Low-rank Sachdev-Ye-Kitaev models

Motivated by recent proposals of experimental realization of fast scramblers, we study a family of solvable variants of the ($q=4$) Sachdev-Ye-Kitaev model in which the rank and eigenvalue distribution of the coupling matrix Jij,kl are tuneable. When the rank is proportional to the number of fermions, the low temperature behavior is sensitive to the eigenvalue distribution. We obtain a complete classification of the possible non-Fermi liquid quantum phases. These include two previously studied phases whose fermion scaling dimension depends continuously on the rank; we show that they are maximally chaotic, but necessitate an extensively degenerate or negative semidefinite coupling matrix. More generic distributions give rise to ``almost Fermi liquids'' with a scaling dimension Δ=1/2, but which differ from a genuine Fermi liquid in quasiparticle decay rate, quantum Lyapunