Method Of Recurrence Relations Research Articles

The Memory Function of an Impurity in Mass and Hooke Constant in a Diatomic Chain

The memory function of an impurity with different mass and Hooke constant in a classical diatomic chain is studied by means of the recurrence relations method. The Laplace transform of the memory function has two pairs of resonant poles and three branch cuts. The poles contribute a cosine and the cuts contribute acoustic and optic branches, which are expressed as a convolution of a difference of two sines and an expansion of even-order Bessel functions. The expansion coefficients are integrals of Jacobian elliptic function sn(u) along the real axis in a complex u+iv plane for the acoustic branch, and integrals of nd(v) along a contour parallel to the imaginary axis of the plane for the optical branch, respectively. Different special cases are discussed in detail. It shows that a perfect monatomic or diatomic chain and such a chain with a mass impurity share the same memory function except for a constant factor, and that the pole contribution exists only if the impurity has both different mass and Hooke constant.

Spin dynamics of an Ising chain with bond impurity in a tilt magnetic field

Spin dynamics of quantum Ising system has been the subject of condensed matter physics in the past few decades. In this work, the bond-impurity effects on dynamics of the one-dimensional quantum Ising model with both transverse (Bzi) and longitudinal magnetic field (LMF, Bxi) was studied in the high-temperature limit by the recurrence relations method. Three cases, i.e., the LMF effect on the dynamics of the pure Ising model without impurity (Bxi≠0), the weak- and strong-bond impurity effect on the dynamics of the transverse Ising model without LMF (the impurity strength Jj≠Ji, Bxi=0), and their combining effect (Jj≠Ji, Bxi≠0), were investigated respectively. It is found that the dynamical behaviors of the system depend on both the impurity and the LMF. In general, the central-peak behavior (collective-mode behavior) will be enhanced (depressed) when increasing Jj or Bxi properly. However, when Jj reaches a critical value Jjc , the spin motion will be frozen, i.e., the bond impurity will act as a switch, which may turn off or turn on the spin motion. In contrast, the LMF can active the motion of the frozen spin. Therefore, the dynamical behavior of the Ising system depends on the combining effects of the impurity and the LMF.

Plasmon Resonances of Graphene-Dielectric-Metal Structures Calculated by the Method of Recurrence Relations

Graphene supports surface plasmons in the terahertz range, and compared with noble-metal plasmons, they show an extreme level of field confinement and relatively long propagation distances, with the advantage of being highly tunable via electrostatic field. Nevertheless, its interaction with light is normally rather weak. To obtain a more powerful capability of excite plasmons, a combination of graphene and artificial structures (metamaterials) present a powerful tunability for enhancing light-matter interaction. These features make graphene metamaterials a promising candidate for plasmonics and surface plasmon resonance for biological sensors. In this work, we study the plasmon spectra in a finite number of graphene layers on a metallic-dielectric substrate surrounded by materials with different dielectric constants. It is shown that using standard electromagnetic boundary conditions and solving the recurrence relation (a suitable alternative to transfer matrix method) for the coefficients of the electric potential between graphene layers, an explicit effective dielectric function of the metamaterial can be obtained giving the plasmon dispersion relations. It is found that the metal-dielectric-layered graphene structure supports both high-energy optical plasmon oscillations and out-of-phase low-energy acoustic charge density excitations. Experimentally, the Kretschmann configuration can be used to excite the surface plasmon resonances. It is based on the observation of a sharp minimum in the reflection coefficient versus angle (or wavelength) curve.

Recent Advances in the Calculation of Dynamical Correlation Functions

We review various theoretical methods that have been used in recent years to calculate dynamical correlation functions of many-body systems. Time-dependent correlation functions and their associated frequency spectral densities are the quantities of interest, for they play a central role in both the theoretical and experimental understanding of dynamic properties. In particular, dynamic correlation functions appear in the fluctuation-dissipation theorem, where the response of a many-body system to an external perturbation is given in terms of the relaxation function of the unperturbed system, provided the disturbance is small. The calculation of the relaxation function is rather difficult in most cases of interest, except for a few examples where exact analytic expressions are allowed. For most of systems of interest approximation schemes must be used. The method of recurrence relation has, at its foundation, the solution of Heisenberg equation of motion of an operator in a many-body interacting system. Insights have been gained from theorems that were discovered with that method. For instance, the absence of pure exponential behavior for the relaxation functions of any Hamiltonian system. The method of recurrence relations was used in quantum systems such as dense electron gas, transverse Ising model, Heisenberg model, XY model, Heisenberg model with Dzyaloshinskii-Moriya interactions, as well as classical harmonic oscillator chains. Effects of disorder were considered in some of those systems. In the cases where analytical solutions were not feasible, approximation schemes were used, but are highly model-dependent. Another important approach is the numericallly exact diagonalizaton method. It is used in finite-sized systems, which sometimes provides very reliable information of the dynamics at the infinite-size limit. In this work, we discuss the most relevant applications of the method of recurrence relations and numerical calculations based on exact diagonalizations. The method of recurrence relations relies on the solution to the coefficients of a continued fraction for the Laplace transformed relaxation function. The calculation of those coefficients becomes very involved and, only a few cases offer exact solution. We shall concentrate our efforts on the cases where extrapolation schemes must be used to obtain solutions for long times (or low frequency) regimes. We also cover numerical work based on the exact diagonalization of finite sized systems. The numerical work provides some thermodynamically exact results and identifies some difficulties intrinsic to the method of recurrence relations.

Dynamics of the spin-1/2 Ising two-leg ladder with four-spin plaquette interaction and transverse field.

Multiple-spin coupling is a key tool for investigating magnetic quantum systems. In particular, models with four-spin interactions became of great interest since they have been applied to describe material properties such as superconductivity. In this framework, the dynamics of the spin-1/2 Ising two-leg ladder with four-spin plaquette interaction in a transverse field is investigated. By means of the recurrence relations method, the first four recurrants are determined exactly for the general model, while five exact recurrants are calculated for two particular models. In all cases, higher-order recurrants are computed through extrapolation to obtain time-dependent autocorrelation functions and spectral densities in the long-time regime and infinite-temperature limit. It is found that the relaxation functions decay slower than the expected behavior of the one-dimensional transverse Ising model. The spectral lines, characterized by a Lorentzian behavior in the central body and a Gaussian shape in the tails, display exchange narrowing as the coupling intensities increase.

A monatomic chain with an impurity in mass and Hooke constant

In this paper, a diatomic chain composed of classical harmonic oscillators and an impurity with different mass and Hooke constant is studied by means of the recurrence relations method. The Laplace transform of the momentum autocorrelation function of the impurity has three pairs of resonant poles and three separate branch cuts; however, one pair of poles was found nonphysical, so only two pairs of poles contribute to the momentum autocorrelation function. The three branch cuts give the acoustic and optical branches that are given by a convolution of a sum of sines and an expansion of even-order Bessel functions. The expansion coefficients are integrals of a Jacobian elliptic function along the real axis in a complex plane for the acoustic branch and those along a contour parallel to the imaginary axis for the optical branch, respectively. The ergodicity of momentum of the impurity is also discussed.

Dynamics of the one-dimensional isotropic Heisenberg model with Dzyaloshinskii–Moryia interaction in a random transverse field

The dynamics of the one-dimensional spin-1∕2 isotropic Heisenberg model with Dzyaloshinskii–Moriya (DM) interaction in the presence of a random magnetic field perpendicular to the DM axis (a transverse field) is investigated by using the method of recurrence relations. A bimodal probability distribution function for this transverse field is considered. The first four recurrants, which are used to obtain the short-time expansion for the time-dependent correlation functions, are exactly obtained as function of the probabilities and of the transverse field. A recent proposed extrapolation procedure for the higher-order recurrants is used in order to extend the results to longer times. In the limit of infinite temperature the relaxation and the spectral density functions are computed for several values of the external field and different probability distributions. Depending on the probabilities and on the external fields one has stronger and faster oscillations in the relaxation functions, and a suppression of the central peak in the spectral density.

