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Physical Review Letters

Long-time dynamics via direct summation of infinite continued fractions

Zhi-Xiong Cai, Surajit Sen, S. D. Mahanti

Published Mar 16, 1992• 33 citations
Publisher

Abstract

Mori theory leads to in(finite) continued fractions [I(F)CF's] which upon inverse Laplace transformation (ILT) give the dynamical correlations in Hamiltonian systems. We propose a direct summation method to evaluate these ICF's, e.g., 1/[z+${\mathrm{\ensuremath{\Delta}}}_{1}$/(z+... \ensuremath{\infty})], by replacing them with FCF's with poles L=${10}^{\mathrm{\ensuremath{\zeta}}}$, 2\ensuremath{\le}\ensuremath{\zeta}\ensuremath{\le}5, for ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{\ensuremath{\mu}}}$=${\mathrm{\ensuremath{\mu}}}^{\mathrm{\ensuremath{\varphi}}}$, \ensuremath{\varphi}2. Long-time dynamics is obtained upon an ILT of the ICF for 0\ensuremath{\le}t\ensuremath{\le}\ensuremath{\tau} with \ensuremath{\tau}=f(\ensuremath{\varphi},\ensuremath{\zeta}) being large. Our studies on dynamical correlations for boundary spins in S=1/2 XY chains agree very well with a recent exact solution for these correlations.

DOI
10.1103/physrevlett.68.1637

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Mori theory leads to in(finite) continued fractions [I(F)CF's] which upon inverse Laplace transformation (ILT) give the dynamical correlations in Hamiltonian systems. We propose a direct summation method to evaluate these ICF's, e.g., 1/[z+\ensuremathΔ1{\mathrm{\ensuremath{\Delta}}}_{1}\ensuremathΔ1​/(z+... \ensuremath{\infty})], by replacing them with FCF's with poles L=10\ensuremathζ{10}^{\mathrm{\ensuremath{\zeta}}}10\ensuremathζ, 2\ensuremath{\le}\ensuremath{\zeta}\ensuremath{\le}5, for \ensuremathΔ\ensuremathμ{\mathrm{\ensuremath{\Delta}}}_{\mathrm{\ensuremath{\mu}}}\ensuremathΔ\ensuremathμ​=\ensuremathμ\ensuremathφ{\mathrm{\ensuremath{\mu}}}^{\mathrm{\ensuremath{\varphi}}}\ensuremathμ\ensuremathφ, \ensuremath{\varphi}2. Long-time dynamics is obtained upon an ILT of the ICF for 0\ensuremath{\le}t\ensuremath{\le}\ensuremath{\tau} with \ensuremath{\tau}=f(\ensuremath{\varphi},\ensuremath{\zeta}) being large. Our studies on dynamical correlations for boundary spins in S=1/2 XY chains agree very well with a recent exact solution for these correlations.
The Mori-Lee formalism for solving the Liouville (or Heisenberg) equation of motion for Hermitian systems demonstrates that the Laplace-transformed dynamical correlations in canonical ensembles can be written as (in)finite continued fractions. We show that a model-independent direct summation method allows accurate numerical evaluation of all known classes of these (in)finite continued fractions that arise in dynamics problems and thus provides a powerful technique to study the dynamics of many-body and few-body systems. Some studies on dynamical correlations in s=1/2 quantum spin chains are cited as applications of the method presented.
The continued-fraction formalism demonstrates that the Laplace-transformed dynamical correlations can be expressed as infinite continued fractions (ICF's). We propose a generalization of the dynamical convergence method (GDCM) of J. Hong and M. H. Lee [Phys. Rev. Lett. 55, 2375 (1985)] of perturbatively evaluating insoluble ICF's when a closely related ICF is exactly soluble. The proposed method overcomes an existing limitation of the dynamical convergence method which involves accurate computing of ratios of small numbers. The limitations are surmounted by exploiting an ``inversion'' property of ICF's. The GDCM allows computationally fast and simple perturbative evaluation of insoluble ICF's with up to ${10}^{\ensuremath{\xi}}$, \ensuremath{\xi}\ensuremath{\le}6, levels of the insoluble ICF when a related ICF is soluble. The method appears to be appropriate for studies of asymptotic behavior of dynamical correlations described by slowly converging and nonconverging ICF's, which are otherwise insoluble, when closely related soluble ICF's exist. The desirable feature of the GDCM is that the computation times required to solve the ICF's are unrelated to the details of the convergence properties. The method has been applied to recalculate the dynamical-spin pair correlation of a recently studied classical XX spin cluster. This application is described in this work.
Two representation of the Laplace transforms of time autocorrelation functions, namely the moment expansion and Mori's continued fraction representation, are studied from the point of view of their convergence, with the aim of obtaining new general properties of time autocorrelation functions. First, the relation between them is established by using mathematical techniques in the case of a general dynamical variable, and a direct method, useful for applications, in the case of Hermitian variables. The mathematical structure associated to Mori's generalized random forces is investigated and it is shown that these random forces can be obtained by a Schmidt orthogonalization of the sequence of initial time derivatives of the dynamical variable considered. Then, the convergence criteria for both representations are examined and an illustration is given with the exactly solvable case of an isotopic impurity in a linear chain of coupled harmonic oscillators. Finally, the question of knowing whether the continued fraction expansion is convergent for any Harmitian dynamical variable and any system is discussed, with its implications for the general behaviour of time autocorrelation functions.
We demonstrate that in Hamiltonian many-body systems at equilibrium, any kind of time dependent correlation function c(t) can always be expanded in a series of (complex) exponential functions of time when its Laplace transform C̃(z) has single poles. The characteristic frequencies can be identified as the eigenfrequencies of the correlation. This is done without introducing the concepts of fluctuating forces and memory functions, due to Mori and Zwanzig and extensively used in the literature in the last decades. Our method is based on a different projection technique in the Hilbert space S of the system and shows that appropriate approximations of the exponential series are related to the contraction of S to a finite, usually small, number of dimensions. The time dependence of correlation functions is always described in detail by a multiple-exponential functionality also at long times. This result is therefore also valid for correlation functions of many-body Hamiltonian systems for which a power-law dependence, observed in restricted time ranges and predicted to be the asymptotic one, can be considered at most as a useful approximate modeling of long-time behavior.
We present a study on the nature of the dynamical on-site spin-pair correlation in an s=1/2 isotropic Heisenberg chain at T=\ensuremath{\infty} using the Mori-Lee formalism for solving the Heisenberg equation of motion and the direct summation method of evaluating infinite continued fractions. Our results are consistent with existing short-time analyses and support a power-law decay of the spin-pair correlations. Our study reveals that the frequency spectra corresponding to the on-site dynamical correlations can be seen as consisting of a Gaussian central peak presumably arising from the XY part of the Hamiltonian and a second broad finite frequency peak, which probably arises from the Ising part of the Hamiltonian and is slightly shifted toward a higher frequency than the Ising frequency.