Bogoliubov Theory and Bose Quasiparticles

Quadratic fluctuations about a dilute condensate mix particle creation with hole annihilation. A canonical Bogoliubov transformation diagonalizes that quadratic Hamiltonian and gives $E_k=\sqrt{\epsilon_k(\epsilon_k+2gn_0)}$: phonons at $k\xi\ll1$, free particles at $k\xi\gg1$, and a gapless Goldstone mode enforced by symmetry.

Required background. Use the matched weak-gas saddle, the Goldstone pole argument, and Dyson/self-energy conventions.

Bogoliubov quadratic Hamiltonian

Write $a_{\mathbf0}\simeq\sqrt{N_0}$ in the symmetry-breaking representation and retain terms quadratic in $a_{\mathbf k\ne0}$. With $\epsilon_k=k^2/(2m)$ and $\mu=gn_0$,

$$K_2=\sum_{\mathbf k\ne0}\left[ (\epsilon_k+gn_0)a_{\mathbf k}^\dagger a_{\mathbf k} +\frac{gn_0}{2}\left(a_{\mathbf k}^\dagger a_{-\mathbf k}^\dagger+a_{\mathbf k}a_{-\mathbf k}\right) \right].$$

Set $a_{\mathbf k}=u_k b_{\mathbf k}-v_k b^\dagger_{-\mathbf k}$ with $u_k^2-v_k^2=1$. Cancellation of anomalous terms yields

$$E_k=\sqrt{\epsilon_k(\epsilon_k+2gn_0)}, \quad u_k^2=\frac12\left(\frac{\epsilon_k+gn_0}{E_k}+1\right), \quad v_k^2=\frac12\left(\frac{\epsilon_k+gn_0}{E_k}-1\right).$$

The commutator condition is the normalization check; the positive root is selected by stability. At small momentum, $E_k\simeq ck$ with $c=\sqrt{gn_0/m}$. At large momentum, $E_k=\epsilon_k+gn_0+O(k^{-2})$. These limits independently test both the factor of two and the chemical-potential subtraction Altland and Simons 2023, § 5.2, pp. 242–257.

Depletion and observable response

The quasiparticle vacuum contains bare particles:

$$n_{\mathrm{dep}}=\int_{\mathbf k}v_k^2 =\frac{8}{3\sqrt\pi}\,n_0\sqrt{n_0a^3}$$

in three dimensions at zero temperature. The small ratio $n_{\mathrm{dep}}/n_0$ is a self-consistency check, not a definition of superfluid density.

The density operator couples with amplitude $u_k-v_k$, giving the leading static structure factor

$$S(k)=\frac{\epsilon_k}{E_k}.$$

Thus $S(k)\sim k/(2mc)$ in the phonon regime and tends to one in the particle regime. The compressibility sum rule reproduces $c^2=gn/m$.

In Nambu language, exact gaplessness requires the Hugenholtz–Pines relation $\mu=\Sigma_{11}(0)-\Sigma_{12}(0)$, not the separate vanishing of either self-energy. The original theorem and its assumptions are given in Hugenholtz and Pines 1959, pp. 489–506.

Validity

This is the leading quadratic theory for $na^3\ll1$ and momenta below the interaction-range scale. It is not exact at strong depletion, in one dimension, or near a critical point. Using $n$ or $n_0$ in a subleading expression without matching the perturbative order can move terms between orders. The renormalized correction is derived next.

Exercises

Verify from the formulas for $u_k$ and $v_k$ that $u_k^2-v_k^2=1$ and that $S(k)=\epsilon_k/E_k$ has the correct high-momentum limit.

Solution

Subtracting the two coherence factors cancels the common ratio and leaves one. Since $E_k/\epsilon_k=\sqrt{1+2gn_0/\epsilon_k}\to1$ for $k\to\infty$, $S(k)\to1$, as required for resolving individual particles at short wavelength.

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.
• N. M. Hugenholtz and David Pines, “Ground-State Energy and Excitation Spectrum of a System of Interacting Bosons,” Physical Review 116 (1959) 489–506, doi:10.1103/PhysRev.116.489.