Weakly interacting Bose gas ($n_{\mathrm{ex}}/n\ll 1$)

A dilute condensed Bose system whose interaction is sufficiently weak for a small-depletion expansion. Its low-energy excitations are described by a Bogoliubov transformation rather than free particles. The homogeneous quadratic theory is controlled only while noncondensate occupation remains small compared with the condensate and its excitation energies are stable.

In units $\hslash =1$, polar fields rewrite the coherent-state action asafter removing the periodic total density derivative. The potential has positive-density minimum ${\rho }_0=\mu /g$ for $\mu >0$. Density curvature gives a finite static amplitude cost, while phase changes cost only derivatives. Number-phase conjugacy couples their dynamics; the two fields do not automatically represent two independent excitation branches.

Expand the density-phase action of a Bose gas around ${\rho }_0=\mu /g>0$ and a constant phase. Density and phase are coupled by $i\delta \rho {\partial }_{\tau }\varphi$, and retaining density gradients gives the full quadratic Bogoliubov spectrum. Dropping those gradients is a long-wavelength approximation. A small-fluctuation Gaussian extension around ${\rho }_0$ does not exactly remove the original nonnegative-density constraint.

In the smooth zero-winding long-wave sector, integrate the density Gaussian integral by completing $g(\delta \rho +i{\partial }_{\tau }\varphi /g{)}^2/2$. Its field-independent normalization leaves the displayed action. The real-frequency pole is ${\omega }^2=g{\rho }_0{k}^2/m$, so the phonon speed is $\sqrt{g{\rho }_0/m}$. Keeping density gradients yields ${\omega }^2=g{\rho }_0{k}^2/m+k^4/(4m^2)$, matching the Bogoliubov spectrum.

A periodic complex field permits $\varphi (\beta ,\mathbf{r})-\varphi (0,\mathbf{r})=2\pi w(\mathbf{r})$. Its uniform-density first-order term is therefore a winding-dependent imaginary action, not generally a removable global constant. It vanishes in the smooth zero-winding sector used for Gaussian phonons. Density defects and vortices require more than that smooth-sector approximation.

Conservation of particle number corresponds to global circle group phase rotations of a nonrelativistic Bose field. A positive-density uniform mean-field saddle chooses a phase, with degenerate saddle manifold a circle. The associated Goldstone boson is the long-wavelength phase mode. This is a thermodynamic symmetry-selection statement; an exact finite-volume number eigenstate does not itself choose a phase.

The quasiparticle vacuum contains physical particles at nonzero momentum because the Bogoliubov transformation mixes creation and annihilation operators. Their occupation is $v_k^2=(A_k/E_k-1)/2$, and the depletion density is its momentum sum divided by the volume. At nonzero temperature an additional $(A_k/E_k)n_B(E_k)$ contribution is required; quantum depletion alone is a zero-temperature expression.

The stable quadratic spectrum is $E_k=\sqrt{{ϵ}_k({ϵ}_k+2n{V}_k)}$. If $V_k\to V_0>0$, its low-momentum form is $\hslash |k|\sqrt{n{V}_0/m}$, a linear sound branch. For contact repulsion the healing length separates the phonon and particle regimes. General momentum-dependent interactions can change the higher-energy shape.

At fixed total particle number, $N_0=N-{\sum }_{k\ne 0}{b}_k^{†}{b}_k$. Expanding the condensate interaction energy subtracts $n{V}_0$ from the normal excited-particle coefficient, cancelling its direct Hartree term. The resulting quadratic coefficients are $A_k={ϵ}_k+n{V}_k$ and $B_k=n{V}_k$, with an opposite-momentum anomalous pair term. Replacing $N_0$ by $N$ before making this correction loses the cancellation.