Dealing with truncation in Monte Carlo renormalization-group calculations.

Abstract

We use the $d=2$ Ising model as a concrete example to show that truncation of the derivative matrix $T$ to a subspace $S$ of short-range interactions needs some justification, since each row of $T$ contains arbitrarily large elements in the long-range sector. We point out that despite this a small parameter justifying truncation may exist, we show how to find it, and we perturbatively correct the errors due to truncation, all without leaving $S$. The bulk of our analysis carries over to other Monte Carlo renormalization-group studies, particularly the $d=3$ Ising model.