  The issue of ergodicity is often underestimated. The presence of
zero-frequency excitations in bosonic Green's functions determine the
appearance of zero-frequency momentum-dependent quantities in correlation
functions. The implicit dependence of matrix elements make such quantities also
relevant in the computation of susceptibilities. Consequently, the correct
determination of these quantities is of great relevance and the
well-established practice of fixing them by assuming the ergodicity of the
dynamics is quite questionable as it is not justifiable a priori by no means.
In this manuscript, we have investigated the ergodicity of the dynamics of the
$z$-component of the spin in the 1D Heisenberg model with anisotropic
nearest-neighbor and isotropic next-nearest-neighbor interactions. We have
obtained the zero-temperature phase diagram in the thermodynamic limit by
extrapolating Exact and Lanczos diagonalization results computed on chains with
sizes $L = 6 \div 26$. Two distinct non-ergodic regions have been found: one
for $J^\prime/J_z \lesssim 0.3$ and $|J_\perp|/J_z < 1$ and another for
$J^\prime/J_z \lesssim 0.25$ and $|J_\perp|/J_z = 1$. On the contrary,
finite-size scaling of $T \neq 0$ results, obtained by means of Exact
diagonalization on chains with sizes $L = 4 \div 18$, indicates an ergodic
behavior of dynamics in the whole range of parameters.
