99 2 Blockspin Cluster Algorithms for Quantum Spin Systems

Cluster algorithms are developed for simulating quantum spin systems like the one-and two-dimensional Heisenberg ferro-and anti-ferromagnets. The corresponding two-and three-dimensional classical spin models with four-spin couplings are maped to blockspin models with two-blockspin interactions. Clusters of blockspins are updated collectively. The efficiency of the method is investigated in detail for one-dimensional spin chains. Then in most cases the new algorithms solve the problems of slowing down from which standard algorithms are suffering. Two-dimensional quantum spin systems are relevant for the description of the un-doped anti-ferromagnetic precursor insulators of high-T c superconductors. Presumably understanding the physics of the superconductors requires an understanding of their precursor insulators. This already is a nontrivial problem, which most likely does not allow for a complete analytic solution. Therefore it is natural to use a numerical approach to compute the properties of these materials. At present different methods are used in numerical studies of quantum spin systems (for a recent review see for example ref.[1]). Small systems can be solved completely by a direct diag-onalization of the hamiltonian. For larger systems one can use Monte-Carlo methods. For this purpose the finite temperature partition function of the d-dimensional quantum spin system is expressed as a pathintegral of a (d + 1)-dimensional classical system of Ising-like spin variables with four-spin couplings. The classical system is then simulated on a euclidean time lattice with lattice spacing ǫ using importance 1