Excitation spectrum of Heisenberg spin ladders

Heisenberg antiferromagnetic spin ‘‘ladders’’ (two coupled spin chains) are low-dimensional magnetic systems which for S=1/2 interpolate between half-integer-spin chains, when the chains are decoupled, and effective integer-spin one-dimensional chains in the strong-coupling limit. The spin-1/2 ladder may be realized in nature by vanadyl pyrophosphate, (VO${)}_2$${\mathrm{P}}_2$${\mathrm{O}}_7$. In this paper we apply strong-coupling perturbation theory, spin-wave theory, Lanczos techniques, and a Monte Carlo method to determine the ground-state energy and the low-lying excitation spectrum of the ladder. We find evidence of a nonzero spin gap for all interchain couplings ${\mathit{J}}_{\mathrm{\perp }}$>0. A band of spin-triplet excitations above the gap is also analyzed. These excitations are unusual for an antiferromagnet, since their long-wavelength dispersion relation behaves as (k-${\mathit{k}}_0$${)}^2$ (in the strong-coupling limit ${\mathit{J}}_{\mathrm{\perp }}$≫J, where J is the in-chain antiferromagnetic coupling). Their band is folded, with a minimum energy at ${\mathit{k}}_0$=π, and a maximum between ${\mathit{k}}_1$=π/2 (for ${\mathit{J}}_{\mathrm{\perp }}$=0) and 0 (for ${\mathit{J}}_{\mathrm{\perp }}$=∞). We also give numerical results for the dynamical structure factor S(q,ω), which can be determined in neutron scattering experiments. Finally, possible experimental techniques for studying the excitation spectrum are discussed.