Spin ladders with nonmagnetic impurities

We investigate antiferromagnetic spin ladders with nonmagnetic impurities by variational and numerical (Lanczos and DMRG) methods. The interaction between the two unpaired spins opposite to the impurities is described by an effective exchange interaction $J_{eff}$, the magnitude of which depends on the impurity distance. The magnitude of $J_{eff}$ is different for unpaired spins at the edges of an open ladder and in the bulk. This difference is related to the different distribution of the unpaired spin into the bulk of the ladder. The numerical results are interpreted using matrix product states. Using the DMRG we calculate the spectrum of low-lying energy levels for up to 6 impurities and find that these spectra can be reproduced assuming pair interactions with an accuracy of better than 10\%. We discuss the filling of the ladder gap with impurity states and argue that in the thermodynamic limit the spin ladder with a finite concentration of impurities always shows a Curie susceptibility at low temperatures.