First off, I would like to point out that solid state physics is not like quantum mechanics or maybe QFT in that you can articulate (nearly) the whole
theory under a mathematical formulation, starting from a set of axioms and going on. In an excerpt from the last reference we can read the following:
While in related fields, such as Statistical Mechanics and
Atomic Physics, many key problems are readily formulated in unambiguous
mathematical form, this is less so in Condensed Matter Physics, where some say that rigor is "probably impossible and certainly unnecessary". By carefully selecting the most important questions and formulating them as well-defined mathematical problems, and then solving a good number of them, Lieb has demonstrated the quoted opinion to be erroneous on both counts. What is true, however, is that many of these problems turn out to be very hard. It is not unusual that they take a decade (even several decades) to solve.
The theoretical developments in condensed matter are, I think, to a large extent motivated by experimental
observations. Phenomenological models are built that with time are set on a more rigorous formulation. Besides, we build models that we think can explain the observations with each model requiring a specific type of math. Once we can explain the rough characteristics, then we include more and more details. There is always a balance of what we want to reproduce and how simple (and enlightening) the model is. The more details the more involved the mathematical model is so it increasingly requires more advanced math. Sometimes we need advanced math from the beginning though. So, you will probably not find a single mathematical treatment of everything but several models scattered all around. Therefore, I think it is best to first read general books of solid state to find what problems exist and then pick the model you like the most. In the following a set of topics within solid state theory and some references for the most rigorous treatments I found are presented.
You will find many mathematical models to play with, like Thomas-Fermi theory, DFT, tight-binding, Hubbard, ...
- An introduction to First-Principles Simulations of Extended Systems by Fabio Finocchi, Jaceck Goniakowski, Xavier Gonze, Cesare Pisani. Handbook of Numerical Analysis, Vol. X, p. 377.
- Computational Quantum Chemistry: A primer by Eric Cances, Mireille Defranceschi, Werner Kutzelnigg, Claude Le Bris, Yvon Maday, part III, Handbook of Numerical Analysis, Vol. X, p. 3.
- A seminar designed for mathematicians: MSRI-LBNL 2016 Summer School on Electronic Structure Theory. Lecture videos are available online.
- Dynamical Theory of Crystal Lattices - M. Born, K. Huang.
There are phenomenological theories (BCS theory, Ginzburg-Landau theory, ...) but there is not yet an established theory. Researchers are trying to use QFT to explain the phenomena.
- Introduction to superconductivity, Tinkham.
Quantum Hall effect:
This field is still in a nascent state, is evolving and active.
- Lieb, Elliott H. Condensed matter physics and exactly soluble models. Selecta of Elliott H. Lieb. Edited by B. Nachtergaele, J. P. Solovej and J. Yngvason. Springer-Verlag, Berlin, 2004. x+675 pp