A Non-Linear Field Analysis of the Haldane Gap Problem for Quantum Spin Chains

Abstract

In 1983 Haldane¹ suggested that antiferromagnetic quantum spin chains, where each spin is integer, have a finite energy gap in their spectrum whereas chains made up of half-odd-integer spins are gapless. Numerical studies were undertaken to try and confirm² that there was indeed a gap for chains with spins S = 1. There were, however, at least initially, numerical convergence problems which clouded the issue and Bethe Ansatz-like approaches seemed to indicate that there might not be a gap when the spins were integral. In this latter approach, however, the interactions between the spins were not of the same form as those discussed by Haldane. Two decades earlier Lieb, Schultz and Mattis³ had provided a rigorous proof that there was no gap for half-odd-integer spins but the method was shown to fail for integer spins. Spin wave theory for the simplest antiferromagnetic systems was developed by Anderson⁴, Ziman⁵ and Kubo⁶ long ago and the situation was reviewed by Nagamiya et al.⁷