Create Hamiltonians in condensed matter with group theory

[Amentia]

Hello,

I am currently struggling to understand how one can write a Hamiltonian using group theory and change its form according to the symmetry of the system that is considered. The main issue is of course that I have no real experience in using group theory.

So to make my question a bit less general, take for example this article:

https://journals.aps.org/prb/abstract/10.1103/PhysRevB.6.3836

What I would like to understand is how we see that some matrices transform as a given representation, for example here they say the spin 1/2 matrices transform as Gamma 4 for the Td symmetry (3rd page). And how to use it to write the final Hamiltonian we want to obtain? I would like to be able to do that with any direction of magnetic field, stress or whatever applied to a given structure with any initial symmetry...

All the articles I try to understand that make use of group theory for such purposes just state the result as if there were obvious just by looking at the matrix of interest and the character table. Which must be the case but I do not see it.

I do not know if my question is clear. But any help is welcome.

Best regards!

 

[Dr Transport]

Hamiltonian's are invariant with respect to the operations of the group, so only certain combinations of the group elements are allowed. Check out any text on group theory and quantum mechanics (Tinkham is a good start but is terse).

 

[Amentia]

Hello, thank you for your answer. Do you know any book or good pdf that would have exercises with detailed answers on this topic? (find a hamiltonian from group theory)

 

[Cthugha]

I am not exactly sure what you are looking for. Most of this is about knowing the symmetry of perturbations and checking what symmetry the combination of the perturbation and the initial symmetry yields. A very detailed look at group theory is given in the lecture notes of a course taught long ago by Mildred Dresselhaus at MIT. Course notes are still available at MIT (http://web.mit.edu/course/6/6.734j/www/group-full02.pdf), but it has more than 700 pages.

The standard book containing all the tables at a glance would be "Properties of the thirty-two point groups" by Koster. It has been digitized by Google and is available via the Hathi Trust (https://catalog.hathitrust.org/Record/001114463).

 

[Dr Transport]

Look at this article by JM Luttinger, https://journals.aps.org/pr/abstract/10.1103/PhysRev.102.1030. You don't use group theory to write the Hamiltonian, you apply is after the fact to see what solutions are viable and their symmetries given the symmetry of the problem (cubic, tetragonal, etc...)

 

[Amentia]

Thank you for all the references, this is exactly what I am interested in. In fact, I followed a short introduction on group theory and I can see the symmetry elements in a crystal or a molecule. But it is less clear when I have to consider the magnetic field, the spin matrices, a photon, a quasiparticle, etc. Does it follow the same logic? I do not understand how I should use the character tables in those cases.

 

[Dr Transport]

You have to use the properties of the magnetic field, with all of that you are getting into the regime of magnetic group theory.