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Physical review. B, Condensed matter

Thermodynamic properties of a quantum chain with nearest-neighbor anharmonic interactions

Alessandro Cuccoli, V. Tognetti, Ruggero Vaia

Published May 1, 1990• 34 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.41.9588

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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 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 approach is used for determining quantum corrections to the free energy of nonlinear systems. An effective potential, to be inserted in the configurational integral, is constructed by a new variational scheme which gives the quantum modification of the potential due to the anharmonic part together with the frequency renormalization. The single anharmonic oscillator is studied first. Then the extension to fields is presented and a new general expression for the partition function is given. The method is applied to the sine-Gordon chain and explicit calculations of the specific heat are shown.
The path-integral method is used for determination of the quantum corrections to the free energy of nonlinear systems. All quantum effects of the harmonic part of the potential are considered and a variational principle is used to account for the quantum corrections due to the anharmonic part. Correct renormalized frequencies are obtained at any temperature and an effective potential to be inserted in the configurational integral is found. A new general expression for the partition function at any temperature in the low-coupling limit is obtained.
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.
This is a continuation of an effort to reduce to the solution of a characteristic value problem the rigorous calculation of thermodynamic properties of systems in which the intermolecular forces are sufficiently short ranged so that practically the entire potential energy of the system results from interactions between nearest neighbors. The partition function of such a system can be expressed in terms of the largest characteristic value of a linear operator equation and finally as the ratio of partition functions of systems with relatively few particles whose potential energy functions differ somewhat from those in the original system. A method of evaluating grand partition functions and one of introducing interactions between more distant neighbors are discussed. The general theory is applied to the calculation of magnetization, internal energy, and specific heat of two-dimensional ferromagnets on the basis of the Ising model. There seems to exist a λ-point phase transition in the change from ferromagnetic to nonferromagnetic states.