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Journal of the Physical Society of Japan

The Haldane Gap of the S = 1 Heisenberg Antiferromagnetic Chain

Hiroki Nakano, Hiroto Tadano, Norikazu Todoroki, Tôru Sakai

Published Jun 1, 2022• 10 citations
Publisher

Abstract

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.

DOI
10.7566/jpsj.91.074701

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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.
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→∞.
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 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.
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.