Time-dependent Green Function Research Articles

Complex time evolution in tensor networks and time-dependent Green's functions

Complex time evolution in tensor networks and time-dependent Green's functions

Кинетические коэффициенты в формализме временны́х функций Грина при конечной температуре

Обсуждается микроскопическое обоснование возникновения диссипации в модельной ферми- или бозе-системе со слабым локальным взаимодействием. Рассматривается динамика равновесных флуктуаций в формализме Келдыша-Швингера, обсуждается связь диссипации с пинчевыми сингулярностями диаграмм теории возмущений. С помощью уравнения Дайсона определен и вычислен в двухпетлевом приближении параметр диссипации. Показано, что он является прямым аналогом кинетического коэффициента Онзагера и связан с наличием затухания в квазичастичном спектре.

Critical dynamics of the superfluid phase transition: Multiloop calculation of the microscopic model.

Results and method of a three-loop renormalization-group calculation in the model of a Bose gas with a local density-density interaction in the formalism of time-dependent Green functions at finite temperature are presented. The results provide support to the recent conjecture [J. Honkonen, M.V. Komarova, Y.G. Molotkov, and M.Y. Nalimov, Nucl. Phys. B 939, 105 (2019)0550-321310.1016/j.nuclphysb.2018.12.015; Y.A. Zhavoronkov, M.V. Komarova, Y.G. Molotkov, M.Y. Nalimov, and J. Honkonen, Theor. Math. Phys. 200, 1237 (2019)0040-577910.1134/S0040577919080142] that the dynamics of the superfluid phase transition is described by a model which belongs to the same universality class as the stochastic model A.

Self-energy method for time-dependent spectral functions of the Anderson impurity model within the time-dependent numerical renormalization group approach

The self-energy method for quantum impurity models expresses the correlation part of the self-energy in terms of the ratio of two Green's functions and allows for a more accurate calculation of equilibrium spectral functions than is possible directly from the one-particle Green's function [Bulla et al., J. Phys.: Condens. Matter 10, 8365 (1998)], for example, within the numerical renormalization group method. In addition, the self-energy itself is a central quantity required in the dynamical mean field theory of strongly correlated lattice models. Here, we show how to generalize the self-energy method to the time-dependent situation for the prototype model of strong correlations, the Anderson impurity model. We use the equation of motion method to obtain closed expressions for the local Green's function in terms of a time-dependent correlation self-energy, with the latter being given as a ratio of a one-particle time-dependent Green's function and a higher-order correlation function. We benchmark this self-energy approach to time-dependent spectral functions against the direct approach within the time-dependent numerical renormalization group method. The self-energy approach improves the accuracy of time-dependent spectral function calculations, and the closed-form expressions for the Green's function allow for a clear picture of the time-evolution of spectral features at the different characteristic time-scales. The self-energy approach is of potential interest also for other quantum impurity solvers for real-time evolution, including time-dependent density matrix renormalization group and continuous-time quantum Monte Carlo techniques.

Time-dependent Green's function generation with Monte Carlo for transient analysis of source-driven systems

The transient analysis of nuclear reactors is an important issue from a safety point of view. The deterministic methods, by considering different approximations, are extensively used to solve the neutron transport equations. Nowadays, by the continuous increase of computational resources, the feasibility of utilization of pure Monte Carlo methodology in the nuclear reactor transient analysis, without using different approximations, is being increased. In the present work, a detailed Dynamic Monte Carlo (DMC) methodology is developed and employed to investigate the transient analysis of source driven nuclear systems. Analytically solvable time and space and time dependent problems are introduced and solved to verify variance reduction tools used in the DMC methodology. Then, the DMC method is utilized to generate time dependent Green's functions for transient analysis of the source-driven systems (SDSs) studied in literature. The equilibrium and post perturbation transient responses of source driven systems are obtained by utilizing the Green's functions generated for those systems. The newly developed novel method is applied to different transient benchmark problems of SDSs. The performance and validity of the proposed method are discussed through qualitative and quantitative comparisons with the results given for the test and benchmark problems.

