World Line Monte Carlo

The Monte carlo methods extensively used in the study of classical systems cannot be directly applied to quantum systems. Suzuki[14] generalized a method previously proposed by trotter [] and used to to demonstrate that every $d$-dimensional quantum spin system can be mapped onto a $(d+1)$-dimensional problem, similar to the Ising model, suggesting that the Monte Carlo method could be used on the resulting classical model. The first results of such calculations were presented by Suzuki et al. [], who carried out simulations for the heisenberg model in 1D and the $XY$ model in one and two dimensions. Employing a generalization of the method, de Raedt and Lgendijk [] and Hirsch et al [,] performed similar calculations for fermionic systems. Mor erecently, Reger and Young [] and makivic and Ding [] perfected the method for spin systems in two dimensions.

In this section we describe the World Line Monte Carlo method for quantum spin systems in one dimension.