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arXiv:1909.02178 · Journal of the Physical Society of Japan

Haldane Gaps of Large-S Heisenberg Antiferromagnetic Chains and Asymptotic Behavior

Hiroki Nakano, Norikazu Todoroki, Tôru Sakai

Published Oct 4, 2019• 15 citations
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Abstract

The one-dimensional Heisenberg antiferromagnets of large-integer-$S$ spins are studied; their Haldane gaps are estimated by the numerical diagonalization method for $S=5$ and $6$. 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 $S=5$ and $6$ are estimated to be $0.000050 \pm 0.000005$ and $0.0000030 \pm 0.0000005$, respectively. The asymptotic formula of the Haldane gap for $S\rightarrow\infty$ is examined from the new estimates to determine the coefficient in the formula more precisely.

DOI
10.7566/jpsj.88.114702

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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.
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→∞.
The antiferromagnetic Heisenberg spin chain with integer spin has a short-range magnetic order and an excitation energy gap above the ground state. This so-called Haldane gap is proportional to the exchange coupling constant J of the Heisenberg chain. In this study, we discuss recent results about the spin dependence of the Haldane gap and conjecture an analytical formula valid asymptotically for large spin values. We then discuss the robustness of the edge states of the spin-1 chain by studying a spin-1 ladder using the density matrix renormalization group (DMRG) algorithm. We show that the peculiar hidden topological order of the spin-1 chain disappears smoothly by increasing the ladder rung coupling without any intervening phase transition. This is evidence for the fragile character of the topological order of the spin-1 chain.
A new method of determining the spectra of simple quantum-mechanical spin Hamiltonians by direct Hamiltonian-matrix diagonalization is presented. The method is illustrated by applying it to the isotropic one-dimensional Heisenberg antiferromagnet, for which calculations were performed for longer spin chains than by previous direct Hamiltonian-matrix diagonalizations. For spin s=1/2 it was possible to accurately determine the ground- and first-excited-state energies for chains of length N=32 and 64. For s=1 the longest chain for which a similar computation was performed was N=32; the singlet-triplet Haldane gap is found to be 0.421\ifmmode\pm\else\textpm\fi{}0.005 which is consistent with earlier Monte Carlo results. Long-spin-chain calculations are performed iteratively by solving smaller-chain problems, and constructing from their eigenvalues, and eigenstates, the longer-spin-chain basis and Hamiltonian matrices. The computations were performed on a small computer, which suggests that much larger problems could be handled with a parallel-processing supercomputer.
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 $\ensuremath{\Delta}(2)$ is estimated accurately as $(0.0876\ifmmode\pm\else\textpm\fi{}0.0013)J.$ 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.