The Ideal Bose Gas and Bose–Einstein Condensation

Bose–Einstein condensation occurs when the excited states of a Bose gas can hold only a finite density at chemical potential approaching the one-particle ground energy. In a homogeneous gas with quadratic dispersion this happens at nonzero temperature only for spatial dimension $d>2$; finite systems show a smooth crossover, not a singular phase transition.

Required background. Use second-quantized Bose fields for mode occupations, thermal density operators for the Bose distribution, and finite-density ensembles for the role of $\mu$.

Saturation of the excited states

Take a uniform box of volume $V$, periodic boundary conditions, and dispersion $\epsilon_{\mathbf k}=k^2/(2m)$ with ground energy set to zero. The grand-canonical occupation is

$$n_{\mathbf k}=\frac{1}{e^{\beta(\epsilon_{\mathbf k}-\mu)}-1}, \qquad \mu\leq0.$$

Separating the zero mode gives $N=N_0+N_{\mathrm{ex}}$. In the thermodynamic limit,

$$\frac{N_{\mathrm{ex}}}{V} =\int_{\mathbf k}\frac{1}{e^{\beta(k^2/2m-\mu)}-1} =\lambda_T^{-d}g_{d/2}(e^{\beta\mu}), \qquad \lambda_T=\sqrt{\frac{2\pi}{mT}},$$

where $g_s(z)=\sum_{\ell\ge1}z^\ell/\ell^s$. At fixed $T$, the largest possible excited-state density is $n_c=\lambda_T^{-d}\zeta(d/2)$. It is finite precisely when $d>2$. Any excess density must then occupy the ground mode. For $d=3$,

$$T_c=\frac{2\pi}{m}\left[\frac{n}{\zeta(3/2)}\right]^{2/3}, \qquad \frac{N_0}{N}=1-\left(\frac{T}{T_c}\right)^{3/2} \quad (T<T_c).$$

This is a density-of-states argument: near zero energy, $\rho(\epsilon)\propto\epsilon^{d/2-1}$, so $\rho(\epsilon)n_B(\epsilon)\sim\epsilon^{d/2-2}$. Infrared convergence requires $d>2$. The derivation and its finite-size qualifications are developed in Pitaevskii and Stringari 2016, ch. 2.

Finite volume and traps

For finite $V$, $N_0=[e^{-\beta\mu}-1]^{-1}$ and $\mu$ remains strictly negative. No partition-function nonanalyticity occurs. A useful finite-size statement is instead that $N_0$ becomes order $N$ in a sequence of boxes at fixed density. Replacing the sum by an integral misses corrections controlled by the level spacing $\Delta\epsilon\sim1/(mL^2)$.

A harmonic trap changes the low-energy density of states. In three dimensions, $\rho(\epsilon)\propto\epsilon^2/(\omega_x\omega_y\omega_z)$ and the ideal trapped-gas result is $N_{\mathrm{ex}}=\zeta(3)(T/\bar\omega)^3$. This is a different thermodynamic scaling limit, $N\to\infty$, $\bar\omega\to0$ with $N\bar\omega^3$ fixed; it must not be inserted into the uniform formula.

What condensation does and does not establish

The ideal calculation establishes macroscopic ground-mode occupation and, in the homogeneous thermodynamic limit, a macroscopic eigenvalue of the one-body density matrix. It does not establish an interacting equation of state, a critical velocity, vortex rigidity, or a nonzero helicity modulus. Those independent notions are separated on the condensation and superfluidity page. The standard field-theory treatment makes the same distinction Altland and Simons 2023, § 5.2, pp. 242–257.

Exercises

Show directly from the small-$k$ integrand that a uniform ideal gas has no finite-temperature condensate for $d=2$.

Solution

At $\mu\to0$, $n_B(k^2/2m)\simeq2mT/k^2$. The number integral behaves as $\int_0 \mathrm dk\,k^{d-1}/k^2$. For $d=2$ it is $\int_0\mathrm dk/k$, which diverges logarithmically. Thus excited states never saturate at finite density in the infinite uniform system.

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.
• Lev P. Pitaevskii and Sandro Stringari, Bose–Einstein Condensation and Superfluidity, Oxford University Press (2016), ch. 2, doi:10.1093/acprof:oso/9780198758884.001.0001.