordon equations. First, we study the Dirac equation in curved spacetime for massive and massless spin-1/2 fermions. We then separate the Dirac equation into sets of radial and angular equations. Using the analytical solution of the angular equation, the Schrödinger-like wave equations with potentials are derived by decoupling the radial wave equations via the tortoise coordinate. We also consider the Rarita–Schwinger equation for massless spin-3/2 fermions and derive the one-dimensional Schrödinger wave equation with gauge-invariant effective potential. For bosons, we study the Klein–Gordon equation in the regular non-minimal EYM BH. Afterward, semi-analytic methods were used to calculate the fermionic and bosonic GFs. Finally, we discuss the graphical behavior of the obtained effective potentials and bounds on the GFs. According to graphs, the GF is highly influenced by the potential’s shape, which is determined by the parameterization of the model. This is in line with quantum theory.

Volume Integral Equations and Single-Trace Formulations for Acoustic Wave Scattering in an Inhomogeneous Medium

AbstractWe study frequency domain acoustic scattering at a bounded, penetrable, and inhomogeneous obstacleΩ−⊂Rd\Omega^{-}\subset\mathbb{R}^{d},d=2,3d=2,3. By defining constant reference coefficients, a representation formula for the pressure field is derived. It contains a volume integral operator, related to the one in the Lippmann–Schwinger equation. Besides, it features integral operators defined on∂Ω−\partial\Omega^{-}and closely related to boundary integral equations of single-trace formulations (STF) for transmission problems with piecewise constant coefficients. We show well-posedness of the continuous variational formulation and asymptotic convergence of Galerkin discretizations. Numerical experiments in 2D validate our expected convergence rates.

A statistical mechanical model for non-perturbative regimes

Thanks to Feynman graphons, as mathematical tools in dealing with Dyson–Schwinger equations, we formulate a new statistical mechanical model for the study of equilibrium states and observables associated with the solution space of quantum motions in a strongly coupled gauge field theory underlying the evolution of running coupling constants, time and temperature.

Quark condensate and magnetic moment in a strong magnetic field

This paper studies the quark condensate, magnetic moment, magnetic polarization, and magnetic susceptibility in a strong external magnetic field by employing the Dyson–Schwinger equations (DSE). The results show these physical quantities as functions of the magnetic field. We note that the quark’s spin polarizations are approximately proportional to the magnetic field magnitude. For comparison, we investigate the magnetic moments and susceptibility of the nucleon in the constituent quark model framework and demonstrate that both these quantities increase as the magnetic field rises.

Finite-volume effects in baryon number fluctuations around the QCD critical endpoint

We present results for the volume dependence of baryon number fluctuations in the vicinity of the (conjectured) critical endpoint of QCD. They are extracted from the nonperturbative quark propagator that is obtained as a solution to a set of truncated Dyson–Schwinger equations of (2+1)-flavor QCD in Landau gauge, which takes the backcoupling of quarks onto the Yang–Mills sector explicitly into account. This well-studied system predicts a critical endpoint at moderate temperatures and rather large chemical potential. We investigate this system at small and intermediate finite, three-dimensional, cubic volumes and study the resulting impact on baryon number fluctuations and ratios thereof up to fourth order in the region of the critical endpoint. Due to the limitations of our truncation, the results are quantitatively meaningful only outside the critical scaling region of the endpoint. We find that the fluctuations are visibly affected by the finite volume, particularly for antiperiodic boundary conditions, whereas their ratios are practically invariant.

