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arXiv:nucl-th/9907063

Distribution of matrix elements of random operators

Mahir Saleh Hussein, Mauricio Porto Pato

Published Jul 15, 1999• 0 citations
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Abstract

It is shown that an operator can be defined in the abstract space of random matrices ensembles whose matrix elements statistical distribution simulates the behavior of the distribution found in real physical systems. It is found that the key quantity that determines these distribution is the commutator of the operator with the Hamiltonian. Application to symmetry breaking in quantum many-body systems is discussed.

DOI
10.48550/arxiv.nucl-th/9907063

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It is shown that an operator can be defined in the abstract space of random matrices ensembles whose matrix elements statistical distribution simulates the behavior of the distribution found in real physical systems. It is found that the key quantity that determines these distribution is the commutator of the operator with the Hamiltonian. Application to symmetry breaking in quantum many-body systems is discussed.
We demonstrate the connection between an operator's matrix element distribution and entangling power via numerical simulations of random, pseudo-random, and quantum chaotic operators. Creating operators with a random distribution of matrix elements is more difficult than creating operators that reproduce other statistical properties of random matrices. Thus, operators that fulfill many random matrix statistical properties may not generate states of high multi-partite entanglement. To quantify the randomness of various statistical distributions and, by extension, entangling power, we use properties of interpolating ensembles that transition between integrable and random matrix ensembles.
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.
We explore connections between an operator's matrix-element distribution and its entanglement generation. Operators with matrix-element distributions similar to those of random matrices generate states of high multipartite entanglement. This occurs even when other statistical properties of the operators do not coincide with random matrices. Similarly, operators with some statistical properties of random matrices may not exhibit random matrix element distributions and will not produce states with high levels of multipartite entanglement. Finally, we show that operators with similar matrix-element distributions generate similar amounts of entanglement.
The statistics of gaps between quantum energy levels is a hallmark criterion in quantum chaos and quantum integrability studies. The relevant distributions corresponding to exactly integrable vs. fully chaotic systems are universal and described by the Poisson vs. Wigner-Dyson curves. In the transitional regime between integrability and chaos, the distributions are much less universal and have not been understood quantitatively until now. We point out that the relevant statistics that controls these distributions is that of the matrix elements of the nonintegrable perturbation Hamiltonian in the energy eigenbasis of the unperturbed integrable system. With this insight, we formulate a simple random matrix ensemble that correctly reproduces the level spacing distributions in a variety of test systems. For the distribution of matrix elements appearing in our construction, we furthermore discover surprising universal features: across a variety of physical systems with diverse degrees of freedom, these distributions are dominated by simple power laws.
The study of quantum statistical mechanics begins with a review of the basic principles of quantum mechanics. Schrödinger’s equation is introduced and Eigenstates (or stationary states) are defined. Model probability for quantum statistics is assumed to have a uniform distribution in phases. Wave functions for many-body systems are defined. The density matrix is introduced. The Planck entropy and the microcanonical ensemble are defined. The differences between classical and quantum statistical mechanics are all based on the differing concepts of a microscopic ‘state’. While the classical microscopic state (specified by a point in phase space) determines the exact position and momentum of every particle, the quantum mechanical state determines neither; quantum states can only provide probability distributions for observable quantities.
In the time evolution of isolated quantum systems out of equilibrium, local observables generally relax to a long-time asymptotic value, governed by the expectation values (diagonal matrix elements) of the corresponding operator in the eigenstates of the system. The temporal fluctuations around this value, response to further perturbations, and the relaxation toward this asymptotic value are all determined by the off-diagonal matrix elements. Motivated by this nonequilibrium role, we present generic statistical properties of off-diagonal matrix elements of local observables in two families of interacting many-body systems with local interactions. Since integrability (or lack thereof) is an important ingredient in the relaxation process, we analyze models that can be continuously tuned to integrability. We show that, for generic nonintegrable systems, the distribution of off-diagonal matrix elements is a Gaussian centered at zero. As one approaches integrability, the peak around zero becomes sharper, so the distribution is approximately a combination of two Gaussians. We characterize the proximity to integrability through the deviation of this distribution from a Gaussian shape. We also determine the scaling dependence on system size of the average magnitude of off-diagonal matrix elements.