Goldstone modes and Anderson-Higgs mechanism in the context of BCS theory

Question

I'm already familiar with the "Mexican hat" story of Anderson-Higgs mechanism in Landau-Ginzberg theory. However, I have never seen anyone talking about Goldstone and Higgs mechanism in the context of BCS theory of superfluidity and superconductivity. I need the answer to the following questions:

1. In a BCS superfluid, what does the Goldstone mode creation operator look like, in terms of BCS quasi-paritcle operators $\alpha_k=u_k a_k-v_k a^\dagger_{-k}$ ($a_k, a^\dagger_k$ are fermion annihilation/creation operators of momentum $k$)?

2. In BCS superconductivity, what is the operator representation of the gauge boson $A^\mu(k)$ and Higgs boson $H(k)$?

Or to put it another way, say if I want to create a Goldstone boson (with momentum $k$) in superfluid BCS ground state $|\Phi_{BCS}\rangle$, what does the resulting excited state look like?

Answer 1

$\def\kk{{\bf k}} \def\rr{{\bf r}} \def\ii{{\rm i}} \def\ee{{\rm e}} \def\qq{{\bf q}}$ I can't give a complete answer but I will attempt to address question 1). (To be absolutely clear, that means I will talk only about a BCS-type superfluid which is electrically neutral and therefore supports a Goldstone mode. In the charged case, the Goldstone mode is lifted to the plasma frequency by the Anderson-Higgs mechanism.) The original BCS theory actually does not contain a Goldstone mode. This is because one assumes a static condensate described by the order parameter $$\Delta(\rr) = \frac{1}{V}\sum_{\kk,\qq} \ee^{\ii\qq\cdot\rr} \langle a_{\kk+\qq/2,\uparrow}a_{-\kk+\qq/2,\downarrow}\rangle.$$ Here I'm considering a homogeneous $s$-wave superfluid with periodic boundary conditions in a volume $V$. In the ground state, one has a spatially constant order parameter $\Delta(\rr) = {\rm const.}$, which means that Cooper pairs condense into states with zero centre-of-mass momentum $\qq =0$.

The typical BCS mean-field treatment predicts an elementary excitation spectrum consisting of gapped quasiparticles, which are created by breaking apart condensed pairs. However, in an uncharged superfluid, the low-energy excitations actually correspond to gapless collective oscillations of the Cooper-pair condensate, i.e. the Goldstone mode. This leads to a time- and space-varying order parameter $\Delta(\rr,t)$ describing a macroscopic number of Cooper pairs carrying non-zero centre-of-mass momentum $\qq\neq 0$. In other words, the Goldstone mode excitations correspond to the entire condensate being coherently displaced slightly in momentum space. But since in BCS theory the condensate is a classical variable (the mean field), there is no operator describing the dynamics of $\Delta(\rr,t)$. Nevertheless it is possible to compute the spectrum of its excitations using a dynamical extension of BCS theory where the mean-field is time-dependent. In the end, this procedure turns out to be equivalent to the random-phase approximation. A quite comprehensive study along these lines was done by Combescot et al.

A full quantum-mechanical treatment of the Goldstone mode can of course be performed by going beyond mean-field theory. However, usually this is done in the path integral formalism, where there are no operators whatsoever. In this case, the Goldstone mode is an excitation of a Hubbard-Stratanovich field, which is introduced in the Cooper pair channel and used to integrate out the bare fermion field. Quantum fluctuations of the pair condensate are described as Gaussian (or higher-order) fluctuations around the saddle-point describing the BCS ground state. A good original reference for this formalism is Engelbrecht et al. and references therein (unfortunately behind a paywall), although there are many more modern treatments that can also be found by Googling. The topic of fluctuations in BCS-type neutral superfluids is currently very active because of experiments on the BCS-BEC crossover in ultracold atomic gases.

Answer 2

In a superfluid Bose-Einstein condensate, the Goldstone excitations are oscillating perturbations of the order parameter $\psi(\mathbf{r},t)$ above its equilibrium value $\psi_0$. In the long-wavelength limit, they are just sound waves in a superfluid, described by a linearized Gross-Pitaevskii equation.

In a BCS superconductor, the Goldstone modes are oscillations of a Cooper pair condensate. Because of the Coulomb interactions, these excitations are very similar to usual plasma oscillations (bulk plasmons) in a metal, as was first described by P.W. Anderson and G. Rickayzen. In 3D, these modes acquire a gapped dispersion due to long-range character of Coulomb interaction. In contrast, in 2D superconductors the Goldstone modes, as well as usual 2D plasmons, remain gapless.