Cooper pair

A Cooper pair is a correlated pair of fermions associated with superconductivity. The conventional BCS ground state uses opposite-momentum, opposite-spin pair creation operators $c_{k\uparrow }^{†}{c}_{-k\downarrow }^{†}$. Their total momentum is zero for every relative momentum $k$; zero total momentum does not restrict $k$ to zero.

Two added electrons move above a frozen filled Fermi sea and attract within a thin energy shell. The Pauli-blocked continuum has a logarithmic pair susceptibility, so an arbitrarily weak net attractive constant interaction produces a state below the two-particle threshold at zero temperature. This two-added-particle result is not by itself a full many-electron superconducting state.

For shell width $D$, attraction $g>0$ and constant single-spin density of states $\rho$, $1=(g\rho /2)\log [(2D+B)/B]$ gives $B=2D/[\exp (2/(g\rho ))-1]$. The pair energy relative to the free threshold is $-B$. At weak coupling this scale is nonanalytic in $g$.

The pairing Hamiltonian is projected onto empty states in a specified shell above the inert Fermi sea. A state restricted to that shell is an eigenstate of this projected model, not necessarily of a contact interaction that remains unrestricted at all momenta.

A pairing gap is an energy scale for fermionic excitations above a paired superfluid or superconductor ground state. In a simple isotropic paired system the quasiparticle dispersion has the form $E_{\mathbf{k}}=\sqrt{{\xi }_{\mathbf{k}}^2+|\Delta {|}^2}$, where ${\xi }_{\mathbf{k}}$ measures the unpaired single-particle energy relative to the chemical potential. A nonzero minimum gap gives exponentially small fermionic thermal occupations when $k_{B}T\ll |\Delta |$. If the gap varies with direction and has nodes, gapless directions can instead produce power-law thermodynamics. Collective excitations must be counted separately from the gapped fermionic quasiparticles.