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A Bogoliubov quasiparticle is an excitation created by the diagonal mode operator obtained through a Bogoliubov transformation. Its vacuum can contain particles in the original modes because creation and annihilation operators mix. Examples include antiferromagnetic spin waves, Bose-gas phonons and paired-fermion excitations.

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.

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.

The density-phase action of a Bose gas follows directly from $\psi =\sqrt{\rho }{e}^{i\varphi }$:The total derivative integrates to zero for periodic density. HenceA uniform density fluctuation has nonzero quadratic cost $g(\delta \rho {)}^2/2$, whereas a uniform phase change has no energy cost. Thus the radial fluctuation is massive in the static quadratic-action sense, while the phase is massless. The number-phase conjugacy term $i\delta \rho {\partial }_{\tau }\varphi$ couples their dynamics: this does not imply an additional independent gapped quasiparticle branch in the nonrelativistic Bose gas.

The compact phase winding term $i{\rho }_0\int {\partial }_{\tau }\varphi$ vanishes for the smooth zero-winding phonon sector. Globally the phase can wind by $2\pi$ while the complex field remains periodic, so that term should not be discarded across all topological sectors without a further argument.

Put $\rho ={\rho }_0+\delta \rho$ and retain quadratic fluctuations in the smooth zero-winding sector. The quadratic density-phase action isAt wavelengths long compared with the healing length, drop the density-gradient term. Completing the square givesThe real Gaussian density integral, or its equivalent contour shift, contributes only a field-independent determinant. Absorb that normalization and the uniform saddle action into $S_0$. To this quadratic long-wave accuracy,The Gaussian extension of $\delta \rho$ to the full real line is a fluctuation approximation around positive ${\rho }_0$; it is not an exact replacement of the global density constraint. The compact field's vortex/winding sectors also lie beyond this smooth-phonon integral.

The Bose-gas phase-only action is a continuum harmonic chain with Euclidean inverse propagator ${\omega }_n^2/g+{\rho }_0{k}^2/m$. Continuing to real frequency givesRestoring $\hslash$ gives $E_k\sim \hslash c_s|k|$ for a wavenumber $k$. This is the phonon branch of the Bogoliubov spectrum. Keeping the omitted density-gradient term replaces $g$ by $g+k^2/(4m{\rho }_0)$ and yields $E_k^2=c_s^2{k}^2+k^4/(4m^2)$ in the adopted units, so the full quadratic spectrum is $\sqrt{{ϵ}_k({ϵ}_k+2g{\rho }_0)}$ with ${ϵ}_k=k^2/(2m)$.

The phase-only action also tests the condensate assumption. At zero temperature its equal-time phase variance has an infrared contribution $\int d^{d}k/|k|$, logarithmically divergent in one dimension; at positive temperature the zero Matsubara mode gives $\int d^{d}k/k^2$, divergent in dimensions at or below two. Thus this same low-energy theory exposes the regimes where the mean-field condensate cannot describe true thermodynamic long-range order. It permits a finite infrared fluctuation at zero temperature for $d\ge 2$ and at positive temperature for $d>2$, within the remaining weak-coupling assumptions.

Put $A_k={ϵ}_k+n{V}_k$ and $B_k=n{V}_k$, and assume the homogeneous condensate is stable, so $A_k>|B_k|$ for nonzero modes. A real Bogoliubov transformation isThe canonical condition preserves the bosonic canonical commutation relations. Cancellation of pair terms requires $2A_k{u}_k{v}_k=B_k(u_k^2+v_k^2)$. ThereforeThe sign of $v_k$ follows that of $B_k$. Normal ordering the transformed operators gives the bosonic Bogoliubov zero-point shift:Rearranging the last two terms is exactly the requested $-\frac{1}{2}\sum A_k+\sum E_k({\alpha }_k^{†}{\alpha }_k+1/2)$ form. Each momentum appears in the sums, so opposite-momentum pairs are not counted again separately.

The Bogoliubov quasiparticle dispersion is linear at low momentum when $V_k\to V_0>0$:It is a gapless phonon of the condensate, unlike the free-particle quadratic spectrum. For a contact repulsion $V_k=g>0$, the healing length $\xi =\hslash /\sqrt{2mng}$ controls crossover to the particle regime $E_k\simeq {ϵ}_k+ng$. A momentum-dependent potential can modify the higher-energy curve; the sketch uses the contact case and does not assume every interaction has that entire shape.

A phonon has zero chemical potential and energy $\hslash \omega$. The total internal energy is consequentlyWith the change of variable $x=\beta \hslash \omega$ and $\beta \hslash {\omega }_D=T_D/T$, this becomes

Use periodic boundary conditions on the square crystal. Each allowed wavevector occupies area $(2\pi {)}^2/A$ in two-dimensional $k$-space. Including the two phonon polarizations, the number of modes in the disk $|k|\le K$ isThe phonon dispersion relation $\omega =C|k{|}^{\alpha }$ gives $K=(\omega /C{)}^{1/\alpha }$, soDifferentiating the cumulative mode count produces the density of states for a two-dimensional power-law phonon dispersion: