Averaged Green Function Research Articles

A new algorithm for investigating strongly correlated systems using Hubbard model

In this work, a new algorithm introduces for calculating the average Green function of an interacting system that obeys the Hubbard model. This algorithm is applied to investigate the effects of onsite repulsion on the band structure of a square lattice in both the single-site approximations such as dynamical mean field theory (DMFT) and multi-site approximations such as effective medium supercell approximation (EMSCA). The advantages of our algorithm in comparison to the Hirsch-Fye algorithm and also the Blankenbecler, Scalapino, and Sugar (BSS) algorithm are the elimination of instabilities resulting from the Metropolis algorithm in the accepting and rejecting configurations, stability at low temperatures, the elimination of systematic errors resulting from the update of the Green's function in the quantum Monte Carlo process, and considering different probabilities for each possible configuration. Finally, by using our algorithm, it is possible to calculate the interacting three-dimensional system's band structure and the density of states that obey the Hubbard model. We have applied our algorithm to an interactive two-dimensional square lattice. As a result, phase transition boundaries can be easily recognized through calculated bands for different repulsions. Our results show that critical Coulomb repulsion values for Mott transition are uc = 9.05t and uc = 2.4t for DMFT and BEMSCA respectively. This means that DMFT significantly overestimates band splitting due to electrons' Coulomb repulsion. We found by starting at low repulsions and then increasing electrons' Coulomb repulsion, a partially flatted valence band around the center of the first Brillouin zone appears, but disappears at high repulsions.

Critical phase boundary and finite-size fluctuations in the Su-Schrieffer-Heeger model with random intercell couplings

A dimerized fermion chain, described by Su-Schrieffer-Heeger (SSH) model, is a well-known example of 1D system with a non-trivial band topology. An interplay of disorder and topological ordering in the SSH model is of a great interest owing to experimental advancements in synthesized quantum simulators. In this work, we investigate a special sort of a disorder when inter-cell hopping amplitudes are random. Using a definition for $\mathbb{Z}_2$-topological invariant $\nu\in \{ 0; 1\}$ in terms of a non-Hermitian part of the total Hamiltonian, we calculate $\langle\nu\rangle$ averaged by random realizations. This allows to find (i) an analytical form of the critical surface that separates phases of distinct topological orders and (ii) finite size fluctuations of $\nu$ for arbitrary disorder strength. Numerical simulations of the edge modes formation and gap suppression at the transition are provided for finite-size system. In the end, we discuss a band-touching condition derived within the averaged Green function method for a thermodynamic limit.

On the Green's function emergence from interferometry of seismic wave fields generated in high-melt glaciers: implications for passive imaging and monitoring

Abstract. Ambient noise seismology has revolutionized seismic characterization of the Earth's crust from local to global scales. The estimate of Green's function (GF) between two receivers, representing the impulse response of elastic media, can be reconstructed via cross-correlation of the ambient noise seismograms. A homogenized wave field illuminating the propagation medium in all directions is a prerequisite for obtaining an accurate GF. For seismic data recorded on glaciers, this condition imposes strong limitations on GF convergence because of minimal seismic scattering in homogeneous ice and limitations in network coverage. We address this difficulty by investigating three patterns of seismic wave fields: a favorable distribution of icequakes and noise sources recorded on a dense array of 98 sensors on Glacier d'Argentière (France), a dominant noise source constituted by a moulin within a smaller seismic array on the Greenland Ice Sheet, and crevasse-generated scattering at Gornergletscher (Switzerland). In Glacier d'Argentière, surface melt routing through englacial channels produces turbulent water flow, creating sustained ambient seismic sources and thus favorable conditions for GF estimates. Analysis of the cross-correlation functions reveals non-equally distributed noise sources outside and within the recording network. The dense sampling of sensors allows for spatial averaging and accurate GF estimates when stacked on lines of receivers. The averaged GFs contain high-frequency (>30 Hz) direct and refracted P waves in addition to the fundamental mode of dispersive Rayleigh waves above 1 Hz. From seismic velocity measurements, we invert bed properties and depth profiles and map seismic anisotropy, which is likely introduced by crevassing. In Greenland, we employ an advanced preprocessing scheme which includes match-field processing and eigenspectral equalization of the cross spectra to remove the moulin source signature and reduce the effect of inhomogeneous wave fields on the GFs. At Gornergletscher, cross-correlations of icequake coda waves show evidence for homogenized incident directions of the scattered wave field. Optimization of coda correlation windows via a Bayesian inversion based on the GF cross coherency and symmetry further promotes the GF estimate convergence. This study presents new processing schemes on suitable array geometries for passive seismic imaging and monitoring of glaciers and ice sheets.

