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

TL;DR

This study estimates Haldane gaps for large-spin Heisenberg antiferromagnetic chains using numerical diagonalization, providing more precise asymptotic behavior as spin size increases.

Contribution

It introduces a new numerical approach to accurately estimate Haldane gaps for large spins and refines the asymptotic formula for the gap as spin approaches infinity.

Findings

Haldane gaps for S=5 and 6 are estimated as 0.000050 and 0.0000030.

A monotonically decreasing sequence of gap estimates is obtained.

The asymptotic coefficient in the gap formula is determined more precisely.

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.