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Physical review. B, Condensed matter

Dynamical correlation functions of the S=12 nearest-neighbor and Haldane-Shastry Heisenberg antiferromagnetic chains in zero and applied fields

Kim Lefmann, Christian Rischel

Published Sep 1, 1996• 18 citations
Publisher

Abstract

We present a numerical diagonalization study of two one-dimensional $S=\frac{1}{2}$ antiferromagnetic Heisenberg chains, having nearest-neighbor and Haldane-Shastry ($\frac{1}{{r}^{2}}$) interactions, respectively. We have obtained the $T=0$ dynamical correlation function, ${S}^{\ensuremath{\alpha}\ensuremath{\alpha}}(q, \ensuremath{\omega})$, for chains of length $N=8\ensuremath{-}28$. We have studied ${S}^{\mathrm{zz}}(q, \ensuremath{\omega})$ for the Heisenberg chain in zero field, and from finite-size scaling we have obtained a limiting behavior that for large $\ensuremath{\omega}$ deviates from the conjecture proposed earlier by M\"uller et al. For both chains we describe the behavior of ${S}^{\mathrm{zz}}(q, \ensuremath{\omega})$ and ${S}^{\mathrm{xx}}(q, \ensuremath{\omega})$ for selected values of the applied field, and compare with previous work by M\"uller et al. and Talstra and Haldane. Suggestions for future finite-field neutron scattering experiments are made.

DOI
10.1103/physrevb.54.6340

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We present a numerical diagonalization study of two one-dimensional S=12S=\frac{1}{2}S=21​ antiferromagnetic Heisenberg chains, having nearest-neighbor and Haldane-Shastry (1r2\frac{1}{{r}^{2}}r21​) interactions, respectively. We have obtained the T=0T=0T=0 dynamical correlation function, S\ensuremathα\ensuremathα(q,\ensuremathω){S}^{\ensuremath{\alpha}\ensuremath{\alpha}}(q, \ensuremath{\omega})S\ensuremathα\ensuremathα(q,\ensuremathω), for chains of length N=8\ensuremath−28N=8\ensuremath{-}28N=8\ensuremath−28. We have studied Szz(q,\ensuremathω){S}^{\mathrm{zz}}(q, \ensuremath{\omega})Szz(q,\ensuremathω) for the Heisenberg chain in zero field, and from finite-size scaling we have obtained a limiting behavior that for large \ensuremathω\ensuremath{\omega}\ensuremathω deviates from the conjecture proposed earlier by M"uller et al. For both chains we describe the behavior of Szz(q,\ensuremathω){S}^{\mathrm{zz}}(q, \ensuremath{\omega})Szz(q,\ensuremathω) and Sxx(q,\ensuremathω){S}^{\mathrm{xx}}(q, \ensuremath{\omega})Sxx(q,\ensuremathω) for selected values of the applied field, and compare with previous work by M"uller et al. and Talstra and Haldane. Suggestions for future finite-field neutron scattering experiments are made.
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Extensive numerical studies have been performed on Heisenberg antiferromagnetic chains of spin (1/2) (up to N=20), spin 1 (N=14), spin (3/2) (N=10), and spin 2 (N=8). With use of the Lancz\"os technique, primarily, the two lowest-lying eigenvalues have been calculated for all values of wave vector q and all values of magnetization (STz{S}_{T}^{z}STz​) up to saturation for each chain. From a knowledge of the eigenvector corresponding to the lowest eigenstate for each STz{S}_{T}^{z}STz​, the T=0 spin-pair correlation functions have also been calculated as a function of field. We find a most unusual quantum-classical crossover phenomenon. It shows up in greatest detail in the field-dependent dispersion spectra, but the consequences are consistently manifested in the behavior of the T=0 magnetization isotherms and in the correlations both in real space and the Fourier transforms in q space. The additional data relevant to behavior in a field have allowed us to extend previous numerical studies of Heisenberg antiferromagnetic chains with higher spin whose purpose was to examine the validity of the Haldane conjecture. The Haldane conjecture implies that Heisenberg antiferromagnetic chains with integer spin have a gap in their excitation spectrum whereas chains with half-integer spin do not. While no feature of our extended investigations is in conflict with the conjecture, unusual features associated apparently with very slow convergence make the outcome less than conclusive. It appears that calculations on significantly longer chains are required to observe with confidence the large-N asymptotic limiting behavior.
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.
We present a selection rule for matrix elements of local spin operators in the S=1/2 Haldane-Shastry model. Based on this rule we extend a recent exact calculation by Haldane and Zirnbauer of the ground-state dynamical spin correlation function Sab{\mathit{S}}^{\mathit{a}\mathit{b}}Sab(n,t)=〈0\ensuremath{\Vert}Sa{\mathit{S}}^{\mathit{a}}Sa(n,t)Sb{\mathit{S}}^{\mathit{b}}Sb(0,0)\ensuremath{\Vert}0〉 and its Fourier transform Sab{\mathit{S}}^{\mathit{a}\mathit{b}}Sab(Q,E) of this model to a finite magnetic field. In zero field, only two-spinon excitations contribute to the spec tral function; in the (positively) partially spin-polarized case, there are two types of elementary excitations: spinons (\ensuremath{\Delta}Sz{\mathit{S}}^{\mathit{z}}Sz=\ifmmode\pm\else\textpm\fi{}1/2) and magnons (\ensuremath{\Delta}Sz{\mathit{S}}^{\mathit{z}}Sz=-1). The magnons are divided into left- or right-moving branches. The only classes of excited states contributing to the spectral functions are (I) two spinons, (II) two spinons+one magnon, (IIIa) two spinons+two magnons (moving in opposite directions), and (IIIb) one magnon. The contributions to the various correlations are ${\mathit{S}}^{\mathrm{\ensuremath{-}}+}$: (I); Szz{\mathit{S}}^{\mathit{z}\mathit{z}}Szz: (I)+(II); ${\mathit{S}}^{+\mathrm{\ensuremath{-}}}$: (I)+(II)+(III). In the zero-field limit there are no magnons, while in the fully polarized case, there are no spinons. We discuss the relation of the spectral functions to correlations of the Calogero-Sutherland model at coupling \ensuremath{\lambda}=2.
A real-space, zero-temperature renormalization-group method is used to study a Heisenberg antiferromagnetic spin chain with long-range interactions varying as ${(\mathrm{distance})}^{\ensuremath{-}p}$. Simple renormalization-group equations are obtained and studied analytically, and the results are checked against a more accurate numerical calculation. The calculations provide evidence for a phase transition at $p\ensuremath{\approx}1.85$ characterized by a change in the behavior of zero-temperature correlation functions. For $p\ensuremath{\gtrsim}1.85$ the large-distance physics is that of the nearest-neighbor antiferromagnet, while there is a line of fixed points for $p\ensuremath{\lesssim}1.85$. I conjecture that $p\ensuremath{\approx}1.85$ also marks an order-disorder transition at finite temperature such as occurs in the Ising model with power-law interactions.