# Topology and Superconductor Symmetry Breaking

I read an article today on stack exchange titled "Superconductor Symmetry Breaking". The 2016 Nobel Prize was awarded for research on topological phase transitions in the study of superconductors and as an example how KT transition can be used to explain how superconductivity disappears at higher temperatures.

Is symmetry breaking in superconductors ultimately some description of topological spaces or are gauge theories describing spontaneous symmetry breaking in superconductors actually a subset of some deeper topological description of superconductivity.

Or perhaps superconductivity comes in many guises and different aspects may be described by disparate mathematical descriptions.

As a hobbyist I can only ask the experts. I am curious about the topic. Tags maybe not correctly chosen.

• I think your question is a good one, although far too broad to answer in one go. Could I suggest you read physics.stackexchange.com/questions/133780/… and also en.wikipedia.org/wiki/Spontaneous_symmetry_breaking and then come back with a specific focused question ( or a series of them), on the aspects you do not follow.
– user108787
Oct 5, 2016 at 21:11
• Yes your first link was what I read. The problem is I do not have enough background to be asking the question correctly and yes you are right I need to regroup and ask the question in a way that has more meaning. I have one up vote to the answer below mostly because some kind person was good enough to answer but since I do not even understand one complete sentence in the answer I cannot say if someone might just be pulling my leg. That's what I get for only having an M.S. in Math and not physics. bad me. But maybe the answer can help someone else and that's a good thing.
– user86411
Oct 6, 2016 at 3:11
• The answer is genuine, rest assured of that, and not really meant for you, (nor I ), but for those with the background to follow it. You asked a question involving both symmetry and topology so..........As I say, I would suggest taking your time and learning it bit by bit. You are asking questions that required the combined efforts (and associated mistakes and false assumptions), of many talented people for decades, to develop a knowledge of, so even an overview of these subjects may not fit in any single answer box. Best of luck with the background study and "smaller" questions.
– user108787
Oct 6, 2016 at 4:31

Despite being particularly illuminating, the answer by Lawrence B. Crowell is a bit too technical regarding the general nature of the question.

As you guess in your question, superconductivity is a concept with many many different facets. To list a few of them : superconductivity is

• a state of matter characterised by zero resistance flow of electric current and perfect diamagnetism. That's the original observation by Kammerling-Onnes and Meissner & Ochsenfeld in the 10's - 20's of the XX-th century.
• almost a Bose-Einstein condensate of charged particle. This is how London understood superconductivity, leading to the modification of the Maxwell equations in superconductors, and the first description of the superconducting phenomenology: the penetration effect and the notion of fluxoid in non-singly-connected superconductors, see e.g. the book by London [London, F. (1961). Superfluids, volume I: Macroscopic theory of superconductivity. book, Dover Publications, Inc.]
• a second order phase transition driven by a spontaneously symmetry breaking. In particular, the order parameter corresponds to the gap which opens when passing from a gapless metallic phase to a superconducting phase. A model describing the gap effects and useful for drawing simple pictures of the superconducting phenomenology is called the semi-condutor picture of superconductivity.
• an emerging macroscopic quantum phenomena. This can be seen in many ways: the simplest one being that the order parameter is a complex function, i.e. it can be seen as a macroscopic wave function, see e.g. the Ginzburg-Landau formalism
• a phase of matter with phase rigidity and long range order (Note: phase of matter refers to the state of matter, whereas phase rigidity refers to the phase of the macroscopic wave function of the order parameter.) The phase rigidity explains the persistance of current over macroscopic distances, the Josephson effect, the presence of vortex, ... Say quickly it does not give much more insight than the Ginzburg-Landau formalism, but it has a nice name :-)
• a phase with topological defects in the form of vortex and lattice of vortices. This is the famous Abrikosov lattice, historically found from the Ginzburg-Landau formalism. Please do not confound topological defect and topological phase, though for superconductors these two notions are intricate, see below. For a general overview of topological defects, I suggest the review by [Mermin, N. D. (1979). The topological theory of defects in ordered media. Reviews of Modern Physics, 51, 591–648.] unfortunately behind paywall.
• a transmutation phenomena from fermionic electrons to Cooper pairs (in the ground state) and quasi-particles (as excitations). This is the basic idea behind the Bardeen-Cooper-Schrieffer formalism, in addition to the phase transition. At the heart of the BCS formalism lies the guess of a microscopic Ansatz for the wave function of a superconductor in its ground-state (at zero-temperature if you prefer), now called the BCS Ansatz. This Ansatz is nothing but the coherent state one can construct from a Fermi sea (a gas of electrons at zero temperature if you prefer), and it associates pairs of electrons into a pseudo-bosonic state, though the exact nature of a Cooper pair is not really bosonic. So in short the state at zero temparature of a metallic phase is not the usual Fermi sea, but a quantum gas of BCS correlations (perhaps better to say a quantum liquid in that case) when there is attraction between electrons mediated by phonons, the so-called Cooper mechanism. The BCS formalism (especially the reconstruction of BCS ideas by Gor'kov using mathematical tools inherited from quantum-field theory) describes the phase transition, describes the famous isotope effect (demonstrating how the phonons participate to the onset of superconductivity), and allows to demonstrate the validity of the Ginzburg-Landau approach, so it describes as well all the phenomenologies already mentionned above. There is still some debates whether the BCS formalism describes the London momentum effect, but that's an other story full of polemics, so it has nothing to do in this overview.
• an example of a Higgs phenomena. I will not discuss this much, since it is almost completely encoded in the Ginzburg-Landau theory, at least at the basic level I would like to keep this discussion.
• a topological phase of matter (and here comes the beast !)
• ...

and I do not speak of low-dimensional superconductors which you mention in your question, e.g. phase of vortex-antivortex and the Kosterlitz-Thouless phase transition, or the Josephson effects, or the competition between superconducting and magnetic orders, ... just to focus on the general concepts.

