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  Is it possible to couple an odd number of Dirac fermions, at finite density, to a massless gauge field in 2+1d?

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In a beautiful paper by A. N. Redlich on the parity anomaly (PRL 52, 18 (1984), no arXiv version), the author indicates that an odd number of Dirac fermions can never be coupled to a massless gauge field in 2+1d due to the necessity of introducing background Chern-Simons terms to maintain gauge invariance

"The induced topological term $±W[A]$ in $I_{eff}[A]$ is known to produce a mass for the gauge fields. Not only must parity conservation be violated in odd-dimensional theories with an odd number of fermions, but the gauge fields $\textit{must}$ become massive as well",

where $W[A]$ is the Chern-Simons term and $I_{eff}[A]$ the effective action obtained by integrating out the fermion degrees of freedom.

My question is whether this is still true if the fermions are at a finite density, i.e. one adds a chemical potential such that there is a whole Fermi surface of excitations (in this case, it's my understanding that one cannot simply integrate out the fermions to get $I_{eff}[A]$). Is it possible to couple an odd number of Dirac fermions, at finite density, to a massless gauge field in 2+1d?

Note: I'm glossing over issues to do with whether the presence of massless fermions can stabilise 2+1d gauge theories against confinement (I'm assuming here that they can). Also I don't mind whether parity conservation is preserved or not, it's just the mass of the gauge field I'm interested in.    

asked Aug 19 in Theoretical Physics by anonymous [ no revision ]

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