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.
Highlights
The Haldane gap – the energy gap between the unique ground state and the first excited state for the integer-S Heisenberg antiferromagnets in one dimension – is well known; the presence of the gap surprised many condensed-matter physicists when it was originally conjectured in Refs. 1 and 2 by mapping the Heisenberg chain to the nonlinear σ model
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
Summary
The Haldane gap – the energy gap between the unique ground state and the first excited state for the integer-S Heisenberg antiferromagnets in one dimension – is well known; the presence of the gap surprised many condensed-matter physicists when it was originally conjectured in Refs. 1 and 2 by mapping the Heisenberg chain to the nonlinear σ model. The existence of a nonzero gap is widely believed. Extensive studies concerning this phenomenon have contributed considerably to our understanding of the various properties of quantum spin systems. It is known that the density matrix renormalization group (DMRG),5) quantum Monte Carlo (QMC),6) and numerical diagonalization (ND)7) calculations give estimates that agree with each other to five decimal places: ∆/J ∼ 0.41048, where ∆ and J represent the gap value and the strength of the interaction defined later, respectively. For S = 2, after various studies,6, 8–10) the gap estimates from the three approaches agree with each other within errors; ∆/J ∼ 0.089
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