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

+ 2 like - 0 dislike
71 views

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.

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverfl$\varnothing$wThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.