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We study the matrix elements of local operators in the eigenstates of theintegrable XXZ chain and of the quantum-chaotic model obtained by locallyperturbing the XXZ chain with a magnetic impurity. We show that, at frequenciesthat are polynomially small in the system size, the behavior of the variancesof the off-diagonal matrix elements can be starkly different depending on theoperator. In the integrable model we find that, as the frequency$\omega\rightarrow0$, the variances are either nonvanishing (generic behavior)or vanishing (for a special class of operators). In the quantum-chaotic model,on the other hand, we find the variances to be nonvanishing as$\omega\rightarrow0$ and to indicate diffusive dynamics. We highlight whichproperties of the matrix elements of local operators are different between theintegrable and quantum-chaotic models independently of the specific operatorselected.

Low-frequency behavior of off-diagonal matrix elements in the integrable XXZ chain and in a locally perturbed quantum-chaotic XXZ chain