# Cooper pairing from repulsive potential

Suppose the Hamiltonian of a many-electron system consists of a potential which is repulsive : $\langle k_1, k_2 |\hat V |k_1',k_2' \rangle > 0$ where $k_1, k_2, \cdots$ are possible momenta that an electron can have. I want to know whether such potential can lead to cooper pairing. In particular I want to know whether the following statements make sense at all.

Let's consider a similar problem as the one cooper solved that is "consider a pair of electrons which interact above a quiescent Fermi sphere" with the interaction being repulsive here. Let $K \equiv {\{k^i\}}$ denotes the space of all momentum. Let's partition the $K$ in two parts say $K_A \equiv \{ k_A^i \}$ and $K_B \equiv \{ k_B^i \}$. Now consider the paired state $\Psi = 1/(\sqrt{2}) (\Psi_A + e^{i \phi} \Psi_B$) where $\Psi_A$ is a superposition of paired momentum states in $K_A$ i.e $\Psi_A \approx b_1| k^1_A, -k^1_A \rangle + b_2 | k^2_A, -k^2_A \rangle + \cdots$ and similarly $\Psi_B$ is a superposition of paired states of momenta in $K_B$. Now take the matrix element $\langle \Psi| \hat V|\Psi\rangle = 1/2\left(\langle \Psi_A| \hat V|\Psi_A\rangle + \langle \Psi_B| \hat V|\Psi_B + 2*\cos\phi \langle \Psi_A| \hat V|\Psi_B\rangle \right)$, assuming $\langle \Psi_A| \hat V|\Psi_B\rangle$ is real. Now if $\phi = \pi$ and $\hat V$ is such that $\langle k^i_A, -k^i_A| \hat V |k^j_A, -k^j_A \rangle \ll \langle k^i_A, -k^i_A| \hat V |k^j_B, -k^j_B \rangle$ then it looks like the paired state $\Psi$ might be energetically favorable. Here I have suppressed the spins of the fermions.

Please let me know whether something is fundamentally wrong with this ?

• One can study the problem by looking at the BCS gap equation. When the gap in a one-band system is $k$-independent, the gap equation has a non-trivial solution only when the interaction is attractive. If we allow the gap to be anisotropic, or consider a multiband system, even repulsive interactions can yield a non-trivial solution. – leongz Apr 30 '16 at 12:24

Yes, pairing is possible from a repulsive interaction. The reason behind this is that pairing has to occur in a certain angular momentum channel : $l=0$ for s-wave superconductivity, $l=1$ for p-wave, and so on. To see this, you can expand the repulsive $k$-dependent interaction on Legendre polynomials. Check this review that deals with the Kohn-Luttinger mechanism responsible for pairing by the screened Coulomb potential.
• Ok. Thanks for the reference. I have to go through it . But does the mechanism suggested in the question make sense ? If the answer is yes then is it related to the $l \neq 0$ pairing ? – Tuhin Subhra Mukherjee Apr 27 '16 at 13:12
• When the screened Coulomb interaction gets renormalized by the Kohn-Luttinger mechanism, the tail of the potential oscillates (Friedel oscillations) which gives a small attractive part to the potential. With their calculation they get pairing in and odd-$l$ channel for $l$ large enough. – Dimitri Apr 27 '16 at 13:17