Performance analysis of a physically constructed orthogonal representation of imaginary-time Green's function

We show that the imaginary-time Green's functions of a system of quantum charged particles at equilibrium satisfy a set of sum rules which express the screening of a classical external charge. We also provide a new derivation of the perfect screening property ${\mathrm{lim}}_{\mathrm{k}\ensuremath{\rightarrow}0}$${\mathrm{\ensuremath{\epsilon}}}^{\mathrm{\ensuremath{-}}1}$(k)=0 where \ensuremath{\epsilon}(k) is the static dielectric function. The proof relies on exact equations for the Green's functions and certain assumptions of spatial clustering. It is nonperturbative, independent of the statistics, and applies to jellium as well as to multicomponent systems.

Exchange energy and specific heat of the electron gas at low temperature

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.

We develop a causal optimization method that ensures causality in numerical calculations of Green's functions in interacting electron systems. Our method removes noncausality of numerical data by finding causal functions closest to the data. By testing our method with an exactly calculable model and applying it to practical dynamical mean-field calculations, we find that intermediate-frequency behaviors of Green's functions are determined solely by causality, and noncausal statistical errors are removed very efficiently. Furthermore, we demonstrate that numerical calculations of the physical branch of the Luttinger-Ward functional can be stabilized by ensuring causality of the noninteracting Green's function. Our method and findings provide a basis for improving stability and efficiency of numerical simulations of quantum many-body systems.

The problem of finding the Green's function of spin and electromagnetic waves in the sinusoidal superlattice is considered. An analytical expression for the spectral representation of the Green's function has been found in the form of ascending continued fractions, the particular denominators of which are ordinary continued fractions. The Green’s function in the-space has been found by the numerical Fourier transformation of the Greens’s function found in the spectral representation.

Several recent works have introduced highly compact representations of single-particle Green's functions in the imaginary time and Matsubara frequency domains, as well as efficient interpolation grids used to recover the representations. In particular, the intermediate representation with sparse sampling and the discrete Lehmann representation (DLR) make use of low rank compression techniques to obtain optimal approximations with controllable accuracy. We consider the use of the DLR in dynamical mean-field theory (DMFT) calculations, and in particular show that the standard full Matsubara frequency grid can be replaced by the compact grid of DLR Matsubara frequency nodes. We test the performance of the method for a DMFT calculation of ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ at temperature $50\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ using a continuous-time quantum Monte Carlo impurity solver, and demonstrate that Matsubara frequency quantities can be represented on a grid of only 36 nodes with no reduction in accuracy, or increase in the number of self-consistent iterations, despite the presence of significant Monte Carlo noise.

In this study, the authors present an electromagnetic model of a nanodipole array above a double-layer Graphene (DLG) operating at Terahertz. The graphene layers deposited between the dielectric mediums are modelled as surface impedances with conductivities described by the Kubo model. The electromagnetic fields are defined by the spectral representation of the periodic Green's function with impedance condition between the mediums. The main objective of this work is to propose an equivalent and analytical electromagnetic model of a nanodipole array above a DLG, which can be used to describe a surface plasmon resonance sensor that operates at the terahertz frequency. As a way of validating the method, the authors compared the proposed model with a software based on the finite element method. In addition, still, as a form of validation, the authors verified the convergence of the method from the spectral analysis of the proposed Green's Function. As results, spatial and spectral parametric analyses are shown as a function of the concentration of nanodipoles in the array, and the chemical potential applied to the graphene layers. In these results, the authors verify the appearance of plasmonic, optical and acoustic modes, and their contributions in the total field.

I compute the spectral functions of a 1D Holstein polaron moving in a modulated potential, to examine the effects of electron-phonon interactions on band gaps. The imaginary time Green function is computed using diagrammatic quantum Monte Carlo, which exactly sums the diagrammatic series. From the imaginary time Green function, spectral functions are computed. The electron-phonon interaction flattens the electronic dispersion and leads to an increase in the gap at momentum k = π/2. At strong coupling, polaron sidebands form in the gap. These results demonstrate the strong effect that electron-phonon interactions can have on band gaps.

An addition formula for Green's functions

In this note a representation of the discrete Green's function of a compact discretization of a two point boundary value problem of order n 2 is given which among other things allows a direct proff of the convergence (and divergence) properties.

