Review: Quantum Field Theory and Condensed Matter, An Introduction

I just finished my first pass on Shankar’s QFT book, Quantum Field Theory and Condensed Matter, An Introduction. The book ranges from the standard Ising model and path integrals topic to more advanced techniques like bosonization and quantum Hall effect, so it becomes more difficult as you proceed. I went through Chapter 1 to Chapter 15 and the remaining parts just look beyond my capability right now. Still, I feel it a very entertaining reading experience and have learned a lot in the first 15 chapters.

I started this book on Feb 17, 2020, right after I came back to Japan from China. It took me approximately 6 – 10 hours per week reading the book and doing the exercises, and three months later, I end up finishing Chapter 15 on May 18. 

First thing first, you should already know something about field theory in order to understand this book. By field theory, I mean functional integral specifically. For example, in  Chapter 13 where renormalization of $\phi^4$ theory is discussed, Shankar writes down the Green’s function for the free theory without any explanation. For people who are not familiar with functional, this might be hard to follow. As for me, I finished Kardar’s Statistical Physics of Field and the first chapter of Zee’s Quantum Field Theory in a Nutshell, which I think provide a very nice and detailed discussion of functional integral. 

Next let’s talk about the contents of this book. The reason I choose to read this book is that I learned in Kardar’s book that the 2d Ising model, which is purely classical statistical system, can be somehow mapped to a free fermion problem, and thus, becomes solvable. Kardar discusses this in the context of summing over so-called phantom loops in high-temperature expansion of 2D Ising model. This provides a nice picture of what is happening in this model but the derivative contains a few leaps of faith. I’m looking for more formal derivation in Shankar’s book.

In Chapter 2 – 4, Shankar takes the simplest model in statistical physics, 1d Ising model, as an example, to show how to go back and forth between quantum and statistical mechanics. Actually, the mapping itself is interesting enough, since it provides us a different point of view of a problem: we can treat it as statistical or quantum problem as we like! In Chapter 7 – 8, Shankar uses the concepts developed in Chapter 2 – 4 to provide a clear and beautiful exact solution to 2d Ising model on square lattice. The overall procedure is first to map the original statistical problem into its quantum version, then the key step is to apply the Jordan-Wigner transformation so we have a clear anti-commutation relation and expressions of transfer matrices contain quadratic in fermions, which can be easily solved in momentum space. After taking care of the boundary condition, Shankar takes the tau-continuum limit of the model and provides a nice picture of the final solution.

In Kardar’s treatment, the free energy is calculated by summing over high-temperature expansion loops. The analysis uses the techniques in Markov process. In Shankar’s treatment, we can see how the degeneracy of the ground states energy in even and odd sector gives rise to phase transition. I would say both points of view are fantastic and I recommend read both of them!

Other parts of the book talks about various useful techniques. In Chapter 5, the treatment of instanton is extremely nice, way better than Altland’s book. Chapter 6 covers coherent state path integrals for spins, bosons and fermions, a topic confused me for a long time, now clear! Majorana fermions are introduced in Chapter 9, which can be seen as like “half” set of ordinary fermions discussed in Chapter 6. After talking about $\mathbb{Z}_2$ gauges theories in 2 and 3 dimensions, we have a total 6 chapters, 11 – 16, talking about renormalization group method. It is a must-read if you want a detailed study on this topic. I think the most interesting part is that Shankar compares the renormalization in QFT and renormalization group in QFT and critical phenomena, discussing connections of these closed related topics.

Shankar’s QFT book is really nice and he explains everything crystal clear. I will definitely come back and finish the last 5 chapters about bosonization and quantum Hall effect after I become more mature in condensed matter physics.