Magnon- versus Electron-Mediated Spin-Transfer Torque Exerted by Spin Current across an Antiferromagnetic Insulator to Switch the Magnetization of an Adjacent Ferromagnetic Metal

The recent experiment [Y. Wang et al., Science 366, 1125 (2019)] on magnon-mediated spin-transfer torque (MSTT) was interpreted in terms of a picture where magnons are excited within an antiferromagnetic insulator (AFI), by applying nonequilibrium electronic spin density at one of its surfaces, so that their propagation across AFI deprived of conduction electrons eventually leads to reversal of magnetization of a ferromagnetic metal (FM) attached to the opposite surface of AFI. We employ a recently developed time-dependent nonequilibrium Green functions combined with the Landau-Lifshitz-Gilbert equation (TDNEGF+LLG) formalism to evolve conduction electrons quantum-mechanically while they interact via self-consistent back-action with localized magnetic moments described classically by atomistic spin dynamics solving a system of LLG equations. Upon injection of square current pulse as the initial condition, TDNEGF+LLG simulations of FM-polarizer/AFI/FM-analyzer junctions show that reversal of localized magnetic moments within FM-analyzer is less efficient, in the sense of requiring larger pulse height and its longer duration, than conventional electron-mediated STT (ESTT) driving magnetization switching in standard FM-polarizer/normal-metal/FM-analyzer spin valve. Since both electronic, generated by spin pumping from AFI, and magnonic, generated by direct transmission from AFI, spin currents are injected into the FM-analyzer, its localized magnetic moments will experience combined MSTT and ESTT. Nevertheless, by artificially turning off ESTT we demonstrate that MSTT plays a dominant role whose understanding, therefore, paves the way for all-magnon-driven magnetization switching devices with no electronic parts.

Green's function for the Schrödinger equation with a generalized point interaction and stability of superoscillations

In this paper we study the time dependent Schrödinger equation with all possible self-adjoint singular interactions located at the origin, which include the δ and δ′-potentials as well as boundary conditions of Dirichlet, Neumann, and Robin type as particular cases. We derive an explicit representation of the time dependent Green's function and give a mathematical rigorous meaning to the corresponding integral for holomorphic initial conditions, using Fresnel integrals. Superoscillatory functions appear in the context of weak measurements in quantum mechanics and are naturally treated as holomorphic entire functions. As an application of the Green's function we study the stability and oscillatory properties of the solution of the Schrödinger equation subject to a generalized point interaction when the initial datum is a superoscillatory function.

General time-dependent Green's functions of line forces in a two-dimensional, anisotropic, elastic, and infinite solid

In this paper, we derive analytical time-dependent Green's functions in a two-dimensional, anisotropic elastic, and infinite solid. It is based on the Stroh formalism combined with application of the Cauchy's residue theorem. Final expressions of the Green's function are in terms of simple finite line integral from 0 to 2π. The time-dependence of the line forces can be impulsive, Heaviside, or within a given time duration. The space-dependence of the line force is very general, including concentrated or uniformly distributed sources within a given line interval. Green's functions of both displacements and stresses are derived in analytical forms, and are verified against existing results. Numerical examples are presented to demonstrate the effect of source types and material anisotropy on the Green's functions.

Deformation of a spherical, viscoelastic, and incompressible Earth for a point load with periodic time change

SUMMARY Planetary-scale mass redistributions occur on Earth for certain spatiotemporal periods, and these surface mass changes excite the global periodic loading deformations of a viscoelastic Earth. However, the characteristics of periodic viscoelastic deformations have not been well investigated even in a simple earth model. In this study, we derive the semi-analytical Green's functions (fully analytical Love numbers) for long-standing point sources with given periods using a modified asymptotic scheme in a homogeneous Maxwell spherical earth model. Here, the asymptotic scheme is needed in order to obtain accurate semi-analytical time-dependent Green's functions. The amplitudes and phases of the Green's functions may be biased if only the series summations of the Love numbers are used because the influence of viscoelasticity is degree-dependent. We compare the viscoelastic and elastic periodic Green's functions with different material viscosities and loading periods and investigate the amplitude increase percentage and phase delay of the periodic displacement and geoid change. For example, our analysis revealed that the viscosity increases the amplitude by 40–120 per cent and delays the phase approximately −100° to 60° for the displacement and geoid change when bearing a 10-yr loading period, assuming a viscosity of 1018 Pa s and a shear modulus 4 × 1010 Pa.

