Physical review. B, Condensed matter

Frequency moments and spectral shape of quantum chains

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

Published Oct 1, 1992• 36 citations

Abstract

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.

DOI: 10.1103/physrevb.46.8839

Related papers

Spectral shapes of Lennard-Jones chains

Alessandro Cuccoli, V. Tognetti, A. A. Maradudin +2 more

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.

Physical review. B, Condensed matterSep 1, 199325 citations

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

Alessandro Cuccoli, V. Tognetti, Ruggero Vaia

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.

Physical review. B, Condensed matterMay 1, 199034 citations

Dynamic correlations of the classical and quantum Toda lattices

Alessandro Cuccoli, M. Spicci, V. Tognetti +1 more

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.

Physical review. B, Condensed matterApr 1, 199317 citations

Quantum Dynamics in Canonical and Micro-canonical Ensembles. Part II. Tunneling in Double Well Potential

V S Filinov, Yu. E. Lozovik, A V Filinov +2 more

In the second part of this paper on micro-canonical ensemble a new numerical approach for consideration of quantum dynamics and calculations of the average values of quantum operators and time correlation functions in the Wigner representation of quantum statistical mechanics has been developed. The time correlation functions have been presented in the form of the integral of the Weyl's symbol of the considered operators and the Fourier transform of the product of matrix elements of the dynamic propagators. For the latter function the integral Wigner–Liouville's type equation has been derived. The initial condition for this equation has been obtained in the form of the Fourier transform of the Wiener path integral representation of the matrix elements of the propagators at initial time. A numerical procedure for solving this equation combining both molecular dynamics and Monte Carlo methods has been developed. Numerical results have been obtained for series of average values of the quantum operators as well as for the time correlation function characterizing the energy level structure, the momentum flow of tunnelling particles at barrier crossing and the absorption spectra of electrons in a potential well. The developed quantum dynamics method was tested by comparison of numerical results with analytical estimations. Tunnelling transitions and the effect of the quasi stationary state have been considered as the reasons for the peculiarities in behaviour of the time correlation functions and position and momentum dispersions. The possibility of applying the developed approach to the theory of classical wave propagation in random media has also been considered. For classical waves some results have been obtained for Gaussian beam propagation in 2D and 3D waveguides.

arXiv:quant-ph/9811002Physica ScriptaOct 1, 19983 citations

Path-integral approaches to the statistical mechanics of quantum systems: Variational methods for influence functionals

Kurt Allinger

A powerful numerical procedure is developed for calculating temperature-dependent quantities, such as partition functions for quantum systems coupled to thermal or other baths. A variational procedure is worked out to construct model Hamiltonians starting from the general results of the previous paper for influence functionals. Explicit, though intricate, formulas for all relevant quantities are presented. The methods developed are applied to a generalized model for a boson-bath interaction.

Physical Review AJan 1, 19893 citations

Spectral and dynamical properties of strongly correlated systems

Giuseppe Carleo

In the first part of the Thesis we mostly concentrate on spectral properties of strongly correlated systems and on their equilibrium properties. This is accomplished by the general concept of imaginary-time dynamics which we apply to a number of different problems in which different strengths of this approach emerge. In Chapter 1 we introduce the formalism that allows for a connection between the quantum and the classical worlds. The connection is established by means of the imaginary-time quantum evolution which, under certain circumstances, is shown to be equivalent to a classical stochastic process. It is further shown that exact static and spectral properties of correlated systems can be obtained when this mapping is feasible. The relationship between the imaginary-time dynamics in different frameworks such as the path-integral and the perturbative one is also underlined. In Chapter 2 we present a specific implementation of the general ideas previously presented. In particular we introduced an extension to lattice systems of the Reptation Monte Carlo algorithm [30] which benefits of a sampling scheme based on directed updates. Specific improvements over the existing methodologies consist in the unbiased evaluation of the imaginary-time path integrals for bosons and a systematic scheme to improve over the Fixed-node approximation for fermions. Applications to the Hubbard and the Heisenberg models are presented. In Chapter 3 we demonstrate the application of the imaginary-time dynamics to the exact study of spectral properties. Subject of our attention is a highly anharmonic and correlated quantum crystal such as Helium 4 at zero temperature.[33] Concerning this system, we have obtained the first ab-initio complete phonon dispersion in good agreement with neutron spectroscopy experiments. Moreover, we have also studied the density excitations of solid helium in a region of wave-vectors in between the collective (phonon) and the single-particle regimes, where the presence of residual coherence in the dynamics shows analogies between the highly anharmonic crystal and the superfluid phase. In Chapter 4 we introduce a novel method, based on the imaginary-time dynamics, to obtain unbiased estimates of fermionic properties.[34] By means of this method and of a very accurate variational state, we provide strong evidence for the stability of a saturated ferromagnetic phase in the high-density regime of the two-dimensional infinite-U Hubbard model. By decreasing the electron density, we observe a discontinuous transition to a paramagnetic phase, accompanied by a divergence of the susceptibility on the paramagnetic side. This behavior, resulting from a high degeneracy among different spin sectors, is consistent with an infinite-order phase transition scenario. In Chapter 5 the use of imaginary-time dynamics in the context of finite-temperature response functions is highlighted. As an application, we study an intriguing quantum phase featuring both glassy order and Bose-Einstein condensation. [35] We introduce and validate a model for the role of geometrical frustration in the coexistence of off-diagonal long range order with an amorphous density profile. The exact characterization of the response of the system to an external density perturbation is what allows here to establish the existence of a spin-glass phase. The differences between such a phase and the otherwise insulating "Bose glasses" are further elucidated in the Chapter. In the second part of the Thesis we focus our attention on the dynamics of closed systems out of equilibrium. This is accomplished by both non-stochastic exact methods for the dynamics and the introduction of a novel time-dependent Variational Monte Carlo scheme. In Chapter 6 exact diagonalization schemes and renormalization-based methods for one-dimensional systems are introduced. We identify key phenomenological traits resulting from the many-body correlation in closed systems driven sufficiently away from equilibrium.[31] We provide evidences that the dynamics of interacting lattice bosons away from equilibrium can be trapped into extremely long-lived inhomogeneous metastable states. The slowing down of incoherent density excitations above a threshold energy, much reminiscent of a dynamical arrest on the verge of a glass transition, is identified as the key feature of this phenomenon. In Chapter 7 we present an extension to dynamical properties of the Variational Quantum Monte Carlo method.[32] This is accomplished by introducing a general class of time-dependent variational states which is based on the mapping of the many-body dynamics onto an instantaneous ground-state problem. The application of the method to the experimentally relevant quantum quenches of interacting bosons reveals the accuracy and the reliability of the introduced numerical scheme. We indeed obtain for the first time a consistent variational description of the approach to the equilibrium of local observables and underline the origin of the metastability and glassy behavior previously identified. In the very last part we draw our conclusions and show some possible paths for stimulating future research.

Oct 28, 20113 citations