I've tried to adopt a historical classification above, for the sake of convenience. Surely this list is not complete, since we do not understand completely the phenomenon associated with superconductivity (as in high-temperature superconductors, or iron-based superconductors for instance).

One has to note that the different facets listed above describe some of the phenomenologies of superconductivity, but most of them do not describe all the phenomenology. Clearly, the last sentence is tautological, since it's almost clear we do not know yet the complete phenomenology of superconductivity. I've tried to list quickly the phenomenologies described by all approaches.

There is not a lot of documentation about the topological aspects of superconductivity, since it's merely a simple rephrasing of the phenomenology in a new mathematical language. Using the terminology and the concepts of topological phase, one can describe the response of a superconductor under magnetic field, especially the generation of vortices. To the best of my knowledge the Josephson effect and the London momentum effect are not yet described in term of topological state of matter. For more details about the topological phase concepts behind superconductivity you can check the article by

Hansson, T. H., Oganesyan, V., & Sondhi, S. L. (2004). Superconductors are topologically ordered. Annals of Physics, 313, 497–538, also on arXiv:0404327 (open access).

In this paper, the first section is aimed as an introduction to the concepts of topological phases, which the authors demonstrate in the following sections. I think the first section is easy to follow, so I stop my answer here, and I let you come back with more specific questions if you still have some.

Have fun !

Post-scriptum: What Lawrence B. Crowell partly tried to explain (the answer is far more richer) is the so-called Anderson's theorem : as long as there is no time-reversal symmetry breaking interaction in the superconductor (like e.g. magnetic impurities or magnetic field), the superconducting phase is robust. This is one example of a symmetry protected topological state then.

• This answer takes time to appreciate.
– user86411
Jun 11, 2017 at 21:52

The physics is Symmetry Protected Topological (STP) states, and the wikipedia overview of STP states is a decent starter. The physics in part stems from the observation of the quantum Hall and fractional quantum Hall effect. It is physics that stems from a topological order, which can be fractional statistics (anyons etc). Another related physics is the mixing of electric and magnetic monopole charge in a dyon, but as yet that is not detected other than as a sort of Dirac monopole thread through a crystal.

The central physics involves topological insulators, that is very active research topic in condensed matter physics. A topological insulator is a particular instance of a more general concept called a symmetry-protected topological phase of matter (or SPT phase). Consider a d-dimensional hunk of material, where usually D = 3 with a (d-1)-dimensional boundary. The material in an SPT phase has boring physics of the d-dimensional bulk. This insulator occurs because there is an energy gap that is a, obstruction to low-energy propagating excitations. Conversely the physics of the (d-1)-dimensional edge is exotic and exciting, where the boundary or edge might support “gapless” excitations of arbitrarily low energy which can conduct electricity. This more interesting physics exhibited by the edge is a consequence of a symmetry. This physics is stopped or destroyed if the symmetry is broken either explicitly or spontaneously, which is what defines the phase as “symmetry protected.”

The boundary states on the d-1 dimensional is anomalous. In particular with the quantum Hall effect charge on the boundary exhibits a chirality. In effect it appears that charge is not conserved for an observer only paying attention to the boundary. This means this effective field theory is emergent from a d dimensional theory in the bulk. The apparent violation of charge conservation means that $U(1)$ symmetry appears violated.

This is where dyons enter the picture. We know from elementary physics that the electric field obeys $\vec F~=~q\vec E$ and that $\vec F~=~d^2\vec x/dt^2$ that the electric field is invariant under time reversal. However, the magnetic field is different for its force law is $\vec F~=~q\vec v\times\vec B$, where the force again is time reverse invariant, but the velocity is not. This means the magnetic field acquires a sign change with time $t~\rightarrow~-t$. The chirality of the charge carriers (electrons) means there is some magnetic field property The Montenen-Olive duality between electric and magnetic charge $eg~=~2\pi\hbar$. As a result there is some mixing of charge, where this angle is a mixing angle $\theta$. The charge is electric for $\theta~=~0$, magnetic for $\theta~=~\pi$ and back to electric for $\theta~=~2\pi$. These angle numbers are doubled for fermions! As a result for $\theta~=~0$ the bulk is a trivial insulate or vacuum. However, for $\theta~=~\pi$ (bosonic case) or $\theta~=~2\pi$ (fermionic case), which occurs discontinuously on the boundary. This cancels out the chirality of the electrons.

The physics contains the Lagrangian $${\cal L}~=~\epsilon^{\mu\nu\alpha\beta}\partial_\mu(A_\nu\partial_\alpha A_\beta)~=~\epsilon^{\mu\nu\alpha\beta}(\partial_\mu A_\nu\partial_\alpha A_\beta~+~A_\nu\partial_\mu\partial_\alpha A_\beta)$$ $$=~\epsilon^{\mu\nu\alpha\beta}(F_{\mu\nu}F_{\alpha\beta}~+~j^\mu A_\mu).$$ In the last row the first term contains terms $\vec E\cdot\vec B$, which has this apparent symmetry violating principle. The role of the exotic physics in the bulk is to flip the dyon angle to account for this. It is worth noting the theory of the axion plays a similar role with CP violations with QCD. This physics may have very deep roots in the foundations of physics going far beyond solid state physics.