General description of renormalization techniques in real space

1 General description of renormalization techniques in real space 

Iterative renormalizations were introduced around 1970 by Wilson and were widely used by him they were inspired by an earlier article by Kadanoff.5 To perform such renormalizations it is really necessary to use a computer. 

The initial set is described by a model Hamiltonian Jt 0 °o which will be assumed to be classical. Then, the statistical state is defined by the weight W0 = exp[ — JF0 0ct ]. For reasons of simplicity, the factor fi— 1 /kBT is considered as included in the Hamiltonian. Actually, this model Hamiltonian can be rather different from the true Hamiltonian and it may depend on temperature in a complex manner through k. The weight IV0 is associated with the configurations of the systems in other words, it is a function of the values taken by the degrees of freedom. 

The set can also be defined through a new Hamiltonian ct, which is a function of the new degrees of freedom constructed from the initial degrees of freedom °ct. Thus, we form a series of Hamiltonians. iF0 °ct, iF, V, .. , , N nct resulting from iterations defined by setting (for each N)  

This definition can easily be justified in particular we remark that owing to (12.2.1), the partition function 

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