Manifolds Pinned by a High-Dimensional Random Landscape: Hessian at the Global Energy Minimum

We consider an elastic manifold of internal dimension d and length L pinned in a N dimensional random potential and confined by an additional parabolic potential of curvature mu . We are interested in the mean spectral density rho (lambda ) of the Hessian matrix {{mathcal {K}}} at the absolute minimum of the total energy. We use the replica approach to derive the system of equations for rho (lambda ) for a fixed L^d in the N rightarrow infty limit extending d=0 results of our previous work (Fyodorov et al. in Ann Phys 397:1–64, 2018). A particular attention is devoted to analyzing the limit of extended lattice systems by letting Lrightarrow infty . In all cases we show that for a confinement curvature mu exceeding a critical value mu _c, the so-called “Larkin mass”, the system is replica-symmetric and the Hessian spectrum is always gapped (from zero). The gap vanishes quadratically at mu rightarrow mu _c. For mu <mu _c the replica symmetry breaking (RSB) occurs and the Hessian spectrum is either gapped or extends down to zero, depending on whether RSB is 1-step or full. In the 1-RSB case the gap vanishes in all d as (mu _c-mu )^4 near the transition. In the full RSB case the gap is identically zero. A set of specific landscapes realize the so-called “marginal cases” in d=1,2 which share both feature of the 1-step and the full RSB solution, and exhibit some scale invariance. We also obtain the average Green function associated to the Hessian and find that at the edge of the spectrum it decays exponentially in the distance within the internal space of the manifold with a length scale equal in all cases to the Larkin length introduced in the theory of pinning.

Time-dependent spectral functions of the Anderson impurity model in response to a quench with application to time-resolved photoemission spectroscopy

We investigate several definitions of the time-dependent spectral function $A(\omega,t)$ of the Anderson impurity model following a quench and within the time-dependent numerical renormalization group method. In terms of the two-time retarded Green function $G^r(t_1,t_2)$, the definitions differ in the choice of the time variable $t$ with respect to $t_1$ and/or $t_2$. In a previous study [Nghiem {\it et al.} Phys. Rev. Lett. 119, 156601 (2017)], we investigated the spectral function, obtained from the Fourier transform of ${\rm Im}[G^r(t_1,t_2)]$ w.r.t. the time difference $t'=t_1-t_2$, with $t=t_2$. Here, we derivie expressions for the retarded Green function for the choices $t=t_1$ and the average time $t=(t_1+t_2)/2$, within the TDNRG approach. We compare and contrast the resulting $A(\omega,t)$ for the different choices of time reference. Expressions for the lesser, greater and advanced Green functions are also derived within TDNRG for all choices of time reference. The average time lesser Green function $G^<(\omega,t)$ is particularly interesting, as it determines the time-dependent occupied density of states $N(\omega,t)=G^<(\omega,t)/(2\pi i)$, a quantity that determines the photoemission current in time-resolved pump-probe photoemission spectroscopy. We present calculations for $N(\omega,t)$ for the Anderson model following a quench, and discuss the resulting time evolution of the spectral features, such as the Kondo resonance and high-energy peaks. We also discuss the issue of thermalization at long times for $N(\omega,t)$. Finally, we use the results for $N(\omega,t)$ to calculate the time-resolved photoemission current for the Anderson model following a quench (acting as the pump) and study the different behaviors that can be observed for different resolution times of a Gaussian probe pulse.

On the Averaged Green’s Function of an Elliptic Equation with Random Coefficients

We consider a divergence-form elliptic difference operator on the lattice $\mathbb{Z}^d$, with a coefficient matrix that is an i.i.d. perturbation of the identity matrix. Recently, Bourgain introduced novel techniques from harmonic analysis to prove the convergence of the Feshbach-Schur perturbation series related to the averaged Green's function of this model. Our main contribution is a refinement of Bourgain's approach which improves the key decay rate from $-2d+\epsilon$ to $-3d+\epsilon$. (The optimal decay rate is conjectured to be $-3d$.) As an application, we derive estimates on higher derivatives of the averaged Green's function which go beyond the second derivatives considered by Delmotte-Deuschel and related works.

Sound propagation above a flat ground with random spatially varying properties

The acoustic properties of ground surfaces vary in space. Usually, these variations are not accounted for in predictions of outdoor sound propagation for two main reasons. First, they are not known as it is costly to obtain finely sampled spatial measurements of these ground properties. Second, there is no existing simple and quick-to-compute analytical solution for a ground with a spatially-varying admittance, contrary to the well-documented case of a ground with a homogeneous admittance. This imperfect knowledge of the ground characteristics leads to uncertainties about the sound pressure level. This paper aims at characterizing these uncertainties. Propagation of acoustic waves over the ground with a spatially-varying surface admittance is therefore considered. Using the diagram technique, the average Green's function is determined. An asymptotic expression in the form of a Weyl-Van der Pol formula is obtained at long range and at grazing incidence. Modifications of the reflection coefficient and of the surface wave contribution due to randomness are analyzed. Numerical simulations using a linearized Euler equations solver are then carried out. Comparison of the mean pressure obtained from ensemble-average over 200 realizations and from the analytical solution is performed. Finally, the mean intensity and intensity fluctuations are investigated.

