Renormalization Group: Preliminaries

Geoffrey R. Golner

Wilson's exact smooth-cutoff renormalization-group (RG) equation for continuum spin Landau-Ginsburg models is shown to be equivalent to an easily constructed, infinite set of partial differential equations that provide a natural system of successive approximation for numerical calculation. It differs from other RG approaches in that an infinite number of couplings are included at each level of approximation. By way of illustration, preliminary results are presented for Ising-model critical exponents in three dimensions.

Pedro D Sacramento

M. A. Novotny, Hans Gerd Evertz

We report current progress on the synthesis of methods to alleviate two major difficulties in implementing a Monte Carlo Renormalization Group (MCRG) for quantum systems. In particular, we have utilized the loop-algorithm to reduce critical slowing down, and we have implemented an MCRG method in which the symmetries of the classical equivalent model need not be fully understood, since the Renormalization Group is given by the Monte Carlo simulation. We report preliminary results obtained when the resulting MCRG method is applied to the d=1 XXZ model. Our results are encouraging. However, since this model has a Kosterlitz-Thouless transition, it does not yet provide a full test of our MCRG method.

Andrea Carosso, Anna Hasenfratz A, Ethan T. Neil

Gradient flow has proved useful in the definition and measurement of renormalized quantities on the lattice. Recently, the fact that it suppresses high-modes of the field has been used to construct new, continuous RG transformations both analytically and on the lattice, distinct from the usual blocking techniques in spin models and gauge theories. In this work, we discuss such a formulation for scalar field theory, and we present preliminary numerical results for its application to the determination of critical exponents at the Wilson-Fisher fixed point of three-dimensional $ϕ^4$ theory.

G. Benfatto, Giovanni Gallavotti