Bose condensation and Bogoliubov theory

The one-body density matrix records correlations between removing a boson at one point and adding it elsewhere. Bose condensation means one eigenvalue of that matrix grows proportionally to particle number in the thermodynamic limit. For a uniform gas that mode is the zero-momentum state. This definition works even in a number-conserving description where the expectation value of the field operator vanishes. Choosing a condensate phase is a convenient symmetry-breaking description, not a claim that exact particle conservation has failed.

For short-range interactions at low energy, replace microscopic details by a coupling $g=4\pi\hbar^2a_s/m$ fixed by the scattering length $a_s>0$. This replacement requires momenta below the inverse interaction range, and weak depletion in three dimensions is governed by $na_s^3\ll1$. A contact Hamiltonian needs an appropriate ultraviolet prescription beyond the leading formulas used here. The physical scattering length, rather than an arbitrary bare delta-function strength, is the measurable input.

Write the field as a macroscopic condensate contribution plus a fluctuation, and retain the Hamiltonian through quadratic order in nonzero-momentum fluctuations. Interactions couple creation of momentum $\mathbf k$ to creation of $-\mathbf k$, so ordinary single-mode occupation operators no longer diagonalize the quadratic expression. A bosonic Bogoliubov transformation mixes creation and annihilation operators while preserving their commutator. Keeping that algebra is essential: a fermionic rotation would give the wrong normalization.

Diagonalization yields a positive excitation energy for repulsive coupling. At small momentum, the free kinetic energy is much smaller than the interaction energy, and the spectrum becomes linear with sound speed $c=\sqrt{gn_0/m}$. At large momentum within the effective model, it approaches a particle-like branch with an interaction shift. The crossover is smooth. Quadratic theory neglects interactions among these quasiparticles, so it does not by itself predict their finite lifetime or the whole finite-temperature phase diagram.

Condensate fraction measures the macroscopic eigenvalue of a one-body density matrix. Superfluid density measures response to a slow phase twist or moving boundary. They are not interchangeable definitions. Interactions can deplete the condensate while leaving a homogeneous Galilean-invariant ground state fully superfluid at zero temperature. An ideal Bose condensate, conversely, does not acquire a finite Landau critical velocity from a quadratic free-particle dispersion. Response requires its own calculation.

Dimension and order of limits also matter. A homogeneous interacting one-dimensional gas generally lacks conventional long-range condensate order in its thermodynamic ground state, and a uniform two-dimensional gas at positive temperature requires a different treatment of long-wavelength phase fluctuations. Finite traps and finite observation lengths modify these statements. Before using a condensate ansatz, specify geometry, temperature, interaction scale, and the observable being predicted. A useful mean field begins a controlled explanation; it does not erase fluctuations.