Correlated Lloyd model: Exact solution

Exactly solvable model of disordered system representing the generalized Lloyd model with correlated random potential is described. It is shown, that for the model under consideration, the averaged Green's function does not depend on random potential correlation radius and, similarly to the classical Lloyd model, has the form of Green's function of a crystal system, with energy argument supplied by an imaginary part which depends on degree of disorder.

Weak magnetic field solution of Schrödinger equation in two dimensions and continuous density of states

Green function method is applied to the solution of Schrödinger equation in two-dimensions where there is a constant and nearly uniform magnetic field Bo applied perpendicularly and there are infinitesimally weak scatterers within the system. Electronic density of states n(E) obtained from the average Green function go is a continuous function of energy E and not in the form of discrete Landau spikes even in the pure system with no disorder. Self consistent calculation of self-energy gives two possible scattering time values, one leading to resistance increase with Bo while the other causing decrease in resistance.

Lyapunov Exponents of Random Walks in Small Random Potential: The Lower Bound

We consider the simple random walk on Z^d, d > 2, evolving in a potential of the form \beta V, where (V(x), x \in Z^d) are i.i.d. random variables taking values in [0,+\infty), and \beta\ > 0. When the potential is integrable, the asymptotic behaviours as \beta\ tends to 0 of the associated quenched and annealed Lyapunov exponents are known (and coincide). Here, we do not assume such integrability, and prove a sharp lower bound on the annealed Lyapunov exponent for small \beta. The result can be rephrased in terms of the decay of the averaged Green function of the Anderson Hamiltonian -\Delta\ + \beta V.

Strong convergence to the homogenized limit of elliptic equations with random coefficients II

Consider a discrete uniformly elliptic divergence form equation on the $d$ dimensional lattice $\Z^d$ with random coefficients. In [3] rate of convergence results in homogenization and estimates on the difference between the averaged Green's function and the homogenized Green's function for random environments which satisfy a Poincar\'{e} inequality were obtained. Here these results are extended to certain environments with long range correlations. These environments are simply related via a convolution to environments which do satisfy a Poincar\'{e} inequality.

Electromagnetic waves in a randomly inhomogeneous Josephson junction

Spectrum modification and damping of Josephson plasma waves induced by random inhomogeneities of the critical current through the superconductor contact and the averaged Green function of such excitations are analyzed. In the self-consistent approximation that makes it possible to take into account multiple wave scattering on the inhomogeneities, the frequency and damping of averaged waves, as well as position νm and peak width Δν of the Fourier transform imaginary part of the averaged Green function, are determined as functions of wavevector k. The evolution of such functions with the variation of the correlation radius and the relative r.m.s. fluctuations of inhomogeneities is studied. The inhomogeneity-induced wave frequency decrease observed in the long wavelength spectral region qualitatively agrees with the νm behavior. It is established that in the case of “long-range” inhomogeneities, the linear dependence of damping on k changes to the inversely proportional one, and damping tends to zero as k → 0, while Δν at small k attains its maximal values due to nonuniform broadening. In the presence of “short-range” inhomogeneities, the wave damping and Δν are found to be similar functions of k. The results are compared to the numerical calculation data.

EVALUATION OF TRM IN THE COMPLEX THROUGH WALL ENVIRONMENT

Practical interests arising from behind-the-wall target detection, surveillance and reconnaissance etal. claim for high capability of imaging in complicated environments. Time Reversal Mirror (TRM) technique, making use of the principle of reciprocity, emerges as a promising way to deal with such complex problem. In this paper, we investigate TRM in the ultra-wideband (UWB) through wall radar imaging (TWRI) through numerical simulation. The probing region is a square room, with walls of rough surface and random media parameters. TRM is used to image the target settled in the room. We evaluate the degradation of the images when the aperture of the array is decreased or the received signals are contaminated by noises. The back projection (BP) algorithm is employed here as a comparison for imaging quality. For the case in which the random walls are changed between the forward and inverse phase of time reversal, we check the imaging stability and applied an averaged Green's function to improve the imaging quality. Finally, some interesting conclusions are drawn.

