The Hidden BEC in BCS Superconductivity — Epoche B2
Research Note: The Hidden BEC Inside BCS Superconductivity Two routes to a macroscopic quantum state are usually taught as opposites. In Bose-Einstein condensation (BEC), a gas of pre-formed bosons, cooled below a critical temperature, drops a macroscopic fraction of its particles into a single quantum state. In the Bardeen-Cooper-Schrieffer (BCS [1] ) theory of superconductivity [2] , by contrast, there are no bosons to begin with: electrons are fermions, and only a feeble phonon-mediated attraction binds them into Cooper pairs that then condense. The textbook lesson is that these are different worlds. This note argues the reverse — that the mean-field equations of BCS theory already describe a condensate of composite bosons, and that BEC is simply the strong-coupling limit of the same description. To expose the hidden BEC we must first build the two pictures separately, then watch a single dial connect them. What Bose-Einstein condensation actually is Bosons are particles whose many-body wavefunction is symmetric under exchange, so any number of them may share one quantum state. An ideal gas of them, of mass $m_b$ and number density $n_b$, condenses when the thermal de Broglie wavelength grows to the interparticle spacing, at a critical temperature $$ k_B T_c = \frac{2\pi\hbar^2}{m_b}\left(\frac{n_b}{\zeta(3/2)}\right)^{2/3}, $$ where $k_B$ is Boltzmann's constant, $\hbar$ the reduced Planck constant, and $\zeta(3/2)\approx 2.612$ the Riemann zeta value that counts the excited states available. Below $T_c$ the condensate is captured by a single complex field, the order parameter or macroscopic wavefunction $\Psi(\mathbf r)=\sqrt{n_0}\,e^{i\varphi}$: the squared magnitude $n_0$ is the condensate density, and the single shared phase $\varphi$ is what carries a supercurrent. Two properties define this textbook BEC — the bosons are pre-formed (stable whether or not they condense) and dilute (they barely overlap). Keep both in view; the Cooper pair will challenge them. The Fermi sea, and why electrons pair Electrons are fermions: their wavefunction is antisymmetric, so the Pauli principle forbids two of them from occupying the same state. At zero temperature they therefore stack one per state up to the Fermi energy $E_F$, filling the "Fermi sea"; the topmost occupied states carry momentum $\hbar k_F$ and speed $v_F$. A single electron near the surface has kinetic energy $\epsilon_{\mathbf k}=\hbar^2 k^2/2m$, conventionally measured from the chemical potential $\mu$ as $\xi_{\mathbf k}=\epsilon_{\mathbf k}-\mu$, with $\mu\approx E_F$ in a normal metal. Coulomb repulsion notwithstanding, the lattice supplies a weak attraction: a passing electron pulls the positive ions together, and because the heavy ions respond sluggishly they leave a trail of excess positive charge that a second electron finds attractive. Cooper showed in 1956 that against the filled sea even an infinitesimal attraction binds a pair, and the most favourable pair has zero total momentum and opposite spins, $(\mathbf k\uparrow,-\mathbf k\downarrow)$. The BCS mean-field ground state Rather than track every electron-electron collision, BCS replace the interaction by an average, or mean, field — each pair feels the collective effect of all the others. The ground state is then a product, over every pair mode $\mathbf k$, of an "empty" and an "occupied" amplitude: $$ |\Psi_{\mathrm{BCS}}\rangle = \prod_{\mathbf k}\left(u_{\mathbf k}+v_{\mathbf k}\,c^{\dagger}_{\mathbf k\uparrow}c^{\dagger}_{-\mathbf k\downarrow}\right)|0\rangle, \qquad u_{\mathbf k}^2+v_{\mathbf k}^2=1, $$ where $c^{\dagger}$ creates an electron, $|0\rangle$ is the state with no electrons, $v_{\mathbf k}^2$ is the probability that the pair mode $(\mathbf k\uparrow,-\mathbf k\downarrow)$ is occupied, and $u_{\mathbf k}^2=1-v_{\mathbf k}^2$ that it is empty. Minimising the energy fixes these coherence factors as $$ v_{\mathbf k}^2=\frac{1}{2}\left(1-\frac{\xi_{\mathbf k}}{E_{\mathbf k}}\right), \qquad E_{\mathbf k}=\sqrt{\xi_{\mathbf k}^2+\Delta^2}, $$ in which $\Delta$ is the energy gap and $E_{\mathbf k}$ the energy of the elementary excitations. The occupation $v_{\mathbf k}^2$ tells the whole story: far below the Fermi surface ($\xi_{\mathbf k}\ll-\Delta$) it equals $1$, far above ($\xi_{\mathbf k}\gg\Delta$) it equals $0$, and pairing smears the razor-sharp step of the Fermi sea into a crossover of width $\sim\Delta$ centred on $\mu$. Solving for the gap — and for the chemical potential The gap is not a free parameter; it must be consistent with the interaction strength $V$ that produced it (where $V>0$ represents an attraction). Self-consistency yields the BCS gap equation, and conservation of particle number fixes a companion equation for $\mu$: $$ \frac{1}{V}=\sum_{\mathbf k}\frac{\tanh(\beta E_{\mathbf k}/2)}{2E_{\mathbf k}}, \qquad N=\sum_{\mathbf k}\left(1-\frac{\xi_{\mathbf k}}{E_{\mathbf k}}\tanh\left(\frac{\beta E_{\mathbf k}}{2}\right)\right), $$ with $\beta=1/k_B T$ the inverse temperature and $N$ the electron number. In an ordinary metal $\Delta\ll E_F$, so the number equation barely shifts $\mu$ from $E_F$ and is routinely ignored. That habit is exactly what conceals the BEC: once the attraction is strong, $\mu$ moves dramatically, and the two equations must be solved together. The gap is a macroscopic wavefunction Define the pair field $\Psi(\mathbf r)=\langle\psi_\downarrow(\mathbf r)\psi_\uparrow(\mathbf r)\rangle$, the amplitude to find two opposite-spin electrons at the same point. Evaluated in the BCS state it gives $$ \Psi(\mathbf r)=\langle\psi_\downarrow(\mathbf r)\psi_\uparrow(\mathbf r)\rangle\propto\Delta, $$ so the superconducting gap is the order parameter — a complex field with a magnitude and a single phase, precisely the object that describes a BEC. The resemblance runs deeper than an analogy. Because a pair mode holds at most one pair, $\left(c^{\dagger}_{\mathbf k\uparrow}c^{\dagger}_{-\mathbf k\downarrow}\right)^2=0$, each fact