Dyson–Schwinger equations in minimal subtraction

We compare the solutions of one-scale Dyson–Schwinger equations (DSEs) in the minimal subtraction (MS) scheme to the solutions in kinematic momentum subtraction (MOM) renormalization schemes. We establish that the MS-solution can be interpreted as a MOM-solution, but with a shifted renormalization point, where the shift itself is a function of the coupling. We derive relations between this shift and various renormalization group functions and counterterms in perturbation theory. As concrete examples, we examine three different one-scale Dyson–Schwinger equations: one based on the 1-loop multiedge graph in D=4 dimensions, one for D=6 dimensions, and one for mathematical toy model. For each of the integral kernels, we examine both the linear and nine different non-linear Dyson–Schwinger equations. For the linear cases, we empirically find exact functional forms of the shift between MOM and MS renormalization points. For the non-linear DSEs, the results for the shift suggest a factorially divergent power series. We determine the leading asymptotic growth parameters and find them in agreement with the ones of the anomalous dimension. Finally, we present a tentative exact solution to one of the non-linear DSEs of the toy model.

Quantum refractive index for two- and three-dimensional systems

We study the scattering of a plane-wave by two- and three-dimensional potentials — circular and elliptical disks and spherical and spheroidal regions. These regions are filled with Dirac delta functions, and we name these theoretical materials Dirac medium. By solving the Lippmann–Schwinger equation in the position-representation, we calculate the wavefunctions, differential, and total cross-sections as well as the quantum refractive index which characterizes these materials. The quantum refractive index is a complex function of the incident wavenumber k as well as of the coupling strength γ0. For certain special values of k and γ0, its real part can be negative, and its imaginary part can be positive, negative, and even zero, revealing that Dirac medium are the quantum mechanical analogues of certain materials that have electromagnetic properties such as negative extinction coefficient and refractive index.

Homogenization of sound-soft and high-contrast acoustic metamaterials in subcritical regimes

We propose a quantitative effective medium theory for two types of acoustic metamaterials constituted of a large number N of small heterogeneities of characteristic size s, randomly and independently distributed in a bounded domain. We first consider a “sound-soft” material, in which the total wave field satisfies a Dirichlet boundary condition on the acoustic obstacles. In the “sub-critical” regime sN = O(1), we obtain that the effective medium is governed by a dissipative Lippmann–Schwinger equation which approximates the total field with a relative mean-square error of order O(max((sN)2N-1/3, N-1/2)). We retrieve the critical size s ~ 1/N of the literature at which the effects of the obstacles can be modelled by a “strange term” added to the Helmholtz equation. Second, we consider high-contrast acoustic metamaterials, in which each of the N heterogeneities are packets of K inclusions filled with a material of density much lower than the one of the background medium. As the contrast parameter vanishes, δ → 0, the effective medium admits K resonant characteristic sizes (si(δ))1≤i≤K and is governed by a Lippmann–Schwinger equation, which is diffusive or dispersive (with negative refractive index) for frequencies ω respectively slightly larger or slightly smaller than the corresponding K resonant frequencies (ωi (δ))1≤i≤K. These conclusions are obtained under the condition that (i) the resonance is of monopole type, and (ii) lies in the “subcritical regime” where the contrast parameter is small enough, i.e. δ = o(N−2)), while the considered frequency is “not too close” to the resonance, i.e. Nδ1/2 = O(|1 - s/si(δ)|). Our mathematical analysis and the current literature allow us to conjecture that “solidification” phenomena are expected to occur in the “super-critical” regime Nδ1/2|1 - s/si(δ)|-1 → + ∞.

Fermionic Greybody Factors in Schwarzschild Acoustic Black Holes

In Schwarzschild acoustic black hole (SABH) spacetime, we investigate the wave dynamics for the fermions. To this end, we first take into account the Dirac equation in the SABH by employing a null tetrad in the Newman–Penrose (NP) formalism. Then, we consider the Dirac and Rarita–Schwinger equations, respectively. The field equations are reduced to sets of radial and angular equations. By using the analytical solution of the angular equation set, we decouple the radial wave equations and obtain the one-dimensional Schrödinger-like wave equations with their effective potentials. The obtained effective potentials are graphically depicted and analyzed. Finally, we investigate the fermionic greybody factors (GFs) radiated by the SABH spacetime. A thorough investigation is conducted into how the acoustic tuning parameter affects the GFs of the SABH spacetime. Both the semi-analytic WKB method and bounds for the GFs are used to produce the results, which are shown graphically and discussed.

