RGApproachToInteractingFermions: Difference between revisions

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=Introduction=
=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 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.
 
The basic procedure is as follows.  Let us first consider a non-interacting Hamiltonian in momentum space,
 
<math> \displaystyle H=\int\frac{d^d\mathbf{k}}{(2\pi)^d}\epsilon(\mathbf{k})a^{\dagger}(\mathbf{k})a(\mathbf{k}), </math>
 
where <math>a^{\dagger}(\mathbf{k})</math> creates a fermion with wave vector <math>\mathbf{k}</math> and write the associated partition function in path integral form:
 
<math> \displaystyle Z=\int{D[\psi^{\ast},\psi] e^{-S[\psi^{\ast},\psi]}},</math>
 
where the action <math>S</math> is, in momentum and frequency space,
 
<math> \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). </math>
 
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 <math> -\Lambda\leq\epsilon(\mathbf{k})-\mu\leq\Lambda</math>.


=System at half-filling in one dimension=
=System at half-filling in one dimension=

Revision as of 23:07, 30 November 2010

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.

System at half-filling in one dimension

System with spherically symmetric Fermi surface in two or three dimensions

System with non-spherically symmetric Fermi surface in two dimensions

System with nested Fermi surface in two dimensions

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