RGApproachToInteractingFermions

Introduction

The renormalization group (RG) is a very powerful tool in physics. Essentially, it is a way to continuously map a given theory onto other theories possessing the same low-energy physics by successively integrating out "fast", or high-energy, modes. This is expressed in terms of differential equations giving the "flows" of different coupling constants that appear in the theory.

The basic proceedure is as follows. Let us consider a given interacting Hamiltonian, and write the associated partition function in path integral form:

$\displaystyle Z=\int {D[\psi ^{\ast },\psi ]e^{-S[\psi ^{\ast },\psi ]}},$

where the action $S$ is

$\displaystyle S=\int _{0}^{\beta }{d\tau \left[\sum _{\bf {r}}{\psi ^{\ast }({\bf {r}},\tau )\left({\frac {\partial }{\partial \tau }}-\mu \right)\psi ({\bf {r}},\tau )+H[\psi ^{\ast },\psi ]}\right]}$

and $H[\psi ^{\ast },\psi ]$ is the Hamiltonian in normal-ordered form with all the bosonic (fermionic) creation and annihilation operators replaced with complex (Grassman) numbers. Note that we are considering a theory on some underlying lattice, so that ${\bf {r}}$ is the position vector for a point on the lattice.