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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Band structure of the Hamiltonian matrix of a real ‘‘chaotic’’ system: The Ce atom

A. A. Gribakina, V. V. Flambaum, G. F. Gribakin

Published Nov 1, 1995• 19 citations
Publisher

Abstract

The properties and structure of the Hamiltonian matrix of a realistic many-body quantum chaotic system (the rare-earth atom of Ce) are analyzed and compared with those assumed in band random matrix theories. The sparsity of the matrix and the behavior of the mean squared matrix elements 〈${\mathit{H}}_{\mathit{i}\mathit{j}}^{2}$${\mathrm{〉}}_{\mathrm{\ensuremath{\Vert}}\mathit{i}\mathrm{\ensuremath{-}}\mathit{j}\mathrm{\ensuremath{\Vert}}=\mathrm{\ensuremath{\Delta}}}$ as function of the distance \ensuremath{\Vert}i-j\ensuremath{\Vert} from the diagonal are studied. Fitting 〈${\mathit{H}}_{\mathit{i}\mathit{j}}^{2}$${\mathrm{〉}}_{\mathrm{\ensuremath{\Vert}}\mathit{i}\mathrm{\ensuremath{-}}\mathit{j}\mathrm{\ensuremath{\Vert}}=\mathrm{\ensuremath{\Delta}}}$ with the exponent ${\mathit{H}}_{0}^{2}$exp(-\ensuremath{\Delta}/b) yields the bandwidths of b=54 and 79 for the matrices of ${\mathit{J}}^{\mathrm{\ensuremath{\pi}}}$=${4}^{+}$, and ${4}^{\mathrm{\ensuremath{-}}}$ states, respectively.

