Perturbation Theory for Spin Ladders Using Angular-Momentum Coupled Bases

We compute bulk properties of Heisenberg spin-1/2 ladders using Rayleigh-Schr\"odinger perturbation theory in the rung and plaquette bases. We formulate a method to extract high-order perturbative coefficients in the bulk limit from solutions for relatively small finite clusters. For example, a perturbative calculation for an isotropic $2\ifmmode\times\else\texttimes\fi{}12$ ladder yields an eleventh-order estimate of the ground-state energy per site that is within 0.02% of the density-matrix-renormalization-group value. Moreover, the method also enables a reliable estimate of the radius of convergence of the perturbative expansion. We find that for the rung basis the radius of convergence is ${\ensuremath{\lambda}}_{c}\ensuremath{\simeq}0.8,$ with $\ensuremath{\lambda}$ defining the ratio between the coupling along the chain relative to the coupling across the chain. In contrast, for the plaquette basis we estimate a radius of convergence of ${\ensuremath{\lambda}}_{c}\ensuremath{\simeq}1.25.$ Thus, we conclude that the plaquette basis offers the best currently available perturbative approach which can provide a reliable treatment of the physically interesting case of isotropic $(\ensuremath{\lambda}=1)$ spin ladders. We illustrate our methods by computing perturbative coefficients for the ground-state energy per site, the gap, and the one-magnon dispersion relation.