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arXiv:cond-mat/9402043 · Physical review. B, Condensed matter

Finite-lattice extrapolations for a Haldane-gap antiferromagnet

O. Golinelli, Th. Jolicœur, R. Lacaze

Published Aug 1, 1994• 142 citations
arXivHTMLPDF

Abstract

We present results of exact diagonalization of the isotropic antiferromagnetic S=1 Heisenberg chain by the Lanczos method, for finite rings of up to N=22 sites. The Haldane gap G(N) and the ground-state energy per site e(N) converge, with increasing N, faster than a power law. By VBS and Shanks transformations, the extrapolated values are G(\ensuremath{\infty})=0.41049(2) and e(\ensuremath{\infty})=-1.401485(2). The spin-spin correlation function is well fit by exp(-r/\ensuremath{\xi})/ \ensuremath{\surd}r with \ensuremath{\xi}=6.2.

DOI
10.1103/physrevb.50.3037

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We present results of exact diagonalization of the isotropic antiferromagnetic S=1 Heisenberg chain by the Lanczos method, for finite rings of up to N=22 sites. The Haldane gap G(N) and the ground-state energy per site e(N) converge, with increasing N, faster than a power law. By VBS and Shanks transformations, the extrapolated values are G(\ensuremath{\infty})=0.41049(2) and e(\ensuremath{\infty})=-1.401485(2). The spin-spin correlation function is well fit by exp(-r/\ensuremath{\xi})/ \ensuremath{\surd}r with \ensuremath{\xi}=6.2.
The S=1 isotropic antiferromagnetic Heisenberg chain is studied by exact diagonalizations using the Lancz\"os algorithm. Energy gaps, structure factors at k=\ensuremath{\pi}, and staggered susceptibilities at T=0 are calculated for finite rings up to N=16, and extrapolated to an infinite system using Shanks' transformation. The estimated energy gap is 0.411\ifmmode\pm\else\textpm\fi{}0.001, which agrees with the result of Monte Carlo calculations by Nightingale and Bl\"ote. Further, it is found that a finite-size correction decays exponentially and the decay constant corresponds with the correlation length, which is about 5.
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.
We propose two methods of estimating a systematic error in extrapolation to the infinite-size limit in the study of measuring the Haldane gaps of the one-dimensional Heisenberg antiferromagnet with the integer spin up to S=5. The finite-size gaps obtained by numerical diagonalizations based on Lanczos algorithm are presented for sizes that have not previously been reported. The changes of boundary conditions are also examined. We successfully demonstrate that our methods of extrapolation work well. The Haldane gap for S=1 is stimated to be 0.4104789±0.00000130.4104789 \pm 0.00000130.4104789±0.0000013. We successfully obtain the gaps up to S=5, which make us confirm the asymptotic formula of the Haldane gap in S→∞S\to\inftyS→∞.
Highly precise numerical calculations using a variation of the Lanczos method devised by Paige were performed for s=1/2 antiferromagnetic Heisenberg finite chains up to sizes N=28. Several interesting physical quantities, including the ground-state energy, the mass gap, and the spin-wave velocity, were computed and fitted for logarithmic finite-size corrections, as suggested by conformal invariance and Bethe ansatz calculations. A crossover size for a maximum of the spin-wave velocity is predicted for N\ensuremath{\sim}50, well above the current available sizes in a typical Lanczos simulation. A list of our numerical results is given along with extrapolated values using standard algorithms.
An efficient projector Monte Carlo method has been used to calculate the energies of the ground state and the first excited state, and hence the gap between them, of the spin-2 Heisenberg antiferromagnetic chains with linear sizes up to N=32. By direct extrapolation, we found that a Haldane gap exists, \ensuremath{\Delta}E=0.05, and the value of the ground-state energy per site is e0{\mathit{e}}_{0}e0​=-4.7545 as N\ensuremath{\rightarrow}\ensuremath{\infty}.