Cyclic spin exchange in cuprate spin ladders

We investigate the influence of a cyclic spin exchange $J_{\mathrm{cyc}}$ on the one- and two-triplet excitations of an undoped two-leg $S=1/2\mathrm{}$ ladder, using the density-matrix renormalization group (DMRG). The dispersion of the $S=0\mathrm{}$ two-triplet bound state is dramatically reduced by $J_{\mathrm{cyc}}$ due to a repulsion between triplets on neighboring rungs. In $(\mathrm{La},\mathrm{Ca}{)}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ a consistent description of both the spin gap and the $S$=0 bound state requires $J_{\mathrm{cyc}}{/J}_{\perp }\approx 0.20–0.27$ and ${J/J}_{\perp }\approx 1.25–\mathrm{1.35.}$ With these coupling ratios the recently developed dynamical DMRG yields an excellent description of the entire $S=0\mathrm{}$ excitation spectrum observed in the optical conductivity, including the continuum contribution.