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. Here, we will discuss how to perform RG for fermions, which differs in some respects from the case with bosonic fields. This discussion will follow that in Shankar's paper [1]. The most important difference, one which is true of any fermionic system, is that we have a Fermi sea in the ground state of our system. This means that, unlike in the bosonic case, we cannot simply impose a cutoff on the momentum, restricting it to the inside of a sphere of radius $\Lambda$. We will illustrate the correct procedure as we go along for different cases.

System at half-filling in one dimension

We will begin with a simple system - a Fermi gas in one dimension at half filling.

References

[1] R. Shankar, Rev. Mod. Phys. 66, 129 (1994).