Effects of long-wave inhomogeneities of the exchange and elastic force constants in the framework of the coherent potential approximation

The formalism of averaged Green's functions in the continuum is developed for spin waves in ferromagnets with inhomogeneities of the exchange constant and elastic waves in media with inhomogeneities of the force constant. The integral self-consisted equation in the coherent potential approximation is developed for taking into account effects of inhomogeneities of these nonlocal characteristics of the material on the wave spectrum in it. The obtained general results are applied to the study of effects of one-dimensional inhomogeneities of the exchange constant on the wave spectrum in a ferromagnet.

Electronic Structures of Quantum Well of A1-xBx Alloy Semiconductor in Coherent Potential Approximation

The electronic structures of a quantum well constructed from the binary alloy semiconductor A1-xBx are studied by coherent potential approximation (CPA). A tight binding model is used for a single particle (electron, hole, or Frenkel exciton) in a well composed of a rectangular array of Nx×Ny×Nz sites. The effects of diagonal randomness are included as the coherent potential Σ(E), which is assumed to be the same for all sites, and is self-consistently determined with the average Green's function. For slab (∞,∞,Nz) and wire (∞,Ny,Nz) structures, the energy density of states ρ(E) is composed of Nz (or Ny×Nz) subbands with the remains of a two (one)-dimensional van-Hove singularity. When x (or 1-x) is small, a B (A) impurity band always appears on the lower (higher) energy side of the lowest (highest) host subband. The metamorphosis of ρ(E) and the absorption spectrum I(E) due to the creation of a Frenkel exciton by changing the well size and the dimensionality is extensively discussed.

Magnetization Processes in Dilute Metamagnets with Ferro- and Antiferromagnetic Exchange Interactions

Using the Green function method there has been presented the mathematical description of dilute two sub-lattice metamagnets with ferroand antiferromagnetic exchange interactions. The Green function averaging procedure is based on the Cumulant Expansion Technique and includes any localized Green function irrespectively of the decoupling algorithm making the chain system of equations self-consistent. The derived form of the averaged Green function is the base for the analysis of magnetization processes and phase transitions occurring in the system under the influence of external magnetic field. It has been found in particular that the tricritical point and the second-order phase transition vanish when concentrations of magnetic atoms coupled with different sub-lattice points become different.

Physical–mathematical modelling diffusion processes in bodies of random structure using generalized functions and Feynman diagrams

In the paper it is proposed new approach to physical–mathematical modelling mass transfer processes in multiphase randomly nonhomogeneous bodies. An initial-boundary value problem is formulated on the basis of Fick lows. The technique of Feynman diagrams is applied for investigating averaged diffusive fields. A nonlocal equation of diffusion is obtained for an averaged Green function of the disturbed medium. A method is proposed for convergence acceleration of infinite integral Neumann series regarding diffusive processes.

Exchange narrowing of magnetic resonance linewidths in inhomogeneous ferromagnets

The theory of exchange narrowing of ferromagnetic and spin-wave resonance linewidths in ferromagnets with the randomly inhomogeneous magnetic anisotropy is developed. The calculation is carried out by the method of averaged Green functions in the framework of the coherent potential approximation (CPA). One- and three-dimensional inhomogeneities with an arbitrary correlation wave number ${k}_{c}$ are considered (${r}_{c}={k}_{c}^{\ensuremath{-}1}$ is the correlation radius of the inhomogeneities). The method of the approximate solution of the CPA equation is proposed which has the quick convergence. Effects of exchange narrowing of the magnetic resonance lines and shifts of these lines are calculated in the whole region of the values of the correlation wave number ${k}_{c}$. The approximate analytical expressions for the linewidth of the FMR in the limiting cases of the small and large ${k}_{c}$ are obtained. Large narrowing of the FMR and spin-wave linewidths with the decrease of the correlation radius of inhomogeneities is the substantiation of the main advantage of nanocrystalline and amorphous materials over usual polycrystals when they are used at high frequency devices.

An $\mathsf{\epsilon}$ -expansion for small-world networks

I construct a well-defined expansion in $\epsilon=2-d$ for diffusion processes on small-world networks. The technique permits one to calculate the average over disorder of moments of the Green's function, and is used to calculate the average Green's function and fluctuations to first non-leading order in $\epsilon$, giving results which agree with numerics. This technique is also applicable to other problems of diffusion in random media.

A nonlinear theory of turbulent diffusion

It is shown that the well-known conservation laws for magnetic helicity and passive-scalar fluctuation intensity in the case of negligible molecular diffusion require that the hierarchy of nonlinear equations for the averaged Green function and the hierarchy of Bethe-Salpeter-type equations for fluctuation intensity be treated in a mutually consistent manner. These hierarchies are obtained to the sixth order in turbulent velocity correlators. Asymptotic formulas describing the evolution of scalar fluctuations and magnetic field are presented. A number of new effects are revealed that strongly affect diffusion, but are beyond the scope of the frequently used model of a delta-correlated turbulent field.