Citation: Mermin, N. David. "The topological theory of defects in ordered media." Reviews of Modern Physics 51.3 (1979): 591.
Web: https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.51.591
Tags: Foundational, Physical, Defects/boundaries, Pedagogical
This paper is perhaps my favorite paper I have ever read. It gives a completely classical discussion of the topological theory of defects in ordered media. While anyons in quantum phases are typically very hard to observe and appear only in very special circumstances, classical defects appear all the time and are impossible to ignore. Many defects are of topological nature, meaning that they cannot be removed via local surgery. By looking at how the order parameter changes around the defect, it is almost tautological that there is a topological classification of defects in ordered media. The surprising thing isn't just that the classification is true, but also that it is useful!
Trying to understand topological defects without homotopy theory quickly becomes very complicated. For example, there is the case of the biaxial nematic. The biaxial nematic is an innocuous seeming physical system whose order parameter space ends up having nonabelian fundamental group. This means that the fusion rules of the defects are nonabelian, and can be used for topological classical computation!
This paper is a really great read for people trying to understand nonabelian anyons. It might seem that pointlike quasiparticles with nonabelian braiding statistics is some strange quantum phenomenon but this is not the case. All of this is completely classical, and by enforcing homotopy equivalence energetically we recover the Kitaev quantum double model.
There is so much stuff in this paper that it is hard to summarize. It is an absolute pleasure to read. Here are some nice quotes:
"At a minimum, homotopy theory provides the natural language for the description and classification of defects in a large class of ordered systems. Whether it will eventually gain as wide a currency among condensed matter physicists as, for example, the language and theorems of the theory of group representations, depends both on how large that class of systems proves to be, and on how many of the nontrivial topological insights turn out to have direct manifestations in the laboratory".
"The need for such a pedagogical article lies in the nature of the mathematical literature on homotopy theory, which can be approached only with great effort by those with the mathematical background of a typical physicist (as I can testify from painful personal experience). The trouble is that expositions which can be followed by one with such a background —i.e., those to be found in introductory undergraduate topology texts —fail to reach many of the theorems and concepts central to the physical applications. Expositions that do go far enough, however, place the subject in so general a mathematical setting and assume so high a level of mathematical expertise, that it is not at all easy to ferret out those occasional bits of the vastly expanded subject that are physically pertinent."