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arXiv:cond-mat/9903035 · Physical review. B, Condensed matter

Haldane gap for the S=2 antiferromagnetic Heisenberg chain revisited

Xiaoqun Wang, Shaojing Qin, Lu Yu

Published Dec 1, 1999• 35 citations
arXivHTMLPDF

Abstract

Using the density matrix renormalization group (DMRG) technique, we carry out a large scale numerical calculation for the $S=2$ antiferromagnetic Heisenberg chain. Performing systematic scaling analysis for both the chain length L and the number of optimal states kept in the iterations m, the Haldane gap $\ensuremath{\Delta}(2)$ is estimated accurately as $(0.0876\ifmmode\pm\else\textpm\fi{}0.0013)J.$ Our systematic analysis for the $S=2$ chains not only clarifies the controversies arising from various DMRG calculations and Monte Carlo simulations, but also sheds light on how to obtain reliable results from the DMRG calculations for other similar complicated systems.

DOI
10.1103/physrevb.60.14529

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Using the density matrix renormalization group (DMRG) technique, we carry out a large scale numerical calculation for the S=2S=2S=2 antiferromagnetic Heisenberg chain. Performing systematic scaling analysis for both the chain length L and the number of optimal states kept in the iterations m, the Haldane gap \ensuremathΔ(2)\ensuremath{\Delta}(2)\ensuremathΔ(2) is estimated accurately as (0.0876\ifmmode±\else\textpm\fi0.0013)J.(0.0876\ifmmode\pm\else\textpm\fi{}0.0013)J.(0.0876\ifmmode±\else\textpm\fi0.0013)J. Our systematic analysis for the S=2S=2S=2 chains not only clarifies the controversies arising from various DMRG calculations and Monte Carlo simulations, but also sheds light on how to obtain reliable results from the DMRG calculations for other similar complicated systems.
The one-dimensional Heisenberg antiferromagnet of S = 1 spins is studied by numericaldiagonalization method based on the Lanczos algorithm. The Haldane gap of this system isnumerically estimated from finite-size clusters under the twisted boundary condition up tothe 30-site system. The finite-size results are analyzed by a convergence-acceleration technique; the multistep procedure of the convergence acceleration is successfully carried out.The present estimate for the gap magnitude is more precisely than the previously reportedvalue from a Lanczos-diagonalization study. It also agrees well with the result reported ina study based on the density matrix renormalization group calculation. The present studyclarifies that Lanczos-diagonalization calculations can sufficiently contribute for preciselyestimating an energy gap although this method can treat only small clusters.
The structure function S(k) for the s=1, Haldane-gap antiferromagnetic chain is measured accurately using the recent density-matrix renormalization-group method, with a chain length of 100. Excellent agreement with the nonlinear-\ensuremath{\sigma}-model prediction is obtained, both at k\ensuremath{\approxeq}\ensuremath{\pi}, where a single-magnon process dominates, and at k\ensuremath{\approxeq}0, where a two-magnon process dominates. We repeat our calculation with crystal-field anisotropy chosen to model Ni(C2{\mathrm{C}}_{2}C2​H8{\mathrm{H}}_{8}H8​N2{\mathrm{N}}_{2}N2​)2{)}_{2})2​NO2{\mathrm{NO}}_{2}NO2​(ClO4{\mathrm{ClO}}_{4}ClO4​), obtaining good agreement with both field-theory predictions and recent experiments. Correlation lengths, gaps, and velocities are determined for both polarizations.
Low-lying magnon dispersion in a S=1S=1S=1 Heisenberg antiferromagnetic (AF) chain is analyzed using the non-Abelian density-matrix-renormalization-group (DMRG) method. The scattering length ab{a}_{\mathrm{b}}ab​ of the boundary coupling and the intermagnon scattering length aaa are determined. The scattering length ab{a}_{\mathrm{b}}ab​ is found to exhibit a characteristic diverging behavior at the crossover point. In contrast, the Haldane gap $\ensuremath{\Delta}$, the magnon velocity vvv, and aaa remain constant at the crossover. Our method allowed estimation of the gap of the S=2S=2S=2 AF chain to be $\ensuremath{\Delta}=0.0891623(9)$ using a chain length longer than the correlation length $\ensuremath{\xi}$.
The spectral properties, and associated critical singularities, of spin-1 Heisenberg antiferromagnetic chains are of high current interest, both theoretical and experimental. Theoretical studies using the single mode approximation and also a Monte Carlo calculation indicate that the spectrum has a gap at k=π, as predicted by Haldane. They also predict the unusual feature that the excited state continuum is asymmetric about the mid-point of the Brillouin zone, k=π/2. However, neither approach is sufficiently accurate to predict reliably the magnitude of the gap at k=0 which was not discussed by Haldane. Our approach is to use static (spectral) calculations of excited states of chains of N=4 through 14 spins for all values of k, a novel feature not associated with previous finite-size calculations, to discuss the dispersion spectra of the S=1 Heisenberg chain. In addition, we have calculated dynamical properties, in particular S(q,ω), exactly for finite chains up to N=14 spins. This allows us to estimate the gap at k=0 to good accuracy, and we find that to within this accuracy the gap at k=0 is twice the Haldane gap at k=π. This result is important in connection with very recent neutron scattering and magnetization measurements. In addition, we have calculated the correlation length and critical singularities associated with dynamical quantities.
It is conjectured that the Haldane phase of the S=1S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1S=1 antiferromagnetic Heisenberg chain and the S=1/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2S=1/2 ferromagnetic-antiferromagnetic alternating Heisenberg chain is stable against any strength of randomness, because of imposed breakdown of translational symmetry. This conjecture is confirmed by the density matrix renormalization group calculation of the string order parameter and the energy gap distribution.
We report a density-matrix renormalization-group (DMRG) study of S=1/2 antiferromagnetic Heisenberg spin chains. We show the Friedel oscillations in open spin chains within bosonization theory. The bosonization predictions are in excellent agreement with the DMRG calculation. The chain length dependence of the ground state and several low-energy excited state energies are compared with conformal field theory results. Based upon the DMRG calculations for various quantities, the interplay between the system size and the number of optimal states kept in the DMRG is carefully discussed. \textcopyright{} 1996 The American Physical Society.