= Solution
On the physical density-phase contour put $\bar\psi=\sqrt\rho\,e^{-i\phi}$. Direct differentiation gives
$$
\bar\psi\partial_\tau\psi=\frac12\partial_\tau\rho+i\rho\partial_\tau\phi,
\qquad
|\nabla\psi|^2=\frac{|\nabla\rho|^2}{4\rho}+\rho|\nabla\phi|^2.
$$
Integrating the <Laplacian> by parts with the stated bulk boundary convention, and dropping the periodic total <derivative> of $\rho$, produces the <density-phase action of a Bose gas>,
$$
S=\int d\tau\,d^dr\left[i\rho\partial_\tau\phi
+\frac{|\nabla\rho|^2}{8m\rho}+\frac\rho{2m}|\nabla\phi|^2
+\frac g2(\rho-\rho_0)^2-\frac{\mu^2}{2g}\right].
$$
Write $\rho=\rho_0+\delta\rho$ and $\phi=\phi_0+\varphi$, where $\phi_0$ is constant. To quadratic order,
$$
\boxed{S_2=\int d\tau\,d^dr\left[
\frac g2(\delta\rho)^2+i\delta\rho\partial_\tau\varphi
+\frac{|\nabla\delta\rho|^2}{8m\rho_0}
+\frac{\rho_0}{2m}|\nabla\varphi|^2\right].}
$$
The uniform static density fluctuation has positive curvature $g$. Its inverse static susceptibility at <wavevector> $q$ is $A_q=g+q^2/(4m\rho_0)$, which remains nonzero at $q=0$. This is the sense in which the amplitude or density fluctuation is massive. The phase has no term proportional to $\varphi^2$ without <derivatives>; a uniform phase <rotation> costs no <action>. It is the massless direction required by global $U(1)$ <symmetry>.
The term $i\delta\rho\partial_\tau\varphi$ shows <density-phase conjugacy>. It is important to distinguish <static density stiffness versus an excitation gap>: the density and phase do not form two independent normal-mode branches with one gapped and one gapless particle. In this nonrelativistic first-order theory they supply the conjugate variables of one positive Bogoliubov branch, derived below. The omitted cubic terms include $\delta\rho|\nabla\varphi|^2/(2m)$ and corrections from the density-gradient denominator.
There is also a background term $i\rho_0\int d\tau\,d^dr\,\partial_\tau\varphi$. It vanishes for smooth fluctuations with zero temporal winding. Periodicity of $\psi$, however, permits $\varphi(\beta)-\varphi(0)=2\pi w$. The <compact phase winding term> must be retained when such sectors are included; it is not generally the same constant in all sectors. The local Gaussian <phonon> calculation concerns the smooth zero-winding sector and does not describe vortices or points where the density vanishes.
Back to article page
