  We investigate the rate of thermalization of local operators in the
one-dimensional anisotropic antiferromagnetic Heisenberg model with
next-nearest neighbor interactions that break integrability. This is done by
calculating the scaling of the difference of the diagonal and canonical thermal
ensemble values as function of system size, and by directly calculating the
time evolution of the expectation values of the operators with the Chebyshev
polynomial expansion. Spatial and spin symmetry is exploited and the
Hamiltonian is divided in subsectors according to their symmetry. The rate of
thermalization depends on the proximity to the integrable limit. When
integrability is weakly broken thermalization is slow, and becomes faster the
stronger the next-nearest neighbor interaction is. Three different regimes for
the rate of thermalization with respect to the strength of the integrability
breaking parameter are identified. These are shown to be directly connected
with the relative strength of the low and higher energy difference off-diagonal
operator matrix elements in the symmetry eigenbasis of the Hamiltonian. Close
to the integrable limit the off-diagonal matrix elements peak at higher
energies and high frequency fluctuations are important and slow down
thermalization. Away from the integrable limit a strong low energy peak
gradually develops that takes over the higher frequency fluctuations and leads
to quicker thermalization.
