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, which correspond to different phases of the system.   
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 proceedure is as follows.  Let us consider a given interacting Hamiltonian, and write the associated partition function in path integral form:
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>
 
and write the associated partition function in path integral form:


<math> \displaystyle Z=\int{D[\psi^{\ast},\psi] e^{-S[\psi^{\ast},\psi]}},</math>
<math> \displaystyle Z=\int{D[\psi^{\ast},\psi] e^{-S[\psi^{\ast},\psi]}},</math>


where the action <math>S</math> is
where the action <math>S</math> is, in momentum and frequency space,
 
<math> \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 ]}</math>


and <math>H[\psi^{\ast},\psi]</math> 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 <math>\mathbf{r}</math> 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 <math>\beta\rightarrow\infty</math>.  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:
<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>


<math> \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)},</math>
Note that we are working at zero temperature, so that the usual Matsubara frequency sum becomes an integral.


and similarly for <math>\psi^{\ast}(\mathbf{r},\tau)</math>.
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


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

Revision as of 22:58, 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.

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

and write the associated partition function in path integral form:

where the action is, in momentum and frequency space,

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

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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