= Solution
For a dilute three-dimensional <weakly interacting Bose gas> with repulsive short-range scattering, use a contact coupling $g=4\pi\hbar^2a_s/m>0$ and $na_s^3\ll1$. In imaginary time $\tau_E$, the grand-canonical action is
$$
S_E=\int_0^{\beta\hbar}d\tau_E\int d^3x\left[\bar\psi\hbar\partial_{\tau_E}\psi+\frac{\hbar^2}{2m}|\nabla\psi|^2-\mu|\psi|^2+\frac g2|\psi|^4\right],\qquad Z=\int\mathcal D\psi\,e^{-S_E/\hbar}.
$$
The uniform saddle has $n_0=\mu/g$. Its real-time equation is the <Gross–Pitaevskii equation>; quantum depletion means condensate and total density need not coincide exactly.
Writing $\psi=\sqrt{n_0}+\delta\psi$ and retaining quadratic fluctuations gives a Nambu kernel with diagonal $\epsilon_k+gn_0$, anomalous coupling $gn_0$ and $\epsilon_k=\hbar^2k^2/(2m)$. The <Bogoliubov quasiparticle dispersion> is
$$
\boxed{E_k=\sqrt{\epsilon_k(\epsilon_k+2gn_0)},\qquad E_k\simeq\hbar c_s|k|,\quad c_s=\sqrt{gn_0/m}.}
$$
Phase and density fluctuations form the gapless sound mode associated with broken global $U(1)$ symmetry. At large momentum the spectrum crosses towards particle behaviour; the <healing length> is $\xi=\hbar/\sqrt{2mgn_0}$. The depletion integral $\int d^3k\,v_k^2/(2\pi)^3$ is finite in three dimensions and gives $(n-n_0)/n\simeq8\sqrt{na_s^3}/(3\sqrt\pi)$ to leading order. The contact coupling denotes a scattering-length-renormalized interaction; naive ultraviolet zero-point integrals require renormalization.
The condensate phase gives <superfluid velocity> $\mathbf v_s=(\hbar/m)\nabla\theta$ and quantized circulation $\oint\mathbf v_s\cdot d\mathbf l=2\pi\hbar\ell/m$. The <Landau criterion>, $v_c=\min_kE_k/(\hbar k)$, gives $c_s$ for this homogeneous Bogoliubov spectrum. Boundaries and vortex nucleation can cause dissipation below that value, so it is not a universal measured critical speed. Condensation and superfluid stiffness are related but distinct; dimensional fluctuations can remove true condensate order while retaining superfluid responses.
\Image[/past-exam-of-the-quantum-condensed-matter-field-theory-course-of-the-university-of-cambridge/2014/paper-1-qcm2014-bose-dispersion.png]
{title=The dilute-gas Bogoliubov spectrum crosses from sound to particle-like excitations}
{source=paper-1-qcm2014-bose-dispersion.py}
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