I have been thinking about this problem, but cannot get a satisfactory answer.

In CDW phase, if there is Periodic Lattice Distortion (PLD, or Peierls instability), one may think of the order parameter as the expectation value of the corresponding phonon mode: $\Delta = \langle b_{2k_F} + b^{\dagger}_{-2k_F} \rangle$ (the simple 1-d example, see George Gruner's book), maybe in this case, we can think of there is a BEC of that specific phonon mode?

However, if the origin of the CDW is purely electronic(assume no PLD), where one might define the order parameter as: $\Delta_k = \sum_{k'} V_{k,k'} \langle c^{\dagger}_{k'+Q} c_{k'} \rangle$ It seems that, the $U(1)$ symmetry is not broken in this case.(really?) But one can still do a standard Hubbard-Stratonovich transformation to introduce the auxiliary field (order parameter), which can be set to be complex, then assume the phase of the order parameter is fixed ($U(1)$ broken or not?) and then arrive at the $\textit{gap equation}$ and after expanding around the mean field configuration, we can get the Ginzburg-Landau free energy. So how to understand this discrepancy here? Can we think of the gauge transformation is different for different state? (like: $ c_k \rightarrow c_k e^{i \theta_k}, c_{k+Q} \rightarrow c_{k+Q} e^{i \theta_{k+Q}}$)

Hope someone with a deeper understanding about CDW and symmetry breaking can give a detailed explanation.

  • $\begingroup$ The U(1) symmetry is global, you cannot do this transformation. And I think your order parameter is probably wrong (there should be a sum over k ?). A CDW is a symmetry breaking of the (lattice) translation invariance, not of U(1). $\endgroup$ – Adam Sep 8 '16 at 6:55
  • $\begingroup$ The definition of order parameter has been revised; as for the lattice translation symmetry, shouldn't it be changing from one periodicity to another? As for U(1), I'm just wondering if this kind of way(Ginzburg-Landau) to get the effective field theory is proper. thanks for your comments. $\endgroup$ – Chuan Chen Sep 8 '16 at 7:22
  • $\begingroup$ To elaborate on what Adam writes, the CDW occurs with electron-phonon interactions with a symmetry breaking of lattice vibrations at the Fermi energy gap. I have to ask a further question; are you thinking the lattice symmetry breaking can be shifted by some formalism to the symmetry breaking of $U(1)$? $\endgroup$ – Lawrence B. Crowell Sep 8 '16 at 12:20
  • $\begingroup$ For the case where electron-phonon coupling is the driven force, I wonder if it can be understood as a BEC of the corresponding phonon mode? but I'm more interested in the case where the electron-electron interaction plays the role. $\endgroup$ – Chuan Chen Sep 8 '16 at 13:41
  • $\begingroup$ I am going to keep this question as an open window for a while. I have this suspicion this involves new physics. The question seems to be whether the Hubbard-Stratonovich transformation will convert the interaction from a lattice site to site interaction to a general field. I am presuming the field or auxiliary field here is then the phonon field. The HS transformation in a discrete setting appears to be a form of $\Theta$-function, which is a subject I am engaged in. This would suggest that what you are thinking about is a form of the Hitchen's bundle, $\endgroup$ – Lawrence B. Crowell Sep 8 '16 at 19:41

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