Real Space Renormalization in Statistical Mechanics

This paper discusses methods for the construction of approximate real space renormalization transformations in statistical mechanics. In particular, it compares two methods of transformation: the "potential-moving" approach most used in the period 1975-1980 and the "rewiring method" as it has been developed in the last five years. These methods both employ a parameter, called \chi, that measures the complexity of the localized stochastic variable forming the basis of the analysis. Both methods are here exemplified by calculations in terms of fixed points for the smallest possible values of \chi. These calculations describe three models for two-dimensional systems: The Ising model solved by Onsager, the tricritical point of that model, and the three-state Potts model. The older method, often described as lower bound renormalization theory, provides a heuristic method giving reasonably accurate results for critical indices at the lowest degree of complexity, i.e. \chi=2. In contrast, the rewiring method, employing "singular value decomposition", does not perform as well for low \chi values but offers an error that apparently decreases slowly toward zero as \chi is increased. It appears likely that no such improvement occurs in the older approach. A detailed comparison of the two methods is performed, with a particular eye to describing the reasons why they are so different. For example, the old method quite naturally employed fixed points for its analysis; these are hard to use in the newer approach. A discussion is given of why the fixed point approach proves to be hard in this context. In the new approach the calculated the thermal critical indices are satisfactory for the smallest values of \chi but hardly improve as \chi is increased, while the magnetic critical indices do not agree well with the known theoretical values.