Finite-lattice approximations to renormalization groups

Accurate calculations of the thermodynamic properties of a system near its critical point are now possible using finite-lattice approximations to renormalization groups. In order to investigate some features of such approximations, we introduce a linear-renormalization-group transformation appropriate to a system of continuous spins like the Gaussian model of Berlin and Kac. We solve the renormalization-group equations for the Gaussian model exactly, and then study in detail a finite-lattice approximation to the renormalization group for the three-dimensional case. The renormalization-group transformation contains a parameter $b$. The exact transformation has fixed points only for $b=2^{-\frac{5}{2}}$. The approximate transformation has fixed points for a range of values of $b$ around $2^{-\frac{5}{2}}$. Eigenvalues of the transformation, which determine critical exponents, depend on the value of $b$ used in the calculation. We study the problem of identifying the correct value of $b$.