Renormalization Groups

Introduction

This is another introductory post leading towards a numerical renormalization group program. I thought I should expose some generalities before presenting the numerical renormalization group method and the program that implements it. It’s mostly a collection of links rather than a detailed description.

Some links

I already supplied a link at the end of the post about the Kondo Effect but I promised more of them. Here are some of them (more will follow in the next post), starting with one of the most important, The renormalization group: Critical phenomena and the Kondo problem¹. It’s not only about the numerical renormalization group, it’s more general than that. Wilson got a Nobel prize for that work, by the way.

A hint of renormalization² is a general description of renormalization groups. It’s not only about them in condensed matter or statistical physics but also about the renormalization in particle physics, where the idea of renormalization originated. Renormalization groups might be easier to visualize in real space than in the dual space so here is a thesis about Real Space Renormalization Group Techniques and Applications³, with quite a bit of information about the Density matrix renormalization group. Since I mentioned it, here is another one originating from the same time period, 1990s: Functional renormalization group.

Because at the top of the post I added an image of the numerical renormalization group flow to be used as an example, I also point to a review paper about the numerical renormalization group: The numerical renormalization group method for quantum impurity systems⁴. In there you’ll find a similar chart but with different parameters.

Here is another review: Diagonalization- and Numerical Renormalization-Group-Based Methods for Interacting Quantum Systems⁵.

Renormalization

Let’s suppose we have some function that describes a system. For example it might be a Hamiltonian or the action or the partition function. Let’s suppose it’s a Hamiltonian H having a set of parameters K (most of them couplings but it could be also mass, for example). A renormalization group transform is a mapping along with a change of scale. The change of scale needs not to be in the real space, it can be a change of scale in momentum space, a change in the scale of energy.

It’s a mapping that preserves the Hamiltonian form unchanged but now the coupling constants are not constant anymore. The rest is details. Quite complex details, in many cases, though. To see some details you could visit the links I provided.

The mapping can be written in a simpler form that emphasizes the parameters change: .

Fixed Points and Group Flow

The set of parameters of the Hamiltonian is a point in the parameter space. The above transformation applied in a sequence generates a trajectory in the parameter space. That’s the ‘renormalization group flow’. By the way, a renormalization group might not be a group, it might not admit an inverse.

A point in the parameter space where there is an invariance is a fixed point. For a point close to a fixed point, the mapping can be linearized: . Assuming that the linear operator has a complete vector set one can use it to write: where are the eigenvectors. A transformation applied n times gives:

where are the eigenvalues of the linear operator.

For the corresponding parameters get smaller and smaller, they attract the trajectory towards the fixed point. Those are the irrelevant parameters.
For the corresponding parameters get bigger, the trajectory is driven away from the fixed point. Those parameters are the relevant parameters.
The ones that correspond to eigenvalues equal with one are the marginal ones. Because of the nonlinearity they might end up in driving the trajectory either away from the fixed point or towards it.

An example

The image at the top of the post is a numerical renormalization group flow for the Anderson model at half filling. In the chart there is the energy spectrum, scaled along the renormalization group flow.
On the horizontal axis is the step of the renormalization group, on the vertical axis, the energy. Again, the energy is scaled each step (only even steps are charted).

The fixed points are quite visible in the chart. There is more to it, if one uses the program⁶ to chart a Kondo model with proper parameters one can get a quite similar chart, except the beginning part. The universality manifests itself, the model differences do not matter at low energies (corresponding to low temperatures).

So, the regimes are, from left to right, that is, from high energies to low energies, corresponding to high respectively low temperatures:
* Free Orbital regime.
* Local Momentum fixed point.
* Strong Coupling fixed point.

By the way, the flow to a single stable fixed point as in this case is characteristic for systems with a single ground state. There are systems with more than one ground state. In such cases the flow can go into either fixed point, depending on where it starts. One can get quantum phase transitions for such a system.

Conclusion

This is the last ‘theory’ post before presenting the program⁶ for the numerical renormalization group.
I did not write much here, the purpose was just to provide more links to interesting papers. I don’t think I’ll give full explanations in the next post, either, it would be way too much for the purpose of this blog.

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