Using computational techniques, it is shown that pairing is a robust property of hole doped antiferromagnetic (AF) insulators. In one dimension (1D) and for two-leg ladder systems, a BCS-like variational wave function with long-bond spin-singlets and a Jastrow factor provides an accurate representation of the ground state of the t-J model, even though strong quantum fluctuations destroy the off-diagonal superconducting (SC) long-range order in this case. However, in two dimensions (2D) it is argued -and numerically confirmed using several techniques, especially quantum Monte Carlo (QMC) -that quantum fluctuations are not strong enough to suppress superconductivity. 74.20.Mn, 71.10.Fd, 71.10.Pm, 71.27.+a The nature of high temperature superconductors remains an important unsolved problem in condensed matter physics. Strong electronic correlations are widely believed to be crucial for the understanding of these materials. Among the several proposed theories are those where antiferromagnetism induces pairing in the d x 2 −y 2 channel [1]. These approaches include the following two classes: (i) theories based on Resonant Valence Bond (RVB) wave functions, with electrons paired in long spin singlets in all possible arrangements [2,3], and (ii) theories based on two-hole d x 2 −y 2 bound states at infinitesimal doping, formed to minimize the damage of individual holes to the AF order parameter, which condense at finite pair density into a superconductor [4]. However, recent density matrix renormalization group (DMRG) calculations have seriously questioned these approaches since non-SC striped ground states were reported for realistic couplings and densities in the t-J model [5]. Clearly to make progress in the understanding of copper oxides, the 2D t-J model ground state must be fully understood, to distinguish among the many proposals.In this paper, using a variety of powerful numerical techniques, the properties of the t-J model are investigated. Our main result is that in the realistic regime of couplings the 2D t-J model supports a d x 2 −y 2 SC ground state, confirming theories of Cu-oxides based on AF correlations. The t-J model used here is. . stands for nearestneighbor sites, and n i and S i are the electron density and spin at site i, respectively. Our study focuses on the low hole-doping region of chains, two-leg ladders, and square clusters, using different numerical techniques: QMC (pure variational and fixed-node (FN) approximations), DMRG, and Lanczos. Within our QMC approach, it is possible to further improve the variational and FN accuracy by applying a few (p ≤ 2) Lanczos steps to the variational (p = 0) wave function |Ψ V ,. Non-variational estimates of energy and correlation functions can also be extracted with the variance-extrapolation method [6].Our BCS variational wave function is defined as

Spin models that have been proposed to describe dimerized chains, ladders, two dimensional antiferromagnets, and other compounds are here studied when some spins are replaced by spinless vacancies, such as it occurs by Zn doping. A small percentage of vacancies rapidly destroys the spin gap, and their presence induces enhanced antiferromagnetic correlations near those vacancies. The study is performed with computational techniques which includes Lanczos, world-line Monte Carlo, and the Density Matrix Renormalization Group methods. Since the phenomenon of enhanced antiferromagnetism is found to occur in several models and cluster geometries, a common simple explanation for its presence may exist. It is argued that the resonating-valence-bond character of the spin correlations at short distances of a large variety of models is responsible for the presence of robust staggered spin correlations near vacancies and lattice edges. The phenomenon takes place regardless of the long distance properties of the ground state, and it is caused by a "pruning" of the available spin singlets in the vicinity of the vacancies. The effect produces a broadening of the low temperature NMR signal for the compounds analyzed here. This broadening should be experimentally observable in the structurally dimerized chain systems Cu(N O3)2 · 2.5H2O, CuW O4, (V O)2P2O7, and Sr14Cu24O41, in ladder materials such as SrCu2O3, in the spin-Peierls systems CuGeO3 and N aV2O5, and in several others since it is a universal effect common to a wide variety of models and compounds.