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If you consider a massless relativistic fermion theory and you perform a chiral transformation, then you realize that while the classical action remains invariant under this transformation the generating functional does not. The non-conservation of a classical symmetry in the quantum limit is called an anomaly and in the high-energy scenario it appears when we try to regularise the thoery.

What I don't understand is what happens in a condensed matter system (Weyl semimetals) where you have a physical cut-off. What is the anomalous in this case?

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    $\begingroup$ The reason is that things ("anomaly flow") happen "outside" the effective field theory. For example, in Weyl semimetal, the "anomalous" charge flow from one Weyl point to another is simply because these Weyl points are just corners of the same energy band and we decide to ignore that fact in the low-energy field theory. Also see the answer in physics.stackexchange.com/q/177031. $\endgroup$
    – Meng Cheng
    Commented May 4, 2015 at 15:44
  • $\begingroup$ @ Meng Cheng: In the link that you sent me you write "If we apply an electric field, obviously it drives a current" but if the theory has a physical regulator then no current is observed (see pg42 at www.theory.caltech.edu/~preskill/ph230/notes/230Chapter3-Page1-22.pdf ) $\endgroup$
    – Hipparkhos
    Commented May 4, 2015 at 16:34
  • $\begingroup$ It is just a physical fact that if you apply an electric field to a metal, you observe a current. No knowledge of anomaly is needed for this. Now this can not be explained by a regulator in the field theory that does not break the chiral symmetry, however it just reflects the incompleteness of the field theory. $\endgroup$
    – Meng Cheng
    Commented May 4, 2015 at 16:42

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