Effects of a magnetic field on the dynamics of the one-dimensional Heisenberg model with Dzyaloshinskii-Moriya interactions

We use the method of recurrence relations to investigate the dynamics of the one-dimensional spin-$1/2$ Heisenberg model with Dzyaloshinskii-Moriya (DM) interaction in a magnetic field perpendicular to the DM axis. Our results are valid in the high-temperature limit. We determine exactly the first four recurrants, which produce the time-dependent correlation functions. In order to extend our results to longer times, we introduce a new extrapolation procedure for obtaining higher-order recurrants. Our extrapolation mechanism produces good results when compared to the well-known behavior of the relaxation function in the limit of the $\mathit{XY}$ and isotropic Heisenberg models. The relaxation function and the spectral density function are then analyzed for several values of the external field $B$ for the Heisenberg model with DM interactions in one dimension, in the $T\ensuremath{\rightarrow}\ensuremath{\infty}$ limit. We find that the external field produces stronger and faster oscillations in relaxation functions and a suppression of the central peak in the spectral density. That is accompanied by the appearance of a peak centered at a finite frequency, due to enhancement of the collective mode of spins precessing about the external field.

Método das Relações de Recorrência aplicado no problema do elétron em campo magnético constante e uniforme

Resumo Nesse trabalho utilizamos o Método das Relações de Recorrência (MRR) para estudar o comportamento de um elétron na presença de um campo magnético uniforme aplicado ao longo do eixo z de um sistema de referência estabelecido. Encontramos que as funções de autocorrelação temporais nas direções x e y, respectivamente Cx(t) e Cy(t), oscilam periodicamente no tempo, enquanto na direção z, a função de autocorrelação temporal, Cz(t), tem valor constante e igual a um, corroborando o resultado esperado de que o spin do elétron precessa em torno da direção de aplicação do campo magnético uniforme e independente do tempo.

Cut contribution to momentum autocorrelation function of an impurity in a classical diatomic chain

A classic diatomic chain with a mass impurity is studied using the recurrence relations method. The momentum autocorrelation function of the impurity is a sum of contributions from two pairs of resonant poles and three branch cuts. The former results in cosine function and the latter in acoustic and optical branches. By use of convolution theorem, analytical expressions for the acoustic and optical branches are derived as even-order Bessel function expansions. The expansion coefficients are integrals of elliptic functions in the real axis for the acoustic branch and along a contour parallel to the imaginary axis for the optical branch, respectively. An integral is carried out for the calculation of optical branch: ∫0ϕdθ/√((1 − r12 sin2 θ)(1 − r22 sin2 θ)) = igsn−1 (sin ϕ) (r22 > r12 > 1, g is a constant).

A Diatomic Chain with A Mass Impurity

A classic diatomic chain with one mass impurity is studied using the recurrence relations method. The momentum autocorrelation function of the impurity results from contributions of two pairs of resonant poles and three branch cuts. The pole contribution is given by cosine function(s) and the cut contribution is the acoustic and optical branches. The acoustic and optical branches are given by expansions of even-order Bessel function. The expansion coefficients are integrals of elliptic functions in the real axis in a complex plane for the acoustic branch and along a contour parallel to the imaginary axis for the optical branch, respectively. An integral 22 22 1 2 0 d / ( r sin )( r sin ) 1 1 ϕ θ − θ− θ ∫ (r2 2 >r1 2 >1) is evaluated.

Analytical expressions for momentum autocorrelation function of a classic diatomic chain

We use recurrence relations method to study a classical harmonic diatomic chain. The momentum autocorrelation function results from contributions of acoustic and optical branches. By use of convolution theorem, analytical expressions for the acoustic and optical branches are derived as even-order Bessel function expansions. The expansion coefficients are given in terms of integrals of real and complex elliptic functions for the acoustic and optical branches, respectively. Double convolution results respectively in integrals of trigonometric and hyperbolic functions for expansion coefficients of acoustic and optical branches.

Resonance in the polarization spectrum of a two-level atom in a polychromatic field

A numerical solution has been obtained for a two-level atomic system interacting with a 101-component polychromatic field. The analytical method of recurrence relations has confirmed the numerical calculations. The resonance in the polarization spectrum follows the transition frequency.

