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Journal of Physics Conference Series

Precise estimates by finite-size extrapolations of theS=1 Haldane-gapped system

Hiroki Nakano

Published Jan 1, 2010• 0 citations
Publisher

Abstract

We carry out finite-size extrapolations of numerical-diagonalization data of the S =1 Heisenberg chain having a nonzero energy gap between the unique singlet ground state and the first excited state, namely the Haldane gap. Very precise estimates of the ground-state energy per site E g /N = −1.4014840447(39) and the staggered component of the magnetic structure factor S π = 3.864356(31) at T =0 are successfully obtained from the finite-size data of system sizes up to N = 24 under the twisted boundary condition by the sequence interval squeeze method, which was applied to a precise estimation of the Haldane gap by Nakano and Terai [J. Phys. Soc. Jpn. 78 (2009) 014003]. The present estimates are compared with other estimates in previous studies from various methods including the quantum Monte Carlo simulation and the density matrix renormalization group calculation.

DOI
10.1088/1742-6596/200/2/022037

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We carry out finite-size extrapolations of numerical-diagonalization data of the S =1 Heisenberg chain having a nonzero energy gap between the unique singlet ground state and the first excited state, namely the Haldane gap. Very precise estimates of the ground-state energy per site E g /N = −1.4014840447(39) and the staggered component of the magnetic structure factor S π = 3.864356(31) at T =0 are successfully obtained from the finite-size data of system sizes up to N = 24 under the twisted boundary condition by the sequence interval squeeze method, which was applied to a precise estimation of the Haldane gap by Nakano and Terai [J. Phys. Soc. Jpn. 78 (2009) 014003]. The present estimates are compared with other estimates in previous studies from various methods including the quantum Monte Carlo simulation and the density matrix renormalization group calculation.
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 one-dimensional Heisenberg antiferromagnets of large-integer-SSS spins are studied; their Haldane gaps are estimated by the numerical diagonalization method for S=5S=5S=5 and 666. We successfully obtain a monotonically increasing sequence of finite-size energy difference data corresponding to the Haldane gaps from the huge-scale parallel calculations of diagonalization under the twisted boundary condition and create a monotonically decreasing sequence within the range of system sizes treated in this study from the monotonically increasing sequence. Consequently, the gaps for S=5S=5S=5 and 666 are estimated to be 0.000050±0.0000050.000050 \pm 0.0000050.000050±0.000005 and 0.0000030±0.00000050.0000030 \pm 0.00000050.0000030±0.0000005, respectively. The asymptotic formula of the Haldane gap for S→∞S\rightarrow\inftyS→∞ is examined from the new estimates to determine the coefficient in the formula more precisely.
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}.
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→∞.
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.