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arXiv:chao-dyn/9708020 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Structure of eigenstates and local spectral density of states: A three-orbital schematic shell model

Wen-ge Wang, Felix M. Izrailev, Giulio Casati

Published Jan 1, 1998• 53 citations
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Abstract

The average shape of the spectral local density of states (LDOS) and eigenfunctions (EFs) has been studied numerically for a conservative dynamical model (three-orbital Lipkin-Meshkov-Glick model) that can exhibit strong chaos in the classical limit. Attention is paid to the comparison of the shape of the LDOS with that known for random matrix models, as well as to the shape of the EFs, for different values of the perturbation strength. The classical counterparts of the LDOS has also been studied and found to be in remarkable agreement with the quantum calculations. Finally, by making use of a generalization of Brillouin-Wigner perturbation expansion, the form of the long tails of the LDOS and EFs is given analytically and confirmed numerically.

DOI
10.1103/physreve.57.323

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The average shape of the spectral local density of states (LDOS) and eigenfunctions (EFs) has been studied numerically for a conservative dynamical model (three-orbital Lipkin-Meshkov-Glick model) that can exhibit strong chaos in the classical limit. Attention is paid to the comparison of the shape of the LDOS with that known for random matrix models, as well as to the shape of the EFs, for different values of the perturbation strength. The classical counterparts of the LDOS has also been studied and found to be in remarkable agreement with the quantum calculations. Finally, by making use of a generalization of Brillouin-Wigner perturbation expansion, the form of the long tails of the LDOS and EFs is given analytically and confirmed numerically.
For a schematic shell model, we show numerically that, contrary to the behaviour of eigenfunctions, the shapes of the so-called local spectral density of states become close to their forms at extremely strong perturbation (after rescaling) even when the perturbation is relatively weak. The same phenomenon is also found for the random version of the schematic shell model. We suggest that this property of the local spectral density of states may be common to models in which the Hamiltonian matrices in independent particle states have a banded and regular structure.
Quantum–classical correspondence for the average shape of eigenfunctions and the local spectral density of states are well-known facts. In this paper, the fluctuations of the quantum wavefunctions around the classical value are discussed. A simple random matrix model leads to a Gaussian distribution of the amplitudes whose width is determined by the classical shape of the eigenfunction. To compare this prediction with numerical calculations in chaotic models of coupled quartic oscillators, we develop a rescaling method for the components. The expectations are broadly confirmed, but deviations due to scars are observed. This effect is much reduced when both Hamiltonians have chaotic dynamics.
The prediction of the response of a closed system to external perturbations is one of the central problems in quantum mechanics, and in this respect, the local density of states (LDOS) provides an in-depth description of such a response. The LDOS is the distribution of the overlaps squared connecting the set of eigenfunctions with the perturbed one. Here, we show that in the case of closed systems with classically chaotic dynamics, the LDOS is a Breit-Wigner distribution under very general perturbations of arbitrary high intensity. Consequently, we derive a semiclassical expression for the width of the LDOS which is shown to be very accurate for paradigmatic systems of quantum chaos. This Letter demonstrates the universal response of quantum systems with classically chaotic dynamics.
We study the quantum-classical correspondence for systems with interacting spin particles that are strongly chaotic in the classical limit. This is done in the presence of constants of motion associated with the fixed angular momenta of individual spins. Our analysis of the Lyapunov spectra reveals that the largest Lyapunov exponent agrees with the Lyapunov exponent that determines the local instability of each individual spin moving under the influence of all other spins. Within this picture, we introduce a rigorous and simple test of ergodicity for the spin motion, and use it to identify when classical chaos is both strong and global in phase space. In the quantum domain, our analysis of the Hamiltonian matrix in a proper representation allows us to obtain the conditions for the onset of quantum chaos as a function of the model parameters. From the comparison between the quantum and classical domains, we demonstrate that quantum quantities, such as the local density of states (LDoS) and the shape of the chaotic eigenfunctions written in the noninteracting many-body basis, have well-defined classical counterparts. We also find a relationship between the Kolmogorov-Sinai entropy and the width of the LDoS, which is useful for studies of many-body dynamics.
In this paper, we employ a semiperturbative theory to study the statistical structural properties of energy eigenfunctions (EFs) in many-body quantum chaotic systems consisting of a central system coupled to an environment. Under certain assumptions, we derive both the average shape and the statistical fluctuations of EFs on the basis formed by the direct product of the energy eigenbases of the system and the environment. Furthermore, we apply our results to two fundamental questions: (i) the properties of the reduced density matrix of the central system in an eigenstate, and (ii) the structure of the off-diagonal smooth function within the framework of the eigenstate thermalization hypothesis. Numerical results are also presented in support of our main findings.
The Hamiltonian conservative system of two interacting particles has been considered both in classical and quantum description. The quantum model has been realized using a symmetrized two-particle basis reordered in the unperturbed energy. The main attention is paid to the structure of chaotic eigenfunctions (EF's) and to the local spectral density of states (LDOS). A remarkable correspondence has been found for the shapes of EF's and the LDOS in the energy representation to their classical counterparts. Comparison with the band random matrix theory predictions has revealed quite significant differences, which are due to the dynamical nature of the model. On the other hand, a partial agreement is found by inserting randomness ad hoc in the dynamical model for two-body matrix elements. This shows that, at least for small number of particles, care must be taken when classical correlations are neglected. The question of quantum localization in the energy space is discussed for both the dynamical and random models.