Double-sided open periodic structures are analyzed using inhomogeneous plane wave scattering.The leaky and surface wave modes of several unit cells of different structures are computed using the poles of generalized reflection and transmission coefficients of inhomogeneous plane waves in the spectral domain.It is shown that the reflection and transmission coefficients of the zeroth order Floquet mode contain the poles of the Green's function of the complex stratified periodic structure.The properties of evanescent mode amplification as well as super resolution near field imaging in a wire medium are addressed.A balanced leaky wave antenna unit cell with double-sided radiation feature is introduced and it is shown that, in contrast to grounded structures, total absorption in lossless nonchiral double-sided open unit cells is not feasible as long as the behavior of the unit cell is well described by its fundamental mode.

Ocean heat uptake is caused by “excess heat” being added to the ocean surface by air‐sea fluxes and then carried to depths by ocean transports. One way to estimate excess heat in the ocean is to propagate observed sea surface temperature (SST) anomalies downward using a Green's function (GF) representation of ocean transports. Taking a “perfect‐model” approach, we test this GF method using a historical simulation, in which the true excess heat is diagnosed. We derive GFs from two approaches: (a) simulating GFs using idealized tracers, and (b) inferring GFs from simulated CFCs and climatological tracers. In the model world, we find that combining simulated GFs with SST anomalies reconstructs the Indo‐Pacific excess heat with a root‐mean‐square error of 26% for depth‐integrated changes; the corresponding number is 34% for inferred GFs. Simulated GFs are inaccurate because they are coarse grained in space and time to reduce computational cost. Inferred GFs are inaccurate because observations are insufficient constraints. Both kinds of GFs neglect the slowdown of the North Atlantic heat uptake as the ocean warms up. SST boundary conditions contain redistributive cooling in the Southern Ocean, which causes an underestimate of heat uptake there. All these errors are of comparable magnitude, and tend to compensate each other partially. Inferred excess heat is not sensitive to: (a) small changes in the shape of prior GFs, or (b) additional constraints from SF6 and bomb 14C.

We state a general principle for seismic migration/inversion processes: think image point coordinates; compute in surface coordinates. This principle allows the natural separation of multiple travel paths of energy from a source to a reflector to a receiver. Further, the Beylkin determinant (Jacobian of transformation between processing parameters and acquisition surface coordinates) is particularly simple in stark contrast to the common-offset Beylkin determinant in standard single arrival Kirchhoff .A feature of this type of processing is that it changes the deconvolution structure of Kirchhoff operators or the deconvolution imaging operator of wave equation migration into convolution operators; that is, division by Green's functions is replaced by multiplications by adjoint Green's functions.This transformation from image point coordinates to surface coordinates is also applied to a recently developed extension of the standard Kirchhoff inversion method. The standard method uses Green's functions in the integration process and tends to produce more imaging artefacts than alternatives, such as methods using Gaussian beam representations of Green's functions in the inversion formula. These methods point to the need for a true-amplitude Kirchhoff technique that uses more general Green's functions: Gaussian beams, true-amplitude one-way Green's functions, or Green's functions from the two-way wave equation. Here, we present a derivation of a true-amplitude Kirchhoff that uses these more general Green's functions. When this inversion is recast as an integral over all sources and receivers, the formula is surprisingly simple.

A hybrid ray-mode representation of the Green's function in a rectangular cavity is developed using the finite Poisson summation formula. To obtain a numerically efficient scheme for computing the field generated by a point source in a large rectangular cavity, the conventional modal representation of the Green's function is modified in such a way that all the modes near resonance are retained while the truncated remainder of the mode series is expressed in terms of a weighted contribution of rays. For an electrically large cavity, the contribution of rays from distant images becomes small; therefore, the ray sum can be approximated by one or two dominant terms without a loss of numerical accuracy. To illustrate the accuracy and the computational simplification of this ray-mode representation, numerical examples are included with the conventional mode series (summed at the expense of long computation time) serving as a reference.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

We address the problem of analytic continuation of imaginary-frequency Green's functions, which is crucial in many-body physics, using machine learning based on a multi-level residual neural network. We specifically address potential biases that can be introduced due to the use of artificially created spectral functions that are employed to train the neural network. We also implement an uncertainty estimation of the predicted spectral function, based on Monte Carlo dropout, which allows to identify frequency regions where the prediction might not be accurate, and we study the effect of noise, in particular also for situations where the noise level during training is different from that in the actual data. Our analysis demonstrates that this method can indeed achieve a high quality of prediction, comparable or better than the widely used maximum entropy method, but that further improvement is currently limited by the lack of true data that can be used for training. We also benchmark our approach by applying it to the case of SrVO$_3$, where an accurate spectral function has been obtained from dynamical mean-field theory using a solver that works directly on the real frequency axis.