Phase–Density EFT for Bose Superfluids

Writing a neutral Bose field as $\psi=\sqrt n\,e^{i\theta}$ makes density and phase a conjugate pair. A positive static density cost allows the density variable to be eliminated at long wavelengths, giving a phase-only effective action whose two coefficients are the compressibility and superfluid stiffness; their ratio fixes the sound speed.

Required background. Use Bogoliubov theory for the microscopic sound mode and the general treatments of hydrodynamic fields and constitutive data. Helpful background. EFT operator selection explains the derivative expansion.

Density and phase variables

For the contact Bose action, dropping a total derivative gives

$$\mathcal L=-n\,\partial_t\theta -\frac{n}{2m}(\boldsymbol\nabla\theta)^2 -\frac{(\boldsymbol\nabla n)^2}{8mn} +\mu n-\frac{g}{2}n^2.$$

The Berry term shows that $n$ is conjugate to $\theta$. The phase is compact, $\theta\sim\theta+2\pi$. On a site regulator with Cartesian field measure, the change to density and phase gives $d(\operatorname{Re}\psi)\,d(\operatorname{Im}\psi)=\tfrac12\,dn\,d\theta$ at each site, with nonnegative density and one full phase period. The local Jacobian is therefore constant and can be absorbed into the overall normalization. This does not remove the coordinate singularity at zero density or the compact winding sectors. Expand $n=n_0+\pi$ about $\mu=gn_0$. To quadratic order,

$$\mathcal L_2=-\pi\dot\theta-\frac{n_0}{2m}(\nabla\theta)^2 -\frac{g}{2}\pi^2-\frac{(\nabla\pi)^2}{8mn_0}.$$

The density and phase fluctuations form one canonical pair. Their coupled quadratic equations give one Bogoliubov branch, not a separate gapped density excitation; a positive static density cost alone does not establish an additional pole.

At momenta $k\xi\ll1$, the quantum-pressure term is subleading. Completing the square and integrating $\pi$ gives

$$S_{\mathrm{ph}}=\frac12\int\mathrm dt\,\mathrm d^dx \left[\kappa(\partial_t\theta)^2-\Upsilon(\nabla\theta)^2\right], \qquad \kappa=\frac{\partial n}{\partial\mu},\quad \Upsilon=\frac{n_s}{m},$$

with $\kappa=1/g$ and $n_s=n_0$ at this order and at zero temperature. Therefore

$$\omega^2=c^2k^2, \qquad c^2=\frac{\Upsilon}{\kappa}=\frac{gn_0}{m}.$$

This matches the small-$k$ Bogoliubov pole, an independent check of normalization Altland and Simons 2023, § 5.2, pp. 242–257.

Nonlinear completion and cutoff

Galilean invariance organizes the leading action through $X=\mu-\dot\theta-(\nabla\theta)^2/(2m)$, so $\mathcal L=P(X)$ for a zero-temperature irrotational fluid. Expanding the pressure $P$ reproduces the density, compressibility, and phonon self-interactions Son and Wingate 2006, §§ 2–3, pp. 201–209. On a lattice this Galilean identity is absent: stiffness and density are independent.

The smooth phase theory excludes vortex cores, where $n$ vanishes and $\theta$ is singular. It also fails for $k\xi\gtrsim1$, near a Mott transition where amplitude and phase scales mix, and whenever damping or additional conserved modes must be retained. The compact sectors become central on the BKT page.

Exercises

Integrate out $\pi$ from $-\pi\dot\theta-g\pi^2/2$ and check the sign of the kinetic term.

Solution

Complete the square: $-g(\pi+\dot\theta/g)^2/2+(\dot\theta)^2/(2g)$. The Gaussian integral removes the square and leaves a positive coefficient $\kappa=1/g$ for repulsive $g>0$, consistent with positive compressibility.

References

• Alexander Altland and Ben Simons, Condensed Matter Field Theory, 3rd ed., Cambridge University Press (2023), § 5.2, pp. 242–257, doi:10.1017/9781108781244.
• D. T. Son and M. Wingate, “General Coordinate Invariance and Conformal Invariance in Nonrelativistic Physics: Unitary Fermi Gas,” Annals of Physics 321 (2006) 197–224, §§ 2–3, doi:10.1016/j.aop.2005.11.001.