Nonergodic dynamics of the extended anisotropic Heisenberg chain

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 one-dimensional 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 $L=6–26$. Two distinct nonergodic regions have been found: one for $J^{\prime }∕J_z\lesssim 0.3$ and $\mid J_{\perp }\mid ∕J_z<1$ and another for $J^{\prime }∕J_z\lesssim 0.25$ and $\mid J_{\perp }\mid ∕J_z=1$. On the contrary, finite-size scaling of $T\ne 0$ results, obtained by means of Exact diagonalization on chains with sizes $L=4–18$, indicates an ergodic behavior of dynamics in the whole range of parameters.

Figure 1

Zero-temperature ergodicity phase diagram in the $J^{\prime }-J_{\perp }$ plane. Due to the symmetry of the Hamiltonian only the upper half is shown (see in the text). Only two ergodic phases ($E$-I and $E$-II) have been found in the reported parameter space. The others are either nonergodic (NE-I and NE-II) or impossible to conclusively analyze $\left(W\right)$. The latter phase might shrink to a transition line in the bulk limit.

Figure 2

Finite-size scaling in the case of $T=0$ for different points in the phase diagram of Fig. 1. Symbols on panel (a): $+$ corresponds to (NE-II), $▴$ corresponds to (NE-I), $∎$ corresponds to ($E$-I) and $엯$ to ($E$-II) regions of Fig. 1, respectively. On panel (b) different examples from the $W$ region are shown. Hamiltonian couplings are shown in the legend. All energies are expressed in units of $J_z$.

Figure 3

Finite-size scaling of the finite-temperature results for different representative points on the phase diagram of Fig. 1. (a) NE-I phase. (b) NE-II phase. (c), (d) two sample points from the weird $\left(W\right)$ phase, which at $T=0$ show quite distinct finite-size scaling and at $T>0$ behave very similarly. All energies are measured in units of $J_z$. All the data suggest that the bulk system at $T>0$ is always ergodic.