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Example for a quantum anomaly from solid state phys?

  1. May 30, 2010 #1


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    In quantum field theory of high energy physics one encounters so called anomalies like e.g. of the kind discussed in this thread:
    As far as I understood, it basicallymeans that the classical theory has a higher symmetry than the qft. Does anybody know of an example from solid state physics?
  2. jcsd
  3. May 31, 2010 #2
    One example I can think of is actually the emergence of new physics in order to cancel an anomaly.

    In the quantum Hall effect the electrons form a strongly correlated system confined to a 2 dimensional plane and in the presence of a magnetic field aligned perpendicular to this plane. An important aspect of this system is the emergence of a (statistical) gauge field. In the fractional case this gauge field is described through a Chern-Simons term in the action, which is an example of a topological field theory.

    The CS-term contains a gauge structure, which can be traced back to the conservation of charge. A gauge transformation leaves the action invariant everywhere in the (2+1) dimensional spacetime, except at the boundary of the system. The boundary, which is a (1+1) dimensional subspace, breaks the gauge invariance of the action and the theory is therefore anomalous at the boundary. Specifically, performing a gauge transformation gives back the original action, which is gauge invariant, plus a boundary term, which is not gauge invariant. The gauge degrees of freedom are "frozen in" at the boundary.

    There are two ways to deal with this anomaly, which are actually just different viewpoints of the same resolution. The first is to limit the allowed gauge transformations, such that you only consider those transformations which leaves (the value of) the gauge field at the boundary intact. The second is to introduce a boundary term which transforms with respect to a gauge transformation in such way as to cancel the anomaly. In both cases you end up with a dynamical theory at the boundary. It's precisely this boundary term which describes the edge state of the quantum Hall effect. Since CS-theory breaks time invariance, this boundary term turns out to be chiral. In addition, due to the topological nature of the CS-theory this boundary term is also conformal. It's called a Wess Zumino Witten term. What you are dealing with is the existence of one-way moving edge states, which turn out to be the current-carrying states of the quantum Hall effect.

    This is one of the main characterics of the quantum Hall effect from a field theoretic point of view: the emergence of a (2+1) dimensional Chern-Simons term which, by itself, is anomalous, accompanied by a (1+1) dimensional CFT (a WZW term) such that the overal anomaly is canceled again.
  4. May 31, 2010 #3


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    Very interesting! Do you have some reference where to look up the details?
  5. Jun 2, 2010 #4
    Tried to look up a proper source which isn't too technical, but they are hard to find. For info on CS theory applied to the quantum Hall effect you may try this one:


    But that source does not treat the edge states (or the anomaly I am talking about). For that you need to consult:


    Specifically, chapter 3.3 explains in detail how you can derive the theory of the edge states using the Chern-Simons theory in the bulk combined with an anomaly cancellation (I do not know if it is called that way in this article)

    A more general review article would be:


    Although the part on the Chern-Simons theory and the description of the edge states is a lot more technical in this article.

    These articles might be a bit heavy on the technical side if this is your first exposure to the quantum Hall effect.
  6. Apr 13, 2011 #5
    Sorry for responding to this old question, but I think http://arxiv.org/abs/1010.0936" [Broken] (click) paper is exactly what you are looking for. Many different kind of anomalies are applied in a condensed matter physics context, even gravitational anomalies.
    Last edited by a moderator: May 5, 2017
  7. Apr 13, 2011 #6


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    Thank you! I'll give it a look.
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