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, or points which these flows may end at, since these give us some important information about the system. For example, each fixed point corresponds to a certain phase of the system. Since there are many different sets of initial parameters that all flow to the same fixed point, we have an explanation of universality, or the observation that many different systems all possess similar physical properties, such as critical exponents.

The basic procedure is as follows. Let us first consider a non-interacting Hamiltonian in momentum space,

$\displaystyle H=\int {\frac {d^{d}\mathbf {k} }{(2\pi )^{d}}}\epsilon (\mathbf {k} )a^{\dagger }(\mathbf {k} )a(\mathbf {k} ),$

where $a^{\dagger }(\mathbf {k} )$ creates a fermion with wave vector $\mathbf {k}$ 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, in momentum and frequency space,

$\displaystyle S=\int _{-\infty }^{\infty }{\frac {d\omega }{2\pi }}\int {\frac {d^{d}\mathbf {k} }{(2\pi )^{d}}}\psi ^{\ast }(\mathbf {k} ,\omega )(-i\omega +\epsilon (\mathbf {k} )-\mu )\psi (\mathbf {k} ,\omega ).$

Note that we are working at zero temperature, so that the usual Matsubara frequency sum becomes an integral.

We will now go through the first part of any RG calculation - determining the RG transformation. To do this, we first impose a cutoff on the momentum integral, so that the energy is restricted to a range $-\Lambda \leq \epsilon (\mathbf {k} )-\mu \leq \Lambda$.