DOI
10.1103/physreve.52.5667

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arXiv:nlin/0102033•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jun 25, 2001•42 citations
The properties and structure of the Hamiltonian matrix of a realistic many-body quantum chaotic system (the rare-earth atom of Ce) are analyzed and compared with those assumed in band random matrix theories. The sparsity of the matrix and the behavior of the mean squared matrix elements 〈{\mathit{H}}_{\mathit{i}\mathit{j}}^{2}$${\mathrm{〉}}_{\mathrm{\ensuremath{\Vert}}\mathit{i}\mathrm{\ensuremath{-}}\mathit{j}\mathrm{\ensuremath{\Vert}}=\mathrm{\ensuremath{\Delta}}} as function of the distance \ensuremath{\Vert}i-j\ensuremath{\Vert} from the diagonal are studied. Fitting 〈{\mathit{H}}_{\mathit{i}\mathit{j}}^{2}$${\mathrm{〉}}_{\mathrm{\ensuremath{\Vert}}\mathit{i}\mathrm{\ensuremath{-}}\mathit{j}\mathrm{\ensuremath{\Vert}}=\mathrm{\ensuremath{\Delta}}} with the exponent H02{\mathit{H}}_{0}^{2}H02​exp(-\ensuremath{\Delta}/b) yields the bandwidths of b=54 and 79 for the matrices of J\ensuremathπ{\mathit{J}}^{\mathrm{\ensuremath{\pi}}}J\ensuremathπ=4+{4}^{+}4+, and 4\ensuremath−{4}^{\mathrm{\ensuremath{-}}}4\ensuremath− states, respectively.
A theory of strongly interacting Fermi systems of a few particles is developed. At high excitation energies (a few times the single-particle level spacing) these systems are characterized by an extreme degree of complexity due to strong mixing of the shell-model-based many-particle basis states by the residual two-body interaction. This regime can be described as many-body quantum chaos. Practically, it occurs when the excitation energy of the system is greater than a few single-particle level spacings near the Fermi energy. Physical examples of such systems are compound nuclei, heavy open shell atoms (e.g. rare earths) and multicharged ions, molecules, clusters and quantum dots in solids. The main quantity of the theory is the strength function which describes spreading of the eigenstates over many-particle basis states (determinants) constructed using the shell-model orbital basis. A nonlinear equation for the strength function is derived, which enables one to describe the eigenstates without diagonalization of the Hamiltonian matrix. We show how to use this approach to calculate mean orbital occupation numbers and matrix elements between chaotic eigenstates and introduce typically statistical variables such as temperature in an isolated microscopic Fermi system of a few particles.
Phenomena in quantum chaotic systems such as spectral fluctuations are known to be described remarkably well by the random matrix theory (RMT) in small energy intervals and over large time scales. However, in exchange for analytical tractability, RMT has been constructed to leave out an important element of realistic physical systems { the notion of locality. Since quantum information propagates at a finite speed in a spatially-extended quantum chaotic system, there exists time scales before which RMT is effective. How do we describe the non-equilibrium quantum dynamics at intermediate timescales which are much longer than the local relaxation time but too short for RMT to apply? What are the signatures of many-body quantum chaotic systems beyond the description of RMT? In this thesis, we address these questions by analysing minimal models that incorporate a notion of locality and at the same time retain some analytical tractability of RMT. More specifically, we study Floquet quantum circuits that are random in space but periodic in time. These circuits consist of local and random quantum gates that act on a q-state `spin' at each site. After a brief review of classical chaos, quantum chaos, and quantum circuits in Chapter 1, we present three main results. In Chapter 2, we formulate a diagrammatical approach for computing ensemble averages of observables over random unitary circuits, and derive exact results on spectral statistics and dynamical observables for a Floquet circuit in the large-q limit. In Chapter 3, we further investigate the spectral statistics in spatiallyextended many-body quantum chaotic systems by computing the spectral form factor K(t) analytically and numerically. We show that K(t) follows the RMT description at times longer than the Thouless time tTh and identify the scaling of tTh at the large-q limit of a Floquet circuit. In Chapter 4, we discuss eigenstate correlations for spatially-extended many-body quantum chaotic systems, in terms of the statistical properties of matrix elements of local observables. While the eigenstate thermalization hypothesis (ETH) is known to give an excellent description of these quantities, the phenomenon of scrambling and the butterfly effect imply structure beyond ETH. We determine analytically and verify numerically the universal form of this structure at long distances and small eigenvalue separations for Floquet systems.
There is a newly emerging understanding that in the chaotic domain of isolated finite interacting many particle systems smoothed densities define the statistical description of these systems and these densities follow from embedded (two-body) random matrix ensembles and their various deformations. These ensembles predict that the smoothed form of matrix elements of a transition operator between the chaotic eigenstates weighted by the densities at the two ends (i.e. the bivariate strength density) will be a bivariate Gaussian with the bivariate correlation coefficient arising out of the non-commutability of the hamiltonian and the transition operator involved. The ensemble theory extends to systems with a mean-field and a chaos generating two-body interaction (as in nuclei, atoms and diffusive quantum dots). These developments in many-body quantum chaos are described with special reference to one-body transition operators.
The structure of few-body Hamiltonian matrices is studied in the semiclassical regime. Given (A${^}_{1}$(q^,p^),A${^}_{2}$ (q^,p^)), a pair of operators, it can be shown that, under quite general conditions, A${^}_{2}$ takes the form of a banded matrix in the ordered eigenrepresentation of A${^}_{1}$. Moreover, the bandwidth depends only on \ensuremath{\Elzxh} and certain generalized microcanonical averages. In particular, if H=H0{H}_{0}H0​+\ensuremath{\epsilon}V, this implies that the perturbed Hamiltonian is banded in the appropriately ordered unperturbed basis.
A theory of strongly interacting Fermi systems of a few particles is developed.At high excitation energies (a few times the single-particle level spacing) these systems are characterized by an extreme degree of complexity due to strong mixing of the shell-model-based many-particle basis states by the residual twobody interaction.This regime can be described as many-body quantum chaos.Practically, it occurs when the excitation energy of the system is greater than a few single-particle level spacings near the Fermi energy.Physical examples of such systems are compound nuclei, heavy open shell atoms (e.g.rare earths) and multicharged ions, molecules, clusters and quantum dots in solids.The main quantity of the theory is the strength function which describes spreading of the eigenstates over many-particle basis states (determinants) constructed using the shell-model orbital basis.A nonlinear equation for the strength function is derived, which enables one to describe the eigenstates without diagonalization of the Hamiltonian matrix.We show how to use this approach to calculate mean orbital occupation numbers and matrix elements between chaotic eigenstates and introduce typically statistical variables such as temperature in an isolated microscopic Fermi system of a few particles.
Random matrix ensembles defined by a mean-field one body plus a chaos generating random two-body interaction [called embedded ensembles of (1+2)-body interactions] predict for wave functions, in the chaotic domain, an essentially one-parameter Gaussian forms for the energy dependence of the number of principal components (NPC) and the localization length l(H) (defined by information entropy), which are two important measures of chaos in finite interacting many-particle systems. Numerical embedded ensemble calculations and nuclear shell-model results, for NPC and l(H), are compared with the theory. These analyses clearly point out that for realistic finite interacting many-particle systems, in the chaotic domain, wave-function structure is given by (1+2)-body embedded random matrix ensembles.