It has been a long time since my last post. I never expected that the graduate season, even for a Master’s course, would be so busy. I prepared for 1) the JSPS DC1 application (it is a better scholarship, although failed to be selected, sad), 2) master’s thesis for graduation, 3) reapplication of my dormitory KIV (again got rejected), and then 4) took almost a month or more to find a cozy private apartment. Moving to the new place was also very time-consuming. All of a sudden, I’ve entered the doctoral course and lots have changed. Luckily, I finally get to return to my normal physics study and research mode!
From a discrete lattice model to a field theory: why?
When starting to learn the 2D Ising model two years ago, I was very confused why people could even use a very different-looking $\phi^4$ field theory to study it. For example, in Mehran Kardar’s Statistical Physics of Fields, he used the famous Landau-Ginzburg symmetry argument to write down the partition function of the $\phi^4$ field theory. It is of course a well-accepted approach, but I have been very uneasy about this since the first time I heard about it. The leap between a discrete model and a field theory is so large, and there are no quantitative connection between the two descriptions. It would be better if there is a direct way connecting the two.
From the 2D Ising model to the $\phi^4$ field theory through the Hubbard-Stratonovich transformation
As a companion reading of the Wen’s QFT book, the chosen book of the textbook-reading session of our group meeting this semester, I started to scan the Condensed Matter Field Theory by Altland and Simons. Very unexpected, when treating the $\phi^4$ field theory in Chapter 5, he gave a concrete explanation on how the $d$-dimensional Ising model can be described by the $\phi^4$ theory.
I don’t want to repeat his argument here. I just want to mention that a trick called Hubbard-Stratonovich transformation is a key to this derivation. In this transformation, the spin-spin interaction term will be deliberated shifted away, in the expense of the introduction of a real field dynamical variable $\phi$. The summation over all the spin configurations becomes a functional integral over the field configurations, $\phi$. Most remarkably, this approach gives expression of the coupling constants in the $\phi^4$ theory in terms of the original Ising model. For example, the coupling between the quadratic term is like $c_2 \sim 1 – \beta C(0)$, where $\beta$ is the inverse temperature and $C(0) > 0$ the Ising spin coupling. This form of $c_2$ is a key to the second-order phase transition in the Ising model. If you are interested and want to see the detailed argument, please look at the starting part of Chapter 5 of Altland and Simons’s book.
After learning BSC theory of superconductivity and also the Landau-Ginzberg’s treatment (I watched Prof. Sachdev’s lecture videos of Physics 295B in Harvard), I got to know that Gorkov used a very similar method to relate the microscopic BSC theory to the Landau-Ginzberg’s coarse-grained picture. Cool!