Past exam of the superconductivity and Quantum Coherence course of the University of Cambridge / 2012 / Paper 1 / 1 / b

The electron-phonon interaction can produce a retarded attraction. A carrier distorts the ionic lattice; a second carrier can lower its energy in that distortion. Integrating out a phonon gives a frequency-dependent interaction with an attractive low-frequency region. A schematic Matsubara-frequency contribution is $V_{\mathrm{ph}}(\mathbf{q},i{\omega }_n)\propto -|g_{\mathbf{q}}{|}^2{\omega }_{\mathbf{q}}/({\omega }_n^2+{\omega }_{\mathbf{q}}^2)$. It must be combined with the screened Coulomb repulsion, not treated as a statement that the bare electron-electron force is attractive.

Retardation separates the electronic and phonon energy scales and reduces the repulsive contribution in the pairing channel to a Coulomb pseudopotential. In the simplest weak-coupling approximation, a positive net dimensionless pairing strength givesThe formula assumes a phonon cutoff and simplified coupling; strong-coupling calculations need the spectral interaction rather than this one-number replacement. The phonon frequency's dependence on ionic mass motivates an isotope effect, whose actual exponent can be modified by other physics.

An interaction need not be attractive in every momentum channel to support pairing. Spin fluctuations or other electronic correlations can favor a sign-changing gap: in the gap equation, repulsive scattering between two regions can contribute constructively when their gap signs differ. Such mechanisms are candidates for unconventional superconductivity. Observing a d-wave gap identifies symmetry information but does not, by itself, establish a unique microscopic pairing mechanism in a cuprate.