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. We are often interested in determining the fixed points of these flows, which correspond to different phases of the system.

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

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

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 $\mathbf {r}$ is the position vector for a point on the lattice. From this point forward, we will be considering the limit of zero temperature, so that $\beta \rightarrow \infty$. We will also be working in momentum and frequency space, rather than position and imaginary time space, so we introduce the Fourier transform of the fields:

$\displaystyle \psi (\mathbf {r} ,\tau )=\int _{-\infty }^{\infty }{\frac {d\omega }{2\pi }}\int _{\text{B.Z.}}{\frac {d^{d}\mathbf {k} }{(2\pi )^{d}}}\,\psi (\mathbf {k} ,\omega )e^{i(\mathbf {k} \cdot \mathbf {r} -\omega \tau )},$

and similarly for $\psi ^{\ast }(\mathbf {r} ,\tau )$.