Momentum autocorrelation function of a classic diatomic chain

A classical harmonic diatomic chain is studied using the recurrence relations method. The momentum autocorrelation function results from contributions of acoustic and optical branches. By use of convolution theorem, analytical expressions for the acoustic and optical contributions are derived as even-order Bessel function expansions with coefficients given in terms of integrals of elliptic functions in real axis and a contour parallel to the imaginary axis, respectively.

Local Dynamics in an Infinite Harmonic Chain

By the method of recurrence relations, the time evolution in a local variable in a harmonic chain is obtained. In particular, the autocorrelation function is obtained analytically. Using this result, a number of important dynamical quantities are obtained, including the memory function of the generalized Langevin equation. Also studied are the ergodicity and chaos in a local dynamical variable.

Momentum autocorrelation function of an impurity in a classical oscillator chain with alternating masses III. Some limiting cases

The momentum autocorrelation function of a mass impurity in a classic diatomic chain is studied using the recurrence relations method. General expressions for the contributions of branch cuts and resonant poles have been derived and illustrated in previous papers I and II, respectively. In the present paper a series of limiting cases that any one of the three masses m0,m1,m2 approaches to zero or infinity are analyzed. It is found that the cases m0→0 and →(2m2)+ are closely related to each other and that the general expressions for the amplitudes are valid also in the limits λ→0 and ∞. The ergodicity in the case m2→0 is studied and the ratio of two specific infinite products is obtained.

Dynamical class of a two-dimensional plasmonic Dirac system.

A current goal in plasmonic science and technology is to figure out how to manage the relaxational dynamics of surface plasmons in graphene since its damping constitutes a hinder for the realization of graphene-based plasmonic devices. In this sense we believe it might be of interest to enlarge the knowledge on the dynamical class of two-dimensional plasmonic Dirac systems. According to the recurrence relations method, different systems are said to be dynamically equivalent if they have identical relaxation functions at all times, and such commonality may lead to deep connections between seemingly unrelated physical systems. We employ the recurrence relations approach to obtain relaxation and memory functions of density fluctuations and show that a two-dimensional plasmonic Dirac system at long wavelength and zero temperature belongs to the same dynamical class of standard two-dimensional electron gas and classical harmonic oscillator chain with an impurity mass.

Momentum autocorrelation function of an impurity in a classical oscillator chain with alternating masses II. Illustrations

A classic diatomic chain with a mass impurity is studied using the recurrence relations method. The dependence of frequency and amplitude upon two mass ratios is illustrated in detail. By examining the figures we can learn a lot about the model, such as the widths of the acoustic and optical branches as well as the gap between them, the frequency and amplitude of the resonant modes, etc. The momentum autocorrelation functions of the impurity in different regions of the configuration space are numerically calculated and illustrated. A theorem is obtained governing the upper and lower frequencies of the acoustic and optical branches. The envelope of the integrand of the cut contribution is determined. The convergence of numerical calculations of time integration for parameter W to its theoretical value is confirmed.

Self-consistent approach to the description of relaxation processes in classical multiparticle systems

The concept of time correlation functions is a very convenient theoretical tool in describing relaxation processes in multiparticle systems because, on one hand, correlation functions are directly related to experimentally measured quantities (for example, intensities in spectroscopic studies and kinetic coefficients via the Kubo-Green relation) and, on the other hand, the concept is also applicable beyond the equilibrium case. We show that the formalism of memory functions and the method of recurrence relations allow formulating a self-consistent approach for describing relaxation processes in classical multiparticle systems without needing a priori approximations of time correlation functions by model dependences and with the satisfaction of sum rules and other physical conditions guaranteed. We also demonstrate that the approach can be used to treat the simplest relaxation scenarios and to develop microscopic theories of transport phenomena in liquids, the propagation of density fluctuations in equilibrium simple liquids, and structure relaxation in supercooled liquids. This approach generalizes the mode-coupling approximation in the Götze-Leutheusser realization and the Yulmetyev-Shurygin correlation approximations.

Time Evolution in a Two-dimensional Ultrarelativistic-like Electron Gas by Recurrence Relations Method

Time Evolution in a Two-dimensional Ultrarelativistic-like Electron Gas by Recurrence Relations Method