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 ${J}^{\ensuremath{'}}∕{J}_{z}\ensuremath{\lesssim}0.3$ and $\ensuremath{\mid}{J}_{\ensuremath{\perp}}\ensuremath{\mid}∕{J}_{z}<1$ and another for ${J}^{\ensuremath{'}}∕{J}_{z}\ensuremath{\lesssim}0.25$ and $\ensuremath{\mid}{J}_{\ensuremath{\perp}}\ensuremath{\mid}∕{J}_{z}=1$. On the contrary, finite-size scaling of $T\ensuremath{\ne}0$ 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.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 ${J}^{\ensuremath{'}}∕{J}_{z}\ensuremath{\lesssim}0.3$ and $\ensuremath{\mid}{J}_{\ensuremath{\perp}}\ensuremath{\mid}∕{J}_{z}<1$ and another for ${J}^{\ensuremath{'}}∕{J}_{z}\ensuremath{\lesssim}0.25$ and $\ensuremath{\mid}{J}_{\ensuremath{\perp}}\ensuremath{\mid}∕{J}_{z}=1$. On the contrary, finite-size scaling of $T\ensuremath{\ne}0$ 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.

arXiv:cond-mat/0606152Physical Review BSep 26, 200616 citations

Relaxation in an anisotropic Heisenberg model

Ivana Dimitrova, Niklas Jepsen, Jesse Amato-Grill +4 more

Bulletin of the American Physical SocietyJun 4, 20200 citations

Non‐ergodicity of the 1D Heisenberg model

M. Bąk, Adolfo Avella, Francesco Paolo Mancini

Abstract The relevance of zero‐energy functions, coming from zero‐energy modes and present in the structure of bosonic Green's functions, is often underestimated. Usually, their values are fixed by assuming the ergodicity of the dynamics, but it can be shown that this is not always correct. As the zero‐energy functions are connected to fundamental response properties of the system under analysis (specific heat, compressibility, susceptibility, etc.), their correct determination is not an irrelevant issue. In this paper we present some results regarding the zero‐energy functions for the Heisenberg chain of spin‐1/2 with periodic boundary conditions as functions of the number of sites, temperature, and magnetic field. Calculations are pursued for finite chains, using equations of motion, exact diagonalization and Lanczos technique, and the extrapolation to the thermodynamic limit is studied.Abstract The relevance of zero‐energy functions, coming from zero‐energy modes and present in the structure of bosonic Green's functions, is often underestimated. Usually, their values are fixed by assuming the ergodicity of the dynamics, but it can be shown that this is not always correct. As the zero‐energy functions are connected to fundamental response properties of the system under analysis (specific heat, compressibility, susceptibility, etc.), their correct determination is not an irrelevant issue. In this paper we present some results regarding the zero‐energy functions for the Heisenberg chain of spin‐1/2 with periodic boundary conditions as functions of the number of sites, temperature, and magnetic field. Calculations are pursued for finite chains, using equations of motion, exact diagonalization and Lanczos technique, and the extrapolation to the thermodynamic limit is studied.

arXiv:cond-mat/0209527physica status solidi (b)Mar 1, 20038 citations

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 ${J}^{\ensuremath{'}}∕{J}_{z}\ensuremath{\lesssim}0.3$ and $\ensuremath{\mid}{J}_{\ensuremath{\perp}}\ensuremath{\mid}∕{J}_{z}<1$ and another for ${J}^{\ensuremath{'}}∕{J}_{z}\ensuremath{\lesssim}0.25$ and $\ensuremath{\mid}{J}_{\ensuremath{\perp}}\ensuremath{\mid}∕{J}_{z}=1$. On the contrary, finite-size scaling of $T\ensuremath{\ne}0$ 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.