Eliminating critical slowing down in Monte Carlo calculations

We examine methods to improve the major numerical difficulties in lattice field theory. Traditional Metropolis and heat bath Monte Carlo methods in lattice calculations break down whenever one tries to calculate thermodynamic quantities near critical points; this phenomenon is called Critical Slowing Down, (CSD). Recently, alternate methods have been proposed to shorten the relaxation time and thereby, reduce CSD. These methods all modify site-by-site Metropolis and heat bath Monte Carlo to operate on larger spacial scales. One of these newer techniques is to apply multigrid methods to site-by-site Monte Carlo algorithms; another is to stochastically determine clusters of sites on the lattice by simplifying the Hamiltonian until it is determinate.We have applied both techniques to the Ising model and compared the relaxation time constants to those determined by site-by-site Monte Carlo methods and found that they are lower. However, even after we succeeded in vectorizing the algorithms, the computation time needed to calculate each sweep of the lattice is larger than that needed by the site-by-site Monte Carlo methods. The important quantity is the computation time needed to move from one independent configuration to another, which is the time needed to calculate each sweep of the lattice multiplied by the relaxation time constant. The net result of the Multigrid Monte Carlo method is that it is less efficient than regular Monte Carlo whenever the parameters of the algorithm are fixed such that the algorithm satisfies detailed balance. The net effect of the Stochastic Blocking method is an improvement compared to regular Monte Carlo when the coupling constant is close to the critical point. We believe the methods used here can be adapted to lattice gauge theory calculations.U of I OnlyETDs are only available to UIUC Users without author permissio