Calculation of the singlet-triplet gap of the antiferromagnetic Heisenberg model on a ladder

The ground-state energy and the singlet-triplet energy gap of the antiferromagnetic Heisenberg model on a ladder is investigated using a mean-field theory and the density-matrix renormalization group. Spin-wave theory shows that the corrections to the local magnetization are infinite. This indicates that no long-range order occurs in this system. A flux-phase state is used to calculate the energy gap as a function of the transverse coupling, ${\mathit{J}}_{\mathrm{\perp }}$, in the ladder. It is found that the gap is linear in ${\mathit{J}}_{\mathrm{\perp }}$ for ${\mathit{J}}_{\mathrm{\perp }}$≫1 and goes to zero for ${\mathit{J}}_{\mathrm{\perp }}$→0. The mean-field theory agrees well with the numerical results.