Nonmagnetic impurities in spin-gapped and gapless Heisenberg antiferromagnets

We discuss the effects of nonmagnetic impurities (static holes) in the $S=1/2\mathrm{}$ Heisenberg model on a two-dimensional (2D) square lattice, and on two- and three-leg ladders. Results of quantum Monte Carlo simulations show that a free localized moment is induced around an isolated impurity only in the spin-gapped two-leg ladder, in agreement with previous theoretical expectations. The localization length of the $S=1/2\mathrm{}$ impurity moment in a two-leg ladder with isotropic couplings is $\approx 1.5$ lattice spacings. In gapless ladders (odd number of legs) and in the 2D lattice, no free moments are induced. However, in the 2D system, which has antiferromagnetic long-range order at $T=0\mathrm{}$ and a gap for longitudinal fluctuations, the impurity leads to a localized distortion of the magnetization in the direction of the broken symmetry, with total $S^z=1/2.\mathrm{}$ The shape of the distortion is in close agreement with previous spin-wave calculations. For all the systems, a staggered moment is induced by the impurity. For the gapped two-leg ladder this is exponentially localized within a length equal to the spin correlation length of the ladder ($\approx 3.2$ lattice spacings). For the three-leg ladder, the integrated staggered moment diverges as the system size goes to infinity, but the staggered magnetization at a given site appears to vanish. In 2D, we discuss the behavior seen in finite systems both with and without additional symmetry-breaking mechanisms. We also discuss the effect of impurities on the NMR Knight shift and the bulk magnetic susceptibility.