InteractingFermions

Effective field theory and RG approach to interacting Fermions

Introduction

One of the most beautiful aspects of physics is the versatility of ideas and methods. An approach that is developed for a specific problem in a specific field may turn out to be very powerful for a completely different application. One example of such a basic, but extremely powerful tool is the idea of symmetries and in particular broken symmetries. (Broken symmetries are, in spite of their quite misleading name, perfectly valid symmetries only realized in less straightforward way than unbroken symmetries.) In this work, the method of effective field theories (EFTs) and the Renormalization group (RG), applied to interacting fermions, will be discussed. EFTs are typically associated with particle physics, the classical example being the Fermi theory of the $\beta$-decay. However, their usefulness is by far not confined to this field as will be showed in the following. This work is based mainly on the review article by R. Shankar [1] and a TASI lecture by Polchinski [2].

Effective Field theory and the Renormalization group

Basic idea

Consider a quantum field theory that has a characteristic energy scale $E_{0}$. Suppose on is interested only in the physics at $E\ll E_{0}$. In that case, effects at low energy can be described very well by an Effective Field Theory (EFT). The EFT can look very different from the full high energy theory, it can have different interactions and even be written in terms of different fields. One example of an EFT that differs a lot from the underlying full theory is Chiral Pertubation Theory (ChPT), enabling physicists to make prediction in the low energy, strongly coupled regime of QCD. While the fundamental fields are quarks and gluons, the low energy theory is described in terms on pions, Kaons etc. This example also illustrates why EFTs are useful: Not only can they facilitate computations, in cases when the full theory is strongly coupled and perturbation theory breaks down they are the only way to go.

Suppose your theory contains a scalar fields $\phi$. Choose a cutoff $\Lambda$ at roughly $E_{0}$ and write

${\begin{aligned}\phi &=\phi _{H}+\phi _{L}\\{\text{where}}\quad \phi _{H}:&\omega >\Lambda \qquad {\text{high-frequency part of }}\phi \\\phi _{L}:&\omega <\Lambda \qquad {\text{low-frequency part of }}\phi \end{aligned}}$

The action is rewritten as

${\begin{aligned}S(\phi _{L},\phi _{H})=S_{0}(\phi _{L})+S_{0}(\phi _{H})+S_{I}(\phi _{L},\phi _{H})\end{aligned}}$

Where $S_{0}$ is quadratic in the fields and contains either only $\phi _{L}$ or only $\phi _{H}$. The interactions encoded by $S_{I}$ can mix $\phi _{L}$ and $\phi _{H}$. And the generating functional is

${\begin{aligned}\int _{\omega <\Lambda }{\mathcal {D}}\phi _{L}\int _{\omega >\Lambda }{\mathcal {D}}\phi _{H}e^{iS(\phi _{L},\phi _{H})}=\int _{\omega <\Lambda }{\mathcal {D}}\phi _{L}e^{iS_{0}(\phi _{L})}\underbrace {\int _{\omega >\Lambda }{\mathcal {D}}\phi _{H}e^{iS_{I}(\phi _{L},\phi _{H})}e^{iS_{0}(\phi _{H})}} _{\equiv \exp[iS_{\text{eff}}(\phi _{L})]}\end{aligned}}$

where we defined the effective action $S_{\text{eff}}(\phi _{L})$.

If the full action $S(\phi _{L},\phi _{H})$ is known, we can calculate the effective action from the definition above. But even if we have no clue about the full theory, we can still expand in terms of all operators ${\mathcal {O}}_{i}$ that are compatible with the symmetries of the problem:

${\begin{aligned}S_{\text{eff}}=\int d^{D}x\sum _{i}g_{i}{\mathcal {O}}_{i}\end{aligned}}$

This is an infinite sum, but we will see that we can classify the operators by dimensional analysis and that only a handful of operators will turn out to be actually important.