Informal write-ups

These are some casual documents I have prepared on various topics while learning them, either for courses I was attending or for my tutor assistantships. They might be helpful as very quick introductions or as an opinionated conglomeration of the ideas of certain topics.

• The tight-binding model on the 2D square lattice
  This is a quick introduction to the tight-binding problem on a 2D square lattice. I describe how it can be obtained from a more general model, its solution and some of its important properties like the isoenergetic contours and van Hove singularities.

• Topological significance of the crystal momentum
  This is a re-working of R. Rajaraman’s demonstration that the tight-binding dispersion can be obtained by treating the problem of electrons moving in a periodic potential in terms of instantons. This has the advantage that the crystal momentum can be identified as a topological winding number.

• R. Shankar’s RG derivation for Landau Fermi liquid and BCS instability
  This is an expansion of R. Shankar’s derivation (Shankar, 1993) of the Landau Fermi liquid effective Hamiltonian and its possible BCS instability by applying the renormalization group technique on a general 2D interacting fermionic system with a circular Fermi surface. I have added a short introduction to the philosophy of renormalisation group. Certain non-trivial arguments and calculations have also been fleshed out. Where possible, certain parts have been simplified.

• Lightning-quick introduction to single site dynamical mean field theory
  This is a very short introduction to the philosophy and algorithm of dynamical mean field theory (DMFT). I brought these points together and wrote this up mostly to cement my own understanding of the topic. I first discuss the Curie-Weiss mean field theory in the context of the Ising model in order to provide a familiar language, and set up in a slightly different way so that it is easily generalised to DMFT. This might be useful for anyone wanting to know, in brief, what DMFT is, and how it is implemented.