Renormalization Group in Non-Relativistic Quantum Statistics

Dynamic behaviour of a boson gas near the condensation transition in the symmetric phase is analyzed with the use of an effective large-scale model derived from time-dependent Green functions at finite temperature. A renormalization-group analysis shows that the scaling exponents of critical dynamics of the effective multi-charge model coincide with those of the standard model A. The departure of this result from the description of the superfluid transition by either model E or F of the standard phenomenological stochastic models is corroborated by the analysis of a generalization of model F, which takes into account the effect of compressible fluid velocity. It is also shown that, contrary to the single-charge model A, there are several correction exponents in the effective model, which are calculated at the leading order of the ɛ= 4 − d expansion.

Time-dependent Green's function approach to spin transport assisted by nonclassical light

The generation of spin current is of significant importance for advanced spintronic applications. In conventional theory and experiment, microwave sources are usually described by a classical time-varying field in which all fluctuations are neglected. Here, we study the spin current driven by nonclassical light by developing a Green's function method in which both effects of nonclassical light and electron-related interaction can be taken into account. Our theoretical method has more advantages than the linear response theory developed recently as it can be easily extended to the electronic transport problem with interaction. As an example, we calculate the spin current due to the rotating magnetic field in the presence of nonclassical light. It is found that the spin current is sensitive to the state of nonclassical light. Under certain conditions, the transmission and spin current can be smaller than the classical limit, which cannot be broken by a pure classical microwave field. The method developed here and the results reported here will be important for understanding complicated physical phenomena occurring in the hybrid field of spintronics and circuit quantum electrodynamics.

Reversed electromagnetic Vavilov-Čerenkov radiation in naturally existing magnetoelectric media

We consider two semi-infinite magnetoelecteric media separated by a planar interface whose electromagnetic response is described by axion-electrodynamics. The time dependent Green's function characterizing this geometry is obtained by a method which can be directly generalized to cylindrical and spherical configurations of two magnetoelectrics separated by an interface. We establish the far-field approximation of the Green's function and apply these results to the case of a charged particle moving from one medium to the other at a high constant velocity perpendicular to the interface. From the resulting angular distribution of the radiated energy per unit frequency we provide theoretical evidence for the emergence of reversed Vavilov-\v{C}erenkov radiation in naturally existing magnetoelectric media. In the case where one of the magnetolectrics is a 3D topological insulator, ${\rm TlBiSe}_2$ for example, located in front of a regular insulator, we estimate that an average forward Vavilov-\v{C}erenkov radiation with frequency $\sim 2.5 \,\, {\rm eV}$ ($\sim 500\,\, {\rm nm}$) will produce a highly suppressed reversed Vavilov-\v{C}erenkov radiation which can be characterized by an effective frequency in the range of $\sim (4\times 10^{-3}-0.5) \,\, {\rm meV}$. However, this value compares favorably with recent measurements in left-handed metamaterials yielding reversed Vavilov-\v{C}erenkov radiation with frequencies of the order of $(1.2-3.9)\times 10^{-2}\,\, {\rm meV}$.

Time-retarded damping and magnetic inertia in the Landau-Lifshitz-Gilbert equation self-consistently coupled to electronic time-dependent nonequilibrium Green functions

