Real-space Renormalization Group Techniques for Lattice Systems

This thesis explores the application of real-space renormalization group (RG) techniques in the study of critical phenomena in statistical physics. Critical phenomena are characterized by diverging length-scales that manifest with the emergence of correlation functions decaying as power laws both at large spatial and temporal distances and more in general by the presence of singularities in the free energy in the thermodynamic limit. This implies that a macroscopic system close to or at criticality cannot be understood in terms of the properties of its finite subparts. The renormalization group provides a general framework to explain the emergence of singularities in the thermodynamic limit using an iterative procedure involving only an analytic recursion equation. Typically, the implementation of RG at Wilson leads to the so-called proliferation of interactions between degrees of freedom, which is difficult to handle. Some kind of approximation, such as perturbation theory or brute force truncation, is used to simplify the analysis and these approximations are justified a posteriori on the basis of the results obtained. This thesis aims to comprehensively explore and analyze existing RG methods, including decimation, Migdal-Kadanoff bond moving approximation, cumulant approximation, and Monte Carlo renormalisation group methods, in order to gain valuable insights into the critical behavior of various lattice models.