BCS Mean-Field Theory and the Gap Equation

BCS mean-field theory replaces an attractive interaction by a self-consistent pairing-field saddle. For a uniform, spin-balanced, weakly coupled $s$-wave gas it yields quasiparticles with $E_{\mathbf k}=\sqrt{\xi_{\mathbf k}^2+\lvert\Delta\rvert^2}$, coherence factors, coupled gap and number equations, and the universal weak-coupling ratio $2\Delta(0)/T_c=2\pi e^{-\gamma}\simeq3.53$. These are controlled properties of the BCS model when $k_F\lvert a\rvert\ll1$ and $k_Fa<0$; the saddle is neither exact at strong coupling nor evidence for a particular pairing mediator.

Required background. The Cooper instability supplies the attractive eigenchannel. Hubbard–Stratonovich fields supplies the pairing-channel transformation.

Helpful background. Thermal effective potentials supplies the saddle-point thermodynamic logic.

Pairing-field saddle

Take two fermion species with $\xi_{\mathbf k}=\varepsilon_{\mathbf k}-\mu$ and a reduced attractive interaction $-g$ with $g>0$. Introduce

$$\Delta=g\sum_{\mathbf k}\langle c_{-\mathbf k\downarrow}c_{\mathbf k\uparrow}\rangle.$$

In the Nambu basis $\Psi_{\mathbf k}=(c_{\mathbf k\uparrow},c^\dagger_{-\mathbf k\downarrow})^T$, choose the inverse Matsubara propagator

$$\mathcal G^{-1}(\mathbf k,i\omega_n) =i\omega_n\tau_0-\xi_{\mathbf k}\tau_3 -\operatorname{Re}\Delta\,\tau_1 +\operatorname{Im}\Delta\,\tau_2.$$

Its determinant gives poles at $\pm E_{\mathbf k}$, with

$$u_{\mathbf k}^2=\frac12\left(1+\frac{\xi_{\mathbf k}}{E_{\mathbf k}}\right), \qquad v_{\mathbf k}^2=\frac12\left(1-\frac{\xi_{\mathbf k}}{E_{\mathbf k}}\right).$$

The two eigenvalues are particle–hole partners, not two independent physical orbitals. In a fully doubled Nambu basis, a factor $1/2$ in the quadratic form or an explicitly restricted momentum sum removes the duplicate representation. The reduced spin block displayed here instead uses every $\mathbf k$ once with no extra factor $1/2$; that convention gives the grand potential below.

Integrating out the fermions gives the grand-potential density

$$\frac{\Omega(\Delta)}{V} =\frac{\lvert\Delta\rvert^2}{g} +\int_{\mathbf k}(\xi_{\mathbf k}-E_{\mathbf k}) -2T\int_{\mathbf k}\log\!\left(1+e^{-E_{\mathbf k}/T}\right),$$

where $\int_{\mathbf k}\equiv\int\mathrm d^d k/(2\pi)^d$. Stationarity yields

$$\frac1g=\int_{\mathbf k} \frac{1-2f(E_{\mathbf k})}{2E_{\mathbf k}}.$$

At fixed density, this must be solved together with

$$n=\int_{\mathbf k} \left[1-\frac{\xi_{\mathbf k}}{E_{\mathbf k}} \bigl(1-2f(E_{\mathbf k})\bigr)\right].$$

For a three-dimensional contact interaction the first equation is ultraviolet divergent. Matching the bare coupling to the scattering length gives the finite equation

$$-\frac{m}{4\pi a} =\int_{\mathbf k}\left[ \frac{1-2f(E_{\mathbf k})}{2E_{\mathbf k}} -\frac{1}{2\varepsilon_{\mathbf k}} \right].$$

Using an unrenormalized contact $g$ while removing the cutoff changes the problem and is not a harmless approximation.

The matched gap-and-number construction and its crossover limits are developed explicitly in Sá de Melo, Randeria, and Engelbrecht 1993, pp. 3202–3204.

Weak-coupling solution and thermodynamics

If the density of states is approximately constant over a symmetric shell $\omega_c$, the zero-temperature equation becomes

$$\frac{1}{N(0)g} =\int_0^{\omega_c}\frac{\mathrm d\xi}{\sqrt{\xi^2+\Delta_0^2}} \simeq\log\frac{2\omega_c}{\Delta_0},$$

so $\Delta_0=2\omega_c e^{-1/[N(0)g]}$. Linearizing at $T_c$ gives

$$T_c=\frac{2e^\gamma}{\pi}\omega_c e^{-1/[N(0)g]}, \qquad \frac{2\Delta_0}{T_c}=\frac{2\pi}{e^\gamma}.$$

The ratio is universal only within the instantaneous, isotropic, weak-coupling model. Retardation, anisotropy, pair breaking, and strong coupling change it.

At $T=0$, subtracting the normal-state grand potential and using the gap equation gives

$$\frac{\Omega_s-\Omega_n}{V} =-\frac12N(0)\Delta_0^2$$

when $N(0)$ is the single-spin density of states. Differentiation of $\Omega$ supplies entropy and heat capacity. The jump $\Delta C/(\gamma_nT_c)=12/[7\zeta(3)]\simeq1.43$ is again a model result, not a universal diagnostic of mechanism. Bardeen, Cooper, and Schrieffer 1957, §§II–V gives the original construction; Leggett 1980, pp. 13–27 clarifies the paired wavefunction across coupling.

What the saddle establishes

The gap is an energy scale in a specified propagator. In a neutral fluid, the anomalous expectation value transforms under the physical global particle-number $U(1)$; in a charged theory it is gauge covariant rather than itself observable. Robust conclusions come from spectra, free-energy derivatives, phase stiffness and, where charge is present, electromagnetic response, flux quantization, and Josephson interference. Fluctuations are parametrically small in an ordinary weak-coupling three-dimensional metal because $\Delta_0/E_F\ll1$, but they matter near low dimensionality, low density, strong coupling, and phase-ordering transitions.

The structure diagram shows why the mean-field spectrum is an intermediate result rather than the final observable.

BCS self-consistency fixes a paired saddle and its quasiparticle spectrum. It does not by itself establish phase rigidity, Meissner screening, a microscopic mechanism, or topology. Original schematic, not to scale.

See the paired-matter claim test matrix for the observable and negative test appropriate to each stronger claim.

Exercise

Recover the coherence-factor normalization. Show that the positive-energy eigenvector of $\xi\tau_3+\Delta\tau_1$ with real $\Delta>0$ may be normalized so that $u^2+v^2=1$ and $uv=\Delta/(2E)$.

Solution

The eigenvalue equation gives $(E-\xi)u=\Delta v$ after a consistent phase choice. With $u^2=(1+\xi/E)/2$ and $v^2=(1-\xi/E)/2$, normalization is immediate. Their product is $uv=\sqrt{1-\xi^2/E^2}/2=\Delta/(2E)$. Reversing the Nambu basis or the phase of the second component can change the sign of $uv$; observables and the self-consistent equation remain invariant after the same convention change.

References

• Bardeen, J., Cooper, L. N., and Schrieffer, J. R. (1957). “Theory of superconductivity.” Physical Review 108, 1175–1204. doi:10.1103/PhysRev.108.1175.
• 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.
• Sá de Melo, C. A. R., Randeria, M., and Engelbrecht, J. R. (1993). “Crossover from BCS to Bose superconductivity: Transition temperature and time-dependent Ginzburg–Landau theory.” Physical Review Letters 71, 3202–3205. doi:10.1103/PhysRevLett.71.3202.