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arXiv:cond-mat/0209527 · physica status solidi (b)

Non‐ergodicity of the 1D Heisenberg model

M. Bąk, Adolfo Avella, Francesco Paolo Mancini

Published Mar 1, 2003• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1002/pssb.200301688

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We report a detailed analysis of the Drude weights for both thermal and spin transport in one-dimensional spin-$\frac{1}{2}$ systems by means of exact diagonalization and analytic approaches at finite temperatures. Transport properties are studied first for the integrable $\mathrm{XXZ}$ model and second for various nonintegrable systems such as the dimerized chain, the frustrated chain, and the spin ladder. We compare our results obtained by exact diagonalization and mean-field theory with those of the Bethe ansatz, bosonization, and other numerical studies in the case of the anisotropic Heisenberg model both in the gapless and gapped regime. In particular, we find indications that the Drude weight for spin transport is finite in the thermodynamic limit for the isotropic chain. For the nonintegrable models, a finite-size analysis of the numerical data for the Drude weights is presented, covering the entire parameter space of the dimerized and frustrated chain. We also discuss which conclusions can be drawn from bosonization regarding the question of whether the Drude weights are finite or not. One of our main results is that the Drude weights vanish in the thermodynamic limit for nonintegrable models.

arXiv:cond-mat/0304595•Physical review. B, Condensed matter•Oct 22, 2003•198 citations
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.
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 zzz 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−−26L=6--26L=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−−18L=4--18L=4−−18, indicates an ergodic behavior of dynamics in the whole range of parameters.
The effect of boundary scattering of magnons in Heisenberg chains is studied by use of the thermal Green's function technique. In contrast with the usual approach, the periodic boundary condition is not imposed and the space Fourier transform is not used. The decoupled equation of motion for the two‐spin Green's functions is solved exactly as a system of simultaneous difference equations subject to very general boundary conditions. The solutions contain terms representing all the higher order reflections to infinite order of magnons from the two ends. The effect of chain length on the magnon frequency spectrum and its influence on the specific heat are calculated and discussed.
The one-dimensional spin-1/2 Ising-Heisenberg ferromagnets are studied by the Green's function method at finite temperatures. The Green's function is decoupled in terms of the correlation functions of 〈S0z{\mathit{S}}_{0}^{\mathit{z}}S0z​Snz{\mathit{S}}_{\mathit{n}}^{\mathit{z}}Snz​〉 and 〈S0+{\mathit{S}}_{0}^{+}S0+​${\mathit{S}}_{\mathit{n}}^{\mathrm{\ensuremath{-}}}$〉 under the constraint of 〈Sz{\mathit{S}}^{\mathit{z}}Sz〉=0, from which a set of self-consistent equations are derived and solved numerically. The thermodynamic properties such as internal energy, specific heat, and zero-field parallel susceptibility are calculated. Our results predict the correct behavior for the correlation functions and thermodynamic quantities in the entire temperature region, in agreement both with the exact numerical results on the finite chain and the experimental data.
The one-dimensional spin-1 Heisenberg antiferromagnetic chains are studied by a generalized two-time Green's-function method at finite temperature. 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 is set to zero at each site to guarantee the lack of the long-range order in the one-dimensional Heisenberg system. The Green's functions are decoupled in terms of the correlation functions of spin operators. A set of self-consistent equations of the correlation functions are derived and solved numerically. Correlation functions of spin operators, thermodynamic properties such as the internal energy, and specific heat in the entire temperature region are obtained. The Haldane gap in the excitation spectrum appears naturally in the analytical result. When k is zero, there is a gap 2\ensuremath{\Delta}, which is 1.0J and close to the theoretical result of White and Huse. And there is a broad maximum in the temperature dependence of the specific heat. The properties of the thermodynamic quantities are consistent with the numerical results on the finite chains. \textcopyright{} 1996 The American Physical Society.
The one-dimensional Heisenberg model with S = 1/2 is treated with the use of the two-time Green's functions. The hierarchy of the equations of motion of the Green'f functions is decoupled at a stage one-step further than Tyablikov's decoupling. The thermal average of the spin component, ≪ Sz >, is set to zero, becaure the long-range order does not exist in one dimension. Instead, our Green's functions are expressed in terms of the correlation functions cn ≡4≪ S0zSnz >. The Green's function is essentially of the form representing undamped spin waves, whose spactrum depends on c1, c2 and one more parameter. They are determined by the requirement that c1 and c2 should be self-consistent and that c0 should be unity. The self-consistency equations have been solved analytically at high- and low-temperature limits, and also solved numerically in the whole range of the temperature. Thermodynamic quantities have been calculated using these solutions. It has turned out that the theory gives the correct high-temperature expansion for the thermodynamic quantities and the correlation functions. The latter is expressed by cn = (J / 4 kBT )n. In the case of ferromagnetic coupling, the correlation function cn at T = 0 is equal to 1/3 for all n's. This is what is expected from the correct ground state of the ferromagnetic Heisenberg system. The spin-wave spectrum at T = 0 also agrees with the correct one. At T ≪J/ kB, we find that the specific heat goes as T1/2 and the susceptibility goes as T-2. The gross feature of the temperature-dependence of the thermodynamic quantities agrees with Bonner and Fisher. In the case of the antiferromagnetic Heisenberg model, we find c1 = -0.55407 and c2 = 0.16100 at T = 0, which are fairly close to the exact values. The thermodynamic quantities are also in gross agreement with Bonner and Fisher.
We report a detailed analysis of the Drude weights for both thermal and spin transport in one-dimensional spin-12\frac{1}{2}21​ systems by means of exact diagonalization and analytic approaches at finite temperatures. Transport properties are studied first for the integrable XXZ\mathrm{XXZ}XXZ model and second for various nonintegrable systems such as the dimerized chain, the frustrated chain, and the spin ladder. We compare our results obtained by exact diagonalization and mean-field theory with those of the Bethe ansatz, bosonization, and other numerical studies in the case of the anisotropic Heisenberg model both in the gapless and gapped regime. In particular, we find indications that the Drude weight for spin transport is finite in the thermodynamic limit for the isotropic chain. For the nonintegrable models, a finite-size analysis of the numerical data for the Drude weights is presented, covering the entire parameter space of the dimerized and frustrated chain. We also discuss which conclusions can be drawn from bosonization regarding the question of whether the Drude weights are finite or not. One of our main results is that the Drude weights vanish in the thermodynamic limit for nonintegrable models.