Hubbard-Stratonovich treatment of Gross-Pitaevskii
S. Gubser, June 2014
In my lectures, I went through a Hubbard-Stratonovich transformation of the Gross-Pitaevskii
action, following Zee’s work [1], but I did not explain correctly why the Goldstone boson caneventually be treated as a lagrange multiplier enforcing a divergence free condition on thefour-vector whose Hodge dual winds up being the NS-NS three form field strength. Zee’s
treatment is correct but telegraphic. Here I try to walk through the steps more carefully.
Our starting point is the lagrangian
L1 = −ρη̇ −ρ(∇η)2 ,2m(1)

and it’s integrated over R3,1 to get part of the Gross-Pitaevskii action. η is the Goldstoneboson, which in qualitative terms we want to replace with a two-form gauge field B2 . To
this end, we first observe that
L1 = −ξ µ ∂µ η +2m ~2 m ~ ρξ −
ξ − ∇η ,2ρ2ρm(2)

where ξ 0 = ρ and ξ~ is the Hubbard-Stratonovich field. (2) is a straightforward algebraic
equality—no path integral tricks yet!
Now let’s consider what’s going on in the path integral when we perform HubbardStratonovich. First we note that ZZ2 Rm
ρi d4 x L14e= D ξ~ exp i d x L1 +ξ~ − ∇η2ρm

 Z
Z
m ~24µ~.
= D ξ exp i d x −ξ ∂µ η + ξ2ρ
where we have defined the measure and contour so that ZZ
m4
2D ξ~ exp i d x ξ~ = 1 .
2ρ

(3)

(4)

In the first equality of (3), we are assuming that the measure D ξ~ is invariant under a shiftof integration variables ξ~ → ξ~ − ρ ∇η. The second equality in (3) is a slight rearrangementmof (2).

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Our next step is to do the path integral over η:1
ZiDη eRd4 x L1=ZD ξ~Z Z
m ~24µ,Dη exp i d x −ξ ∂µ η + ξ2ρ

(5)where we have assumed that we can switch the order of integration. The first term in square
brackets on the right hand side of (5) can be integrated by parts to give η∂µ ξ µ . Then the ηintegral reduces to a functional delta function enforcing the constraint∂µ ξ µ = 0(6)at every point on R3,1 . We solve (6) by setting1ξ µ = ǫµνλσ Hνλσ6(7)where H3 = dB2 . The reason that (7) works is that (6) is implied by the equality of mixed
partials acting on components of B2 . (7) means in particular thatmX 2m ~2H .
ξ =2ρ4ρ i,j 0ij(8)

2The Hijkterm that is desired as part of the NS-NS two-form dynamics comes from the(0)lagrangian I called L2 in the lecture. To be precise, Hµνλ = Hµνλ + hµνλ where the only non(0)(0)

zero components of Hµνλ are H123 = ρ0 and other components related by index permutations.

ρ0 is the background superfluid density, and hijk describes density fluctuations around it.
We can still set H3 = dB2 . To summarize, (nearly) the whole claim isZ

 Zρ(∇ρ)2 g
422DρDη exp i d x −ρη̇ −(∇η) −− (ρ − ρ0 )
2m8mρ
2
 ZZ(∇hijk )2g µα νβ λγ4,= DB2 exp i d x − η η η hµνλ hαβγ −1248mρ0(9)where η µν = diag{− c12 , 1, 1, 1} and c2 = gρ0 /m.
The main way in which the claim (9) needs improvement is that it neglects the vortex
strings, which sit at the locations where η has 2π monodromy. As explained in lecture,
1For the step (5), what we need is to assume that η is a well-defined real-valued function on R3,1 . This
fails in the presence of vorticity. To proceed in that case, we split η = ηvortex + ηsmooth , where ηvortex isheld fixed, corresponding to some fixed location of vortices in the superfluid, and ηsmooth is the well-defined
real-valued part which we regard as the fluctuating field to be integrated over in (5). Later, we can integrate
over the position of the vortices to get the fully dynamical system of strings plus NS-NS two-form.

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R
including the vortex strings leads to a term 2π Σ b2 in the action, where b2 is the pullback(0)

of B2 (including the part of B2 that gives Hijk ) to the string worldsheet. If we properly
account for the core of the strings (which is a small region of normal phase), there is alsoR√some Nambu-like term −τ1,bare d2 σ − det gab that comes into the action.
There are additional ways in which the claim (9) is slightly tricky and/or incomplete:• The superfluid density ρ0 appearing on the right hand side of (9) should really havebeen ρ, and the proper thing to do is to expand it around ρ0 to get some interactions of

the H3 field with itself. These interactions were uniformly neglected in my lectures, butin principle they are there, and a diagrammatic treatment of them could be developed.
• There’s a non-trivial claim that the Jacobian between DρD ξ~ and the natural measure

DB2 for the two-form gauge field can be neglected. It seems to me that there is notmuch possibility of a problem in the long-distance physics arising from this technicalissue. Of course, to properly describe the path integral over B2 , one must develop some
sort of gauge-fixing technology.
• The higher derivative term in square brackets on the right hand side of (9) gets dropped
once we pass to the infrared limit.

Now let me answer a few questions during or after the lecture which I didn’t fully address:
1. Q: It doesn’t look like η enters simply enough into (2) to be treated as a Lagrange
multiplier. When we integrate η out of (2), doesn’t it result in some extra terms for ξ?
A: Good point! This is why it’s important to make the slight rearrangement of (2)that came up in (3). In the last expression in (3), η appears just once, and after an
integration by parts, you can see that it appears as η∂µ ξ µ .
2. Q: If ρ/m is treated as small, the extra dependence of (2) on η goes away. Are werelying on small ρ/m in order to treat η as a Lagrange multiplier?
A: The slight rearrangement discussed in previous answer shows that η truly becomesa Lagrange multiplier, and we don’t rely on the smallness of ρ/m or any other combination of parameters.
3. Q: Would it be more proper to dualize ξµ = 16 ǫµνλσ Hνλσ without demanding H3 = dB2 ?
A: It should be fine to make the substitution ξµ = 16 ǫµνλσ Hνλσ early on, as a convenientalgebraic shuffling of variables. At some point we must integrate over η, and as soon3

as we do so, it should enforce the condition dH3 = 0, which we can immediately solveas H3 = dB2 since R3,1 is topologically trivial. In short, an early substitution of thissort should result in an equivalent treatment to the one given here.

References[1] A. Zee, “Vortex strings and the antisymmetric gauge potential,” Nucl.Phys. B421
(1994) 111–124.

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