Solvable Strong-Coupling Quantum-Dot Model with a Non-Fermi-Liquid Pairing Transition

We show that a random interacting model exhibits solvable non-Fermi-liquid behavior and exotic pairing behavior. This model, dubbed as the Yukawa-SYK model, describes the random Yukawa coupling between $M$ quantum dots each hosting $N$ flavors of fermions and $N^2$ bosons that self-tune to criticality at low energies. The diagrammatic expansion is controlled by $1/MN$, and the results become exact in a large-$M$, large-$N$ limit. We find that pairing only develops within a region of the $(M,N)$ plane—even though the pairing interaction is strongly attractive, the incoherence of the fermions can spoil the forming of Cooper pairs, rendering the system a non-Fermi liquid down to zero temperature. By solving the Eliashberg equation and the renormalization group equation, we show that the transition into the pairing phase exhibits Kosterlitz-Thouless quantum-critical behavior.

Figure 1

(a) The Feynman diagrams corresponding to the self-energies of the bosons and fermions. (b) The large $N,M$ behavior of the diagrams can be tracked using a double-line formalism. The solid line represents indices $i,j$ and the dashed lines $\alpha ,\beta$.

Figure 2

The gap equation for the interdot pairing. One can show that the interaction for all other channels is repulsive.