The BCS–BEC Crossover

The BCS–BEC crossover continuously connects a weakly bound, overlapping-pair superfluid to a fluid of tightly bound composite bosons when the short-range attraction is tuned through resonance. In a uniform three-dimensional two-component gas, the natural interaction coordinate is $(k_Fa)^{-1}$: negative values describe the BCS side, zero is unitarity, and positive values support a two-body bound state. Pair size, chemical potential, excitation gap, and stiffness evolve differently, so no single “gap” locates the entire crossover.

Required background. BCS gap and number equations, matched scattering length, and the weakly interacting Bose gas supply the two endpoints.

Regulated crossover equations

For equal masses and densities at $T=0$, a useful mean-field interpolation solves

$$-\frac{m}{4\pi a} =\int_{\mathbf k}\left(\frac{1}{2E_{\mathbf k}}-\frac{1}{2\varepsilon_{\mathbf k}}\right),$$ $$n=\int_{\mathbf k}\left(1-\frac{\xi_{\mathbf k}}{E_{\mathbf k}}\right), \qquad E_{\mathbf k}=\sqrt{(\varepsilon_{\mathbf k}-\mu)^2+\Delta^2}.$$

The subtraction matches the contact coupling to $a$ and makes the ultraviolet limit finite. The model tracks the qualitative evolution:

• BCS side, $(k_Fa)^{-1}\ll-1$: $\mu\simeq E_F$, $\Delta/E_F\ll1$, and the pair size greatly exceeds $k_F^{-1}$.
• Unitary region, $(k_Fa)^{-1}=0$: no two-body length remains, so zero-temperature thermodynamics is set by density times universal dimensionless numbers, subject to effective-range and imbalance corrections.
• BEC side, $(k_Fa)^{-1}\gg1$: $\mu\simeq-E_b/2$ with $E_b=1/(ma^2)$, while pairs behave as bosons of mass $2m$ with a residual dimer–dimer interaction.

Mean field does not give quantitatively exact unitary constants or the correct composite-boson scattering length. Those require controlled few-body matching, Monte Carlo, experiment, or another validated method. Leggett 1980, pp. 13–27 establishes the variational interpolation, and Nozières and Schmitt-Rink 1985 develops the normal-state fluctuation extension.

Distinct crossover diagnostics

The minimum fermionic excitation is

$$E_{\min}= \begin{cases} \Delta,&\mu>0,\\ \sqrt{\mu^2+\Delta^2},&\mu<0. \end{cases}$$

The momentum of the minimum moves from near $k_F$ to $k=0$ when $\mu$ changes sign. This is a useful spectral marker within the model, not a thermodynamic phase transition.

A pair-size diagnostic can be formed from the normalized pair wavefunction $\varphi_{\mathbf k}\propto u_{\mathbf k}v_{\mathbf k}=\Delta/(2E_{\mathbf k})$:

$$\xi_{\mathrm{pair}}^2= \frac{\int_{\mathbf k}\lvert\boldsymbol\nabla_{\mathbf k}\varphi_{\mathbf k}\rvert^2} {\int_{\mathbf k}\lvert\varphi_{\mathbf k}\rvert^2}.$$

It decreases from much larger than $k_F^{-1}$ to order $a$. The condensate fraction, contact, sound speed, and stiffness have their own definitions and need not cross at the same coupling.

At finite temperature, pair formation and phase coherence can separate. A suppression of single-particle spectral weight above $T_c$ is often called a pseudogap, but extracting it is method- and probe-dependent. One must rule out inhomogeneity, final-state effects, finite resolution, and other normal-state self-energies. The durable crossover does not require a universal pseudogap boundary.

Control and corrections

At unitarity there is no small interaction parameter. Universality requires $k_Fr_e\ll1$, negligible three-body or loss scales over the observation window, a balanced homogeneous gas, and temperatures quoted relative to $E_F$. Trap data require a local-density or full inhomogeneous forward model.

On the deep BEC side, a dilute expansion is controlled by the composite-boson gas parameter $n_Ba_B^3\ll1$. On the BCS side, $k_F\lvert a\rvert\ll1$ controls weak coupling. The center must be benchmarked rather than declared controlled by continuity.

The interaction coordinate and finite-temperature ordering change in two dimensions, where any short-range attraction supports a bound state and phase coherence is limited by BKT physics. Randeria, Duan, and Shieh 1990, §§II–IV develops that distinct crossover; its two-dimensional formulas should not be inserted into the three-dimensional equations above.

The validity map keeps these three domains and their different controls visible.

Pair size, chemical potential, fermionic gap, and phase stiffness evolve across the crossover but do not define one common boundary. Weak-coupling and dilute-boson endpoints are controlled; unitarity requires benchmark evidence. Original schematic, not to scale.

See the paired-matter claim test matrix for the endpoint and pseudogap tests.

Exercise

Locate the spectral minimum. Minimize $E_k=\sqrt{(k^2/2m-\mu)^2+\Delta^2}$ for $k\ge0$.

Solution

Minimizing $E_k^2$ gives $(k/m)(k^2/2m-\mu)=0$. If $\mu>0$, the minimum occurs at $k=\sqrt{2m\mu}$ and equals $\Delta$. If $\mu<0$, the only allowed stationary minimum is $k=0$, giving $\sqrt{\mu^2+\Delta^2}$. The change at $\mu=0$ is a crossover marker of the mean-field spectrum, not a symmetry-breaking transition.

References

• Leggett, A. J. (1980). “Diatomic molecules and Cooper pairs.” In A. Pekalski and R. Przystawa (eds.), Modern Trends in the Theory of Condensed Matter, pp. 13–27. Springer. doi:10.1007/978-3-642-81409-5_1.
• Nozières, P., and Schmitt-Rink, S. (1985). “Bose condensation in an attractive fermion gas: From weak to strong coupling superconductivity.” Journal of Low Temperature Physics 59, 195–211. doi:10.1007/BF00683774.
• Randeria, M., Duan, J.-M., and Shieh, L.-Y. (1990). “Superconductivity in a two-dimensional Fermi gas: Evolution from Cooper pairing to Bose condensation.” Physical Review B 41, 327–343. doi:10.1103/PhysRevB.41.327.