 Solution

In three dimensions an ideal Bose gas can saturate its excited-state population at chemical potential zero, with the remaining particles forming a Bose-Einstein condensate. The transition follows from the Bose-Einstein distribution integrated against its density of states; below the ideal-gas transition, $n_0/n=1-(T/T_c{)}^{3/2}$. Condensation alone does not supply the finite Landau critical velocity of an interacting superfluid: the ideal quadratic spectrum has ${\inf }_{p}E(p)/p=0$.

By contrast, a weak attractive interaction near a Fermi surface produces the Cooper instability and the BCS gap equation. Its logarithmic enhancement permits pairing at arbitrarily weak attraction in the ideal model. The low-temperature excitations have $E=\sqrt{{\xi }^2+{\Delta }^2}$, and in weak coupling $k_B{T}_c=(e^{{\gamma }_E}/\pi )\Delta (0)\simeq 0.567\Delta (0)$. This transition is controlled by pair formation when phase fluctuations are small; deeply bound pairs instead motivate a Bose-condensation description with their own density and effective mass.

For the given ellipsoidal dispersion, set $q_x=k_x$, $q_y=k_y$ and $q_z=k_z/\sqrt{\gamma }$. Then $\epsilon ={\hslash }^2{q}^2/(2m)$ while the integration measure is $d^{3}k=\sqrt{\gamma }{d}^{3}q$. Thus the anisotropic ideal-gas density of states per unit volume and per internal state isThe full continuum momentum range is assumed; this is an anisotropic three-dimensional gas, not yet a discrete stack of two-dimensional bands.

At the Bose transition, substitute $x=\epsilon /(k_{B}T)$:The integral is $\Gamma (3/2)\zeta (3/2)\simeq 2.315$. Solving for the Bose-Einstein condensation temperature for an anisotropic effective mass givesThe heavier $z$-direction effective mass increases the available excited-state count and lowers the condensation temperature at fixed number density.

For a balanced two-spin Fermi gas, let $n_F$ denote total fermion number density and take $N_s$ to be the single-spin density of states used in the conventional gap equation. Counting the filled ellipsoid givesIf a spin-summed density of states is used instead, the coupling convention in the supplied integral must change accordingly; inserting a factor two into only one side would change the exponent incorrectly.

For ${\epsilon }_c\ll {\epsilon }_F$, replace the density of states across the interaction window by $N_s({\epsilon }_F,\gamma )$. Writing ${\lambda }_0=|g|N_{s0}$, the zero-temperature integral becomesand thereforeThe second expression additionally requires ${\lambda }_0{\gamma }^{1/3}\ll 1$. The explicit density dependence is ${\lambda }_0=|g|m(3{\pi }^2{n}_F{)}^{1/3}/(2{\pi }^2{\hslash }^2)$. This mass-anisotropy enhancement of a weak-coupling BCS gap initially raises the pairing scale, while the ideal Bose condensation scale decreases as ${\gamma }^{-1/3}$.

Large anisotropy weakens transport and coherence in the heavy direction. The separation of pair formation and phase ordering then matters, so a large mean-field gap need not imply a large actual $T_c$. A qualitative sketch has $T_c$ initially rising with the pairing scale and eventually turning down when coherence is the smaller scale. If preformed fermion pairs behave as bosons, use $n_B=n_F/2$ and pair mass about $2m$; in this continuum model this gives $k_B{T}^{\mathrm{BEC}}\simeq 0.218{\epsilon }_{F0}{\gamma }^{-1/3}$.

The turnover cannot be located reliably by taking the weak-coupling formulas to $\gamma \to \infty$: ${\epsilon }_F$ falls, ${\lambda }_0{\gamma }^{1/3}$ grows, the narrow-window assumption fails and the pair chemical potential and mass require a self-consistent treatment. The smooth curve in the figure is explicitly an illustrative interpolation of two extrapolated scales, not such a calculation. If actual decoupled two-dimensional layers are approached, ordinary ideal-gas finite-temperature condensation is absent and interacting phase order requires a different analysis, including the Berezinskii–Kosterlitz–Thouless transition when appropriate.

The cuprates make the distinction between pairing and phase coherence relevant: they are layered and have anisotropic motion, but their strong correlations, lattice band structure and predominantly d-wave superconductivity are outside this attractive continuum $s$-wave model. Anisotropy by itself is therefore not an explanation or quantitative fit of cuprate transition temperatures.