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Physical review. B, Condensed matter

Spectral shapes of Lennard-Jones chains

Alessandro Cuccoli, V. Tognetti, A. A. Maradudin, Arthur R. McGurn, Ruggero Vaia

Published Sep 1, 1993• 25 citations
Publisher

Abstract

We show how the results of classical molecular-dynamics simulations can be used to improve the calculation of quantum spectral densities of anharmonic crystals. In particular we show that the spectral density C(k,\ensuremath{\omega}) of the displacement correlation function for a quantum chain of N atoms interacting through a nearest-neighbor Lennard-Jones potential, can be calculated accurately in the following way. C(k,\ensuremath{\omega}) is expressed in the form of a continued fraction, whose coefficients are given in terms of its even-frequency moments. The latter, up through the sixth, are calculated from an effective potential that includes the effects of quantum fluctuations. The continued fraction is then terminated by the use of a termination parameter that is determined from a fit of the same continued fraction to the spectral density calculated by means of a classical molecular-dynamics simulation.

DOI
10.1103/physrevb.48.7015

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We show how the results of classical molecular-dynamics simulations can be used to improve the calculation of quantum spectral densities of anharmonic crystals. In particular we show that the spectral density C(k,\ensuremath{\omega}) of the displacement correlation function for a quantum chain of N atoms interacting through a nearest-neighbor Lennard-Jones potential, can be calculated accurately in the following way. C(k,\ensuremath{\omega}) is expressed in the form of a continued fraction, whose coefficients are given in terms of its even-frequency moments. The latter, up through the sixth, are calculated from an effective potential that includes the effects of quantum fluctuations. The continued fraction is then terminated by the use of a termination parameter that is determined from a fit of the same continued fraction to the spectral density calculated by means of a classical molecular-dynamics simulation.
The spectral shape of the displacement correlation function of a quantum chain of atoms interacting through a nearest-neighbor potential is approached at all temperatures and wave vectors by the evaluation of the related frequency moments. The latter ones have been obtained, until the sixth one, by using an effective potential derived by a variational approach to the path-integral formulation of the quantum statistical mechanics. This method allows us to reduce the computation of quantum averages of time-independent functions to classical-like space integrals, so that all the tools developed for classical calculations can be again applied. Explicit results for the Lennard-Jones potential are presented and tested against extensive quantum path-integral Monte Carlo simulations, where an improved Trotter extrapolation procedure is also used. The good agreement between the two calculations confirms the apparent strong simplification introduced by the variational method in the evaluation of the quantum averages when the quantum coupling can be treated semiclassically. The reconstruction of the dynamical behavior of the system through the knowledge of a sufficient number of moments appears realistic. Explicit spectral shapes of Lennard-Jones chains are given, showing the relevance of the quantum effects.
The path integral for the quantum partition function of a chain of atoms with nearest-neighbor interaction is calculated by a variational method. This approach turns out to be very rapid and powerful; it is able to describe the quantum properties of the system in the ``solid'' phase. Application with the Lennard-Jones potential is made and explicit results for internal energy and specific heat are shown. Our technique based on the effective potential offers an alternative to the quantum Monte Carlo approach.
The dynamic correlations of classical and quantum Toda lattices are approached by moment expansion. For the classical model, the moments of the spectral shape of the displacement-displacement correlation function are exactly calculated up to the eighth one, while, for the quantum system, their evaluation is limited to the sixth one, using the effective-potential method in low-coupling approximation. The spectral shape is calculated using the continued-fraction expansion. The relevance of quantum effects is clearly shown, in dependence on temperature and quantum coupling. At all wave vectors, the spectral shape presents a single-peak structure, both in the classical and in the quantum regime.
We present the continued fraction method (CFM) as a microscopic approximation to the spectral density of the Hubbard model in the correlated metal phase away from half filling. The quantity expanded as a continued fraction is the single-particle Green's function. Leading spectral moments are taken into account through a set of real expansion coefficients, as known from the projection technique. Further stages are added to the continued fraction, with complex coefficients, thus defining a terminator function. This enables us to treat the entire spectral range of the Green's function on equal footing and determine the energy scale of the Fermi liquid quasiparticles by minimizing the total energy. The solution is free of phenomenological parameters and remains well defined in the strong-coupling limit, near the doping-controlled metal-insulator transition. Our results for the density of states agree reasonably with those from several variants of the dynamical mean-field theory. The CFM requires minimal numerical effort and can be generalized in several ways that are interesting for applications to real materials.
The spectral shape of one-dimensional systems (describing for instance the behaviour of Frenkel excitons) is approached through the exactly solvable model of the XXO Heisenberg quantum spin chain in a transverse magnetic field. Some results for finite size chains concerning 2N-point correlators are presented in detail. In particular, the finite lattice, finite temperature 2-point correlators are explicitly worked out. Moreover, results in closed form are given for 2N-point correlators in the most general situation (finite lattice and thermodynamic limit, finite temperature, finite space and/or time separations). Their relations with frequency moments of the spectral shape are pointed out and the connection with moment expansion through continued fraction representation is given.