Solution of sign and complex action problems with cluster algorithms

Abstract

Two kinds of models are considered which have a Boltzmann weight which is either not real or real but not positive and so standard Monte Carlo methods are not applicable. These sign or complex action problems are solved with the help of cluster algorithms. In each case improved estimators for the Boltzmann weight are constructed which are real and positive. The models considered belong to two classes: fermionic and non-fermionic models. An example for a non-fermionic model is the Potts model approximation to QCD at non-zero baryon density. The three-dimensional three-state Potts model captures the qualitative features of this theory. It has a complex action and so the Boltzmann weight cannot be interpreted as a probability. The complex action problem is solved by using a cluster algorithm. The improved estimator for the complex phase of the Boltzmann factor is real and positive and is used for importance sampling. The first order deconfinement transition line is investigated and the universal behavior at its critical endpoint is studied.

(cont.) An example for a fermionic model with a sign problem are staggered fermions with 2 flavors in 3+1 dimensions. Here the sign is connected to the permutation sign of fermion world lines and is of nonlocal nature. Cluster flips change the topology of the fermion world lines and they have a well defined effect on the permutation sign independent of the other clusters. The sign problem is solved by suppressing those clusters whose contribution to the partition function and observables of interest would be zero. We confirm that the universal critical behavior of the finite temperature chiral phase transition is the one of the three dimensional Ising model. We also study staggered fermions with one flavor in 2+1 dimensions and confirm that the chiral phase transition then belongs to the universality class of the two dimensional Ising model.