Application of the Green's-function Monte Carlo method to the compact Abelian lattice gauge theory

Abstract

We have applied the Green's-function Monte Carlo (GFMC) method to the Hamiltonian formulation of the compact U(1) lattice gauge theory in three and two (space) dimensions on small lattices, 3\ifmmode\times\else\texttimes\fi{}3\ifmmode\times\else\texttimes\fi{}3 and 5\ifmmode\times\else\texttimes\fi{}5. The GFMC method is a Monte Carlo method of finding the ground state of a quantum-mechanical system with many degrees of freedom, by iteration of an integral operator of which the ground state is an eigenstate. An interesting aspect of this method is an importance-sampling technique that makes use of a trial wave function to accelerate convergence of the Monte Carlo estimates. We used two importance functions in these calculations, which were designed to be accurate in the small- and large-coupling limits. These importance functions were optimized by the variational principle; the results of the variational calculations are interesting in their own right. Our Monte Carlo results exhibit evidence of the phase transition of the three-dimensional compact U(1) lattice gauge theory, and indicate the nonexistence of a phase transition in the two-dimensional theory.