Scattering of complex gravitino test fields from a gravitational pulse

Gravitational pulse waves are defined as step functions at the boundaries. In this setting, linearized Rarita–Schwinger equations are precisely solved in Rosen coordinates. It is found that the gravitino’s energy–momentum varies with the sandwich wave’s shape. The gravitino’s energy density will decrease as it crosses the sandwich wave at the test field level since the background won’t change.

Thermodynamic Properties and Critical Behavior of Spin-Polarized Atomic Hydrogen (H↓) Using the Quantum Second-Virial Coefficient

The quantum–mechanical formulations for the second-virial, and the acoustic virial, coefficients are derived for spin-polarized atomic hydrogen (H↓). The dependence of these coefficients on the temperature T as well as the nuclear polarization $$\zeta \;$$ is analyzed. This is done for the first time. The main inputs in these computations are the scattering phase shifts, which are obtained using the Lippmann–Schwinger equation, with the Silvera triplet-state potential. Starting with these phase shifts, comprehensive calculations of the thermophysical properties for this system are performed for T ranging from 1 µK to 100 K, and $$\zeta \;$$ varying from 0 to 1. These properties include, in addition to the quantum second-virial coefficient and the acoustic virial coefficient: the pressure–polarization–temperature $$\left( {P {-} \zeta {-} T} \right)$$ behavior, the entropy (per atom), the speed of sound, the second-virial correction (to the total internal energy per atom and unit density), and the specific heat capacity (per atom and unit density). The Boyle temperature and the Joule inversion temperature are determined. The T-dependence of these thermophysical properties, and related quantities, is explored. As expected, this dependence is most evident in the low-T limit, where quantum effects predominate. The corresponding $$\zeta$$ -dependence becomes noticeable at 80 K and below. It is observed that the virial coefficients tend to decrease with increasing $$\zeta$$ . Comparison of the present results to previous results are included whenever possible. The overall agreement is very good. As T is lowered, the disruptive effects of thermal energy are weakened relative to the attractive interactions between the atoms. Consequently, the H↓ gas makes a transition to a state of higher order and lower entropy—Bose–Einstein condensation (BEC). This is explored carefully.

Electromagnetic Inverse Scattering With an Untrained SOM-Net

Physics-inspired deep learning (DL) methods become a research hotspot in solving inverse scattering problems (ISPs) due to the advantages of imaging quality and speed. However, the generalization ability is still the main bottleneck of DL-based ISP methods. In this article, an untrained SOM-Net (called uSOM-Net, where SOM stands for the subspace-based optimization), which employs our newly proposed SOM-Net as the backbone neural network, is introduced to solve the ISP without training and testing processes. The uSOM-Net not only achieves high reconstruction quality comparable to existing DL-based methods but also owns similar adaptability as the traditional iterative ones. Specifically, the uSOM-Net parameterizes all physical variables in the Lippmann–Schwinger equation by network weights. To overcome the ill-posedness, the uSOM-Net updates the unknown weights with physical losses defined from both the data and the state equations. Besides, prior information from accessible scatterers can also be easily incorporated into the uSOM-Net by adding extra physical constraints with few prior labeled data. The proposed uSOM-Net is verified with both synthetic and experimental examples to demonstrate its superiority over existing ones. The uSOM-Net intends to bridge the gap between traditional iterative ISP methods and DL ones, which introduces a new way of solving ISPs.

Fourier Transform approach to numerical homogenization of periodic media containing sharp insulating and superconductive cracks

In the present paper, we propose a new FFT based numerical homogenization method to compute the effective conductivity of fractured media involving insulating and superconductive cracks. The governing Lippmann Schwinger (LS) equations based on temperature jump and heat flux jump distribution localized on crack lines are formulated and the geometry of sharp cracks is described by exact expressions of form factors. To solve the LS equations, the bi conjugate gradient stabilized method is used and leads to fast convergence. Numerical examples in 2D and 3D show a good accuracy when compared with standard Finite Element Method solutions.