Insights from the Cooper Problem

In the lead-up to the full formulation of the Bardeen-Cooper-Schrieffer Theory of superconductivity (BCS theory), Leon Cooper published a paper entitled Bound Electron Pairs in a Degenerate Fermi Gas (pdf) (referred to colloquially as “The Cooper Problem”). Its utility is not always recognized, but has been stressed by Leggett in his book Quantum Liquids where he says:

It seems not always to be appreciated how useful this “toy” model and simple generalizations of it can be, in particular in giving one a physical feel for which kinds of effects are likely to inhibit (or not) the formation of the superconducting state.

Having solved the Cooper problem in many instances, I tend to agree with Leggett. The Cooper problem, generalized to include the addition of a Zeeman field, shows the detrimental effect of a magnetic field on a Cooper pair. When generalized to include a finite center-of-mass momentum, pair-breaking is again induced.

However, it can also give one an intuition concerning effects that do not inhibit superconductivity. Such a case is where the Zeeman field and the center-of-mass momentum effects “cancel out” to yield a superconducting state (known as the FFLO state). Also, one can realize Anderson’s Theorem (pdf), which states that Cooper pairs are formed from time-reversed partners (as opposed to strictly and -k pairs), a result that is important in understanding the indifference of conventional superconductors to non-magnetic impurities.

Another instance of its usefulness is in understanding the “decoupling” of higher-order pairing (e.g. p-wave, d-wave, etc.). This is discussed in the first chapter of Introduction to Unconventional Superconductivity by Mineev and Somakhin. After solving the problem, one gets a similar result for the binding energy, $\Delta$ to that of the Cooper Problem: $\Delta_l = -2\epsilon_l \exp(-2/N(0)V_l)$

where $l$ is the index labeling the symmetry channel (e.g. l=2 means d-wave) and $\epsilon_l$ denotes an energy cutoff. The result demonstrates that a superconducting state will result when any of the of the angular momentum channels is unstable (at least for a spherical Fermi Surface).

The Cooper Problem: An instructive, easy-to-solve, insightful toy model.