Density-matrix renormalization-group technique with periodic boundary conditions for two-dimensional classical systems

The density-matrix renormalization-group (DMRG) method with periodic boundary conditions is introduced for two-dimensional (2D) classical spin models. It is shown that this method is more suitable for derivation of the properties of infinite 2D systems than the DMRG with open boundary conditions, despite the fact that the latter describes much better strips of finite width. For calculation at criticality, phenomenological renormalization at finite strips is used together with a criterion for optimum strip width for a given order of approximation. For this width the critical temperature of the 2D Ising model is estimated with seven-digit accuracy for a not too large order of approximation. Similar precision is reached for critical exponents. These results exceed the accuracy of similar calculations for the DMRG with open boundary conditions by several orders of magnitude. The method is applied to the calculation of critical exponents of the q = 3,4 Ports model, as well.