Friday, August 2, 2013

The chiral anomaly and topological insulators

Yesterday we had another cake meeting where people "shared their ignorance". 
Mine concerned: what is the relationship between the chiral (parity) anomaly in quantum field theory, edge states, and topological insulators?
I learnt something in the week from this PRL [more string theorists writing about Fermi liquids!], which at least taught me what the relevant "anomaly" equation is for the "violation" of charge conservation.

Ben Powell pointed out that there is a relevant blog post Monopoles passing through flatland by John Preskill. His beautiful post certainly provides the background information I need, in a particularly lucid form. Having read it once I now need to digest it properly. Otherwise, I will still be "ignorant.".


  1. I am in no way an expert on this, but my understanding is as follows. In field theory, anomalies are just symmetries of the classical action which are not preserved by quantum fluctuations (one way to say this is that the path integral measure is not invariant although the action is). These anomalies are typically related to topological obstructions and have very beautiful mathematical structure relating anomalies in various dimensions.

    In the context of D = 2+1 integer quantum Hall effect, anomalies play a role in two different ways. If one considers the edge theory (D=1+1 dimensions), there is a so-called U(1) gauge anomaly. This just means that charge is not conserved and the theory is not consistent by itself, it MUST be the boundary of a higher-dimensional space. Physically this is just the Hall effect, charge flow into the bulk from the boundary (see also the Callan-Harvey paper Preskill mentions). If one considers the D=2+1 dimensional bulk theory instead, the same physics come into play through the so-called parity anomaly. Parity symmetry can be broken by quantum effects, which in different terms means that if you integrate out the electrons there will be a Chern-Simons term in the electromagnetic response (which again give rise to a Hall current). See this chapter of Taylor Hughes PhD thesis for a very simple calculation in a two-band QHE toy-model .

    In D=3+1 dimensions, a theory of fermions coupled to a gauge field can have a so-called Chiral anomaly (sometimes called axial anomaly). This means that if you integrate out the fermions, the electromagnetic response will have an F wegde F term / B.E-term /theta-term (it has many names). Its the term that exist in time-reversal invariant topological insulators, also called the topological magneto-electric effect. Theta = 0 are trivial insulators, while theta = pi are topological.

    It is well-know that this term is a total-derivative of a Chern-Simons term, which means that you will have a Chern-Simons (Quantum Hall) response on the edge of the topological insulator. And from what I said before, the Chern-Simons term is related to Parity anomaly in D=2+1.

    All this structure is captured by the so-called Descent equations. There is the following sequence of relations (k is integer): U(1) gauge anomaly in D=2k --> Parity anomaly in D=2k+1 --> Chiral anomaly in D=2k+2. D is space-time dimension, not spatial dimension.

    There are of course other kinds of anomalies which describe transport effects in other classes of topological insulators/superconductors. For example gravitational anomalies (related to thermal Hall effect), Global anomalies and mixed anomalies. All of this is described very well in .

    I guess the essence is that anomalies are symmetries that are broken by quantum fluctuations. This in turn implies topological charge or thermal response and boundary modes. I am personally still learning, there are some points I am still confused about.

  2. HM,
    Thanks for the detailed comment. It is very helpful.

  3. I'm not sure that the 7 minute ignorance fest worked well as a format, although it may be worth more tries to make sure. The problem seemed to be that if one gets excited about one's ignorance, one has trouble taking in advice in 7 minutes.

    1. I think the problem may have been that rather than being something somewhat simple and well understood, the topics brought up tended to be more like "what I've been wondering about lately". The former being, I suppose, much more work to come up with than the latter.

      I tend to agree that it needs more tries to see if it's ultimately valuable or not.

      For what it's worth, I really enjoyed engaging with each topic for a few minutes. Particularly, I think it was much easier to engage with the topics than it is in the usual 'take 7 minutes to convince us all to read a particular paper'. Of course, it's more difficult to sell a paper than it is to raise a 'simple idea'. Perhaps this is a challenge for the 7 minute paper meetings, to distill it to raising an idea.

      As an example, when you (Seth) talked about the paper where by integrating out one of the nuclear or electronic degrees of freedom, I forget which, but I guess the fast electronic ones?, you ended up with a gauge field which was just a Berry connection. This was cool because there was one concrete concept which was nice to think about for 7(ish) minutes. But then, my paper was about Berry connection too, so perhaps this example sucks.

      Anyway, I think both kinds of 7 minute meetings, if they help the audience to think about a topic for a few minutes and then ponder it in the coming days, that is a successful meeting. However I agree that it would often be the case that the presenter, in the space of only a couple of minutes of feedback, will not make progress with their misunderstanding. Especially when, as you aptly say, they become excited about their ignorance! Nice phrase.

      That was a little disparate, sorry.

    2. I agree we need to refine the format of this meeting.
      Rather than "something complicated I don't understand" maybe we should talk about "something basic I don't understand but think someone in the room can explain to me."
      I set a bad example by talking about the most complicated thing on my list, rather than the simplest, "How does a p-n work?"!