Consider a (2+1)d continuum field theory with the Minkowski action $$\mathcal{L}=|\partial\phi|^2-r|\phi|^2-u|\phi|^4,$$ where $\phi$ is a complex field. The theory undergoes a quantum phase transition by tunning $r$ (this theory can be regarded as the effective description of Bose Hubbard model in the vicinity of the critical point).

In the dual picture, we have $$\mathcal{L}_d=|(\partial-a)\phi_v|^2-r_v|\phi_v|^2-u_v|\phi_v|^4+f^{\mu\nu}f_{\mu\nu}$$ where $\phi_v$ corresponds to the vortex field and is also complex valued. The gauge field is related with the orignal theory by $$j_\mu\sim \epsilon_{\mu\nu\lambda}\partial_\nu a_\lambda$$ where $j_\mu$ is the 3-current associated with the global $U(1)$ symmetry of the original Lagrangian.

The superfluid phase of the original theory corresponds to the gapped phase of $\phi_v$ in the dual theory and the superfluid mode is identified with the gapless mode: photon in the dual theory. The insulating phase of the original theory corresponds to the symmetry breaking phase of $\phi_v$ in the dual theory where vortices have condensed. By Anderson-Higgs mechanism, the gauge field will gain a mass.

My questions are:

  1. When we say vortices condense, are we referring to 'vortex condensation' or 'antivortex condensation' or other cases? It seems there is no preference for vortex or antivortex.

  2. In the insulating phase (vortices condensed in the dual theory), will the vortex superfluid if we inject a vortex into the system?


The tangle of vortex loops in the condensed phase is best thought of from the viewpoint of Euclidean space-time. Then a time slice across a single loop looks like a vortex/anti-vortex pair and so for any gas of closed loops there are are equal numbers of vortices and antivortices.

The "condensation" means that the energy (action) cost per unit length is more than offset by the entropy gain in the number of vortex configurations and so long loops become cheaper than cheap. The way in which such long loops correspond to actual Bose condensation in nicely described in Feynman's Satistical Mechanics book.

I don't understand your second question. It does not make sense grammatically. Is there a word left out or a typo?

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  • $\begingroup$ Hi Mike, thanks for your answer ;). For the second question, what I want to ask is that will the vortices move without friction (like a superfluid) if we inject vortices into the system? (by the way, could you specify which section you are referring to in Feynman's book? thanks.) $\endgroup$ – Ji Zou May 28 '19 at 20:50
  • $\begingroup$ In euclidean space-time nothing moves. The partition function is just sums over configurations of loops in the 3d space. I don't have Feyman's book to hand, but the same material on Bose condensation and long loops is in chapter 11 of my book "The Physics of Quantum Fields" if you have access to a library or an e-copy. $\endgroup$ – mike stone May 28 '19 at 21:14
  • $\begingroup$ Thanks for your information. $\endgroup$ – Ji Zou May 28 '19 at 22:50

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