The conventional Landau-Lifshitz-Gilbert (LLG) equation is a widely used tool to describe dynamics of local magnetic moments, viewed as classical vectors of fixed length, with their change assumed to take place simultaneously with the cause. Here we demonstrate that recently developed [M. D. Petrovi\'{c} {\em et al.}, {\tt arXiv:1802.05682}] self-consistent coupling of the LLG equation to time-dependent quantum-mechanical description of electrons microscopically generates time-retarded damping in the LLG equation described by a memory kernel which is also spatially dependent. For sufficiently slow dynamics of local magnetic moments, the memory kernel can be expanded to extract the Gilbert damping (proportional to first time derivative of magnetization) and magnetic inertia (proportional to second time derivative of magnetization) terms whose parameters, however, are time-dependent in contrast to time-independent parameters used in the conventional LLG equation. We use examples of single or multiple magnetic moments precessing in an external magnetic field, as well as field-driven motion of a magnetic domain wall (DW), to quantify the difference in their time evolution computed from conventional LLG equation vs. TDNEGF+LLG quantum-classical hybrid approach. The faster DW motion predicted by TDNEGF+LLG approach reveals that important quantum effects, stemming from finite amount of time which it takes for conduction electron spin to react to the motion of classical local magnetic moments, are missing from conventional classical micromagnetics simulations. We also demonstrate large discrepancy between TDNEGF+LLG-computed numerically exact and, therefore, nonperturbative result for charge current pumped by a moving DW and the same quantity computed by perturbative spin motive force formula combined with the conventional LLG equation.

Effective large-scale model of boson gas from microscopic theory

An effective large-scale model of interacting boson gas at low temperatures is constructed from first principles. The starting point is the generating function of time-dependent Green functions at finite temperature. The perturbation expansion is worked out for the generic case of finite time interval and grand-canonical density operator with the use of the S-matrix functional for the generating function. Apparent infrared divergences of the perturbation expansion are pointed out. Regularization via attenuation of propagators is proposed and the relation to physical dissipation is studied. Problems of functional-integral representation of Green functions are analyzed. The proposed large-scale model is explicitly renormalized at the leading order.

Analytical solutions for the quantum parametric oscillator from corresponding classical dynamics via a complex Riccati equation

The time-dependent Schrödinger equation for quadratic Hamiltonians has Gaussian wave packets as exact solutions. For the parametric oscillator with frequency ω(t), the width of these wave packets must be time-dependent. This time-dependence can be determined via the solution of a complex nonlinear Riccati equation or an equivalent real nonlinear Ermakov equation. All quantum dynamical properties of the system can easily be constructed from these solutions, e.g., uncertainties of position and momentum, their correlations, tunnelling currents, ground state energies, etc. In addition, the link to the corresponding classical dynamics is supplied via linearization of the Riccati equation to a complex Newtonian equation, actually representing two equations of the same kind: one for the real and one for the imaginary part. If the solution of one part is known, the missing (linear independent) solution of the other can be obtained via a conservation law for the motion in the complex plane. Knowing these two solutions, the solution of the Ermakov equation can be determined immediately plus the explicit expressions for all the quantum dynamical properties mentioned above. Furthermore, these two solutions also provide the time-dependent Green function for the systems propagation for any given initial condition, not only for Gaussians. It is also possible to obtain the corresponding Wigner function and generalized creation and annihilation operators in this way. In searching for problems with analytical solutions, the crucial point is to identify systems that have either a wave packet solution with time-dependent width (in closed form) that obeys a Riccati equation, or a classical parametric oscillator with frequency ω(t) that still enables an analytical solution of the Newtonian equation. Comparing the wave packets with the solution of the diffusion equation – also a Gaussian with time-dependent width – leads to a real Riccati equation. A first attempt at “complexifying” this equation turns out to be problematic. Nevertheless, it provides a clue to time-dependent frequencies inversely proportional to time that lead to Newtonian equations with analytical solutions. Following a detailed analysis of the corresponding quantum dynamics and energetics, possible extensions of the method are indicated.

Translation invariance and antisymmetry in the theory of the nucleon optical model

The first step in any formalism that aims to connect a many-nucleon theory of nucleon-nucleus scattering and the concept of an optical model potential in the sense pioneered by Feshbach is to explain what is meant by the optical model wavefunction. By definition this is a function of a single space coordinate plus a set of single nucleon internal variables. This article gives a critique of the definitio as it is frequently expressed in 2nd Quantisation language and suggest a new definition which is more consistent with the requirements of antisymmetry and translational invariance. A modification of the time-dependent Green's function formalism is suggested.

Photodetachment microscopy in time-dependent fields

Photodetachment of negative ions in combined laser and low-frequency fields is investigated. The time-dependent Green's function method is used for calculation of electron flux at a macroscopic distance from the photodetachment source, typical for a photodetachment microscopy experiment. In calculating the electron flux, we use the stationary phase method for the time integral, equivalent to the semiclassical approximation, to compute the time-dependent wave function. The stationary points ${t}_{1}^{(i)}, i=1,...,n$ correspond to time instances of launching of classical trajectories arriving at the detector at a given spacetime point $(\mathbf{r},t)$. The number of trajectories $n$ contributing to the electron flux at any point in the classically allowed spacetime domain can be controlled by varying the switching interval of the high-frequency laser which initiates the photodetachment process. The divergences inherent in the electron flux in the semiclassical treatment are removed by using the uniform Airy approximation near the caustics.

Nonlinear Luttinger liquid: Exact result for the Green function in terms of the fourth Painlevé transcendent

We show that exact time dependent single particle Green function in the Imambekov–Glazman theory of nonlinear Luttinger liquids can be written, for any value of the Luttinger parameter, in terms of a particular solution of the Painlevé IV equation. Our expression for the Green function has a form analogous to the celebrated Tracy–Widom result connecting the Airy kernel with Painlevé II. The asymptotic power law of the exact solution as a function of a single scaling variable x/\sqrt{t}x/t agrees with the mobile impurity results. The full shape of the Green function in the thermodynamic limit is recovered with arbitrary precision via a simple numerical integration of a nonlinear ODE.

Spatio-temporal imaging of light transport in highly scattering media under white light illumination

Imaging the propagation of light in time and space is crucial in optics, notably for the study of complex media. We here demonstrate the passive measurement of time-dependent Green's functions between every point at the surface of a strongly scattering medium by means of low coherence interferometry. The experimental access to this Green's matrix is essential since it contains all the information about the complex trajectories of light within the medium. In particular, the spatio-temporal spreading of the diffusive halo and the coherent backscattering effect can be locally investigated in the vicinity of each point acting as a virtual source. On the one hand, this approach allows a quantitative imaging of the diffusion constant in the scattering medium with a spatial resolution of the order of a few transport mean free paths. On the other hand, our approach is able to reveal and quantify the anisotropy of light diffusion, which can be of great interest for optical characterization purposes. This study opens important perspectives both in optical diffuse tomography with potential applications to biomedical imaging and in fundamental physics for the experimental investigation of Anderson localization.

Antidamping spin-orbit torque driven by spin-flip reflection mechanism on the surface of a topological insulator: A time-dependent nonequilibrium Green function approach

Motivated by recent experiments observing spin-orbit torque (SOT) acting on the magnetization $\stackrel{P\vec}{m}$ of a ferromagnetic (F) overlayer on the surface of a three-dimensional topological insulator (TI), we investigate the origin of the SOT and the magnetization dynamics in such systems. We predict that lateral F/TI bilayers of finite length, sandwiched between two normal metal leads, will generate a large anti-damping-like SOT per very low charge current injected parallel to the interface. The large values of anti-damping-like SOT are spatially localized around the transverse edges of the F overlayer. Our analysis is based on adiabatic expansion (to first order in $\ensuremath{\partial}\stackrel{P\vec}{m}/\ensuremath{\partial}t$) of time-dependent nonequilibrium Green functions (NEGFs), describing electrons pushed out of equilibrium both by the applied bias voltage and by the slow variation of a classical degree of freedom [such as $\stackrel{P\vec}{m}(t)$]. From it we extract formulas for spin torque and charge pumping, which show that they are reciprocal effects to each other, as well as Gilbert damping in the presence of SO coupling. The NEGF-based formula for SOT naturally splits into four components, determined by their behavior (even or odd) under the time and bias voltage reversal. Their complex angular dependence is delineated and employed within Landau-Lifshitz-Gilbert simulations of magnetization dynamics in order to demonstrate capability of the predicted SOT to efficiently switch $\stackrel{P\vec}{m}$ of a perpendicularly magnetized F overlayer.