VARIATIONAL THEOREM FOR VECTOR-MEAN-FIELD THEORIES OF STATISTICAL TRANSMUTATION

We examine the validity of vector-mean-field theory (VMFT) for statistical transmutation on large lattices with a high density of particles per site (1/2 and 1/4). We take as a difficult test case the representation of hard-core bosons as fermions plus attached flux tubes. We use a variational Monte Carlo method to test the variational properties of the mean-field ground-state wave function against the predictions of the VMFT. We find a discrepancy of order 1 in the thermodynamic limit. This leads us to postulate that a better formulation on a lattice may be that of a renormalized vector-mean-field theory. We show that the renormalization coefficients can be understood by an analysis of the phase fluctuations (whose magnitude we estimate) of the long-range gauge interaction. These phase fluctuations are of order pi on the lattice (thus leading to a breakdown of VMFT on the lattice) while they vanish in a continuum formulation. We give a detailed discussion of the qualitative differences of VMFT on the lattice versus the continuum. In particular, we examine the effect of having lines of zeros (lattice) versus points of zeros (continuum) for the nodes of the many-body wave function. In addition, a remarkable variational theorem is discovered for the ground-state wave function of the VMFT.