Quantum spin chains and the Haldane gap

One-dimensional antiferromagnets have exotic disordered ground states. As was first argued by Haldane (1983), there is an excitation gap for integer, but not half-integer, spin. The authors review the arguments for this behaviour based on field-theory mappings, the Lieb-Schultz-Mattis theorem, exactly solvable models, finite-chain diagonalisation and real experiments.