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The issue of ergodicity is often underestimated. The presence ofzero-frequency excitations in bosonic Green's functions determine theappearance of zero-frequency momentum-dependent quantities in correlationfunctions. The implicit dependence of matrix elements make such quantities alsorelevant in the computation of susceptibilities. Consequently, the correctdetermination of these quantities is of great relevance and thewell-established practice of fixing them by assuming the ergodicity of thedynamics 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 anisotropicnearest-neighbor and isotropic next-nearest-neighbor interactions. We haveobtained the zero-temperature phase diagram in the thermodynamic limit byextrapolating Exact and Lanczos diagonalization results computed on chains withsizes $L = 6 \div 26$. Two distinct non-ergodic regions have been found: onefor $J^\prime/J_z \lesssim 0.3$ 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 Exactdiagonalization on chains with sizes $L = 4 \div 18$, indicates an ergodicbehavior of dynamics in the whole range of parameters.Comment: 6 pages, 7 figure

Nonergodic dynamics of the extended anisotropic Heisenberg chain