Mathematical rigorous introduction to solid state physics

Question

I am looking for a good mathematical rigorous introduction to solid state physics. The style and level for this solid state physics book should be comparable to Abraham Marsdens Foundations of mechanics or Arnols mechanics book for classical mechanics or to Thirrings Physics course for quantum mechanics.

Any recommendations?

Edit: As a reaction to Peter Shor's comment, I try to narrow the scope of the question a bit and give some more specific subareas of solid state physics I am in particular interested in:

• semiconductors and applications
• the quantum hall effect
• superconductivity

Answer 1

The following books discuss rigorous methods in solid state physics:

• "Renormalization group" by G. Benfatto and G. Gallavotti, see this link.
• "Renormalization: an introduction" by M. Salmhofer, see this link.
• "Fermionic functional integrals and the renormalization group", J. Feldman, H. Knorrer and E. Trubowitz, see this link.
• "Non-perturbative renormalization" by V. Mastropietro, see this link.

See also the course by Rivasseau given at the CIME school in Cetraro, September 2010.

Answer 2

The problem with this kind of books is that there is no special mathematics in solid state physics. There are books with titles like "Quantum Field Theory in Solid State Physics" or similar: modern methods in solid state originate from QFT, quantum chemistry and alike. Thus, rigorous introduction may be found there and not in solid state itself.

If you could specify particular topic, probably it would be possible answer your question.

Answer 3

You can wonder about the stability of matter in quantum mechanics or get caught by disorder to learn the rigorous aspects of localization in disordered systems.

EDIT: May 7, 2012

Some more

• Mathematical Physics of Quantum Wires and Devices: From Spectral Resonances to Anderson Localization

• Spin Glasses: A Challenge for Mathematicians: Cavity and Mean Field Models

• Quantum Hall Systems: Braid Groups, Composite Fermions, and Fractional Charge. Also at arxiv

Answer 4

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.

Mathematical crystallography:

• Geometric Crystallography: An Axiomatic Introduction to Crystallography by P. Engel. More here: http://blogs.iucr.net/crystalmath/2013/07/31/what-if-anything-is-mathematical-crystallography/. As to what is interesting now read this article.

Electronic structure:

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.

Lattice dynamics:

• Dynamical Theory of Crystal Lattices - M. Born, K. Huang.

Superconductivity:

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:

• Mathematical Aspects of the Quantum Hall Effect by Jürg Fröhlich.
• Quantization of Hall Conductance for Interacting Electrons on a Torus, Matthew B. Hastings, Spyridon Michalakis, Communications in Mathematical Physics, Volume 334, Issue 1, pp 433–471.

Topological Insulators:

This field is still in a nascent state, is evolving and active.

• Colloquium: Topological insulators

Other:

• 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

Answer 5

In terms of Quantum Field Theory and Topological Quantum Field Theory application in Solid State/Condensed Matter physics, that starts from the basics set-up (like Abelian Chern-Simons theory, B-F theory relevant for the [fractional-]Quantum Hall effect that you mentioned) to the more advanced recent topics on Symmetry protected topological (SPT) states, topologically ordered gauge theories, and symmetry enriched topologically ordered (SET) states, the following References are particularly accessible to Mathematicians and Math-Physicists.

Here are two References:

1. arxiv 1510.07698 Three Lectures On Topological Phases Of Matter by Witten, La Rivista del Nuovo Cimento, 39 (2016) 313-370

2. arxiv 1612.09298 -- Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions by Putrov-Wang-Yau, Annals of Physics 384C (2017) 254-287

A list of new topological field theories and their topological invariants are organized and studied cases by cases in physical relevant to Solid State/Condensed Matter physics 2+1 and 3+1 spacetime dimensions. e.g.

Answer 6

So if you want to understand Qhe and SC rigouresly, you really need to go in field theory language and there are a lot of wonderfull books on that topic.

1. Fradnklin condensed matter field theory this book is mostly related to topological field theory.

2. altland and simons condensed matter field theory so this is more gentle intorduction than the first book, also there are lecture notes of simons in the inetrnet so if you find something not clear in that book go to that lecture notes.

these two books would give an excelent math rigor to soild state physics.

on top of them you can use

3. wens many body book

this is an excelent but very hard book, because it assume that you are very comfortable with QFT, so first you should get your self comfortable with qft by using 2.

in terms of diffucuilty the order is from hardest to easiest is 3 1 2.

these three books are pretty standart in this field, so they should be more than enough in terms of superconductivty and quantum hall effect and much more, I am not sure about semiconductors though.

Answer 7

M.A. Omar, Elementary Solid State Physics, Addison-Wesley, 1993.