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BCS theory describes a superconducting instability caused by attractive interactions between opposite-momentum fermions near a Fermi surface.

John Bardeen, Leon Neil Cooper and John Robert Schrieffer developed BCS theory in 1957, using attraction mediated by phonons.^[ref]

Weak-coupling Cooper pairs overlap. The Cooper instability reorganizes an unpaired Fermi surface; it is not Bose-Einstein condensation of compact, pre-existing molecules. The original paper explicitly distinguishes these pictures.^[ref]

Two complementary clues made the microscopic problem concrete. The superconducting isotope effect connected the transition to phonons.^[ref]

A 1957 transmission experiment used thin lead and tin films as samples in far-infrared and millimeter-microwave measurements. The measured transmission was converted to complex electrical conductivity, normalized to the normal-state value. Comparing film thickness, resistance, annealing and the two metals tested the response's reproducibility. Weak low-photon-energy absorption followed by an onset supported a superconducting energy gap; its extraction required a thin-film electrodynamic model.^[ref]

The later Giaever superconducting tunnelling experiment provided another way to resolve the superconducting energy gap through quantum tunnelling.^[ref] The derivations below explain what the model predicts, including the distinction between a single quasiparticle's minimum energy and the threshold for making two of them.

John Robert Schrieffer coauthored the microscopic BCS theory of conventional superconductivity. Its many-electron ground state coherently combines configurations in which states of opposite spin and momentum are paired. An attractive interaction mediated by lattice vibrations can lower the energy of this collective state, while its excitations have a gap. The theory relates that construction to a phase transition, specific heat, magnetic-field penetration and other measurable properties of a superconductor.^[ref]

A physicist.^[ref]

Awarded in equal shares to John Bardeen, Leon Neil Cooper and John Robert Schrieffer for their jointly developed theory of superconductivity, the BCS theory.^[ref]

A theoretical condensed matter physicist who coauthored the Schrieffer-Wolff transformation with John Robert Schrieffer. Their canonical transformation relates a model with fluctuating impurity charge to an effective model of spin exchange in the weak-mixing limit. This explains how virtual charge processes can produce the low-energy physics associated with the Kondo effect.^[ref]

The Schrieffer-Wolff transformation is a unitary conjugation that decouples selected low- and high-energy sectors to construct an effective Hamiltonian. It is a form of degenerate perturbation theory: states within the retained sector are treated together.

Let an orthogonal projection $P$ commute with $H_0$, put $Q=I-P$, and decompose the perturbation in $H=H_0+ϵV$ asThe first part acts within sectors; the second couples them. For an anti-Hermitian generator $S=ϵS_1+O({ϵ}^2)$, choose $[S_1,H_0]=-V_{od}$. It is the inter-sector coupling, rather than the whole perturbation, that this commutator removes. Projecting the consistently transformed Hamiltonian givesIn an eigenbasis of $H_0$, cross-sector entries obey $(S_1{)}_{mn}=V_{mn}/(E_m-E_n)$. A nonzero separating spectral gap controls these denominators; resonant coupled states belong in a common retained block.^[ref]

For a direct two-level check, take $H_0=\mathrm{diag}(0,\Delta )$, $\Delta >0$, and $V=t|0⟩⟨1|+t^{\ast }|1⟩⟨0|$. The formula predicts the lower energy shift $-{ϵ}^2|t{|}^2/\Delta$. Expanding the exact lower eigenvalue, $(\Delta -\sqrt{{\Delta }^2+4{ϵ}^2|t{|}^2})/2$, gives the same result. A diagonal perturbation would also contribute at first order through $PVP$.

John Robert Schrieffer and P. A. Wolff used the method to relate impurity models underlying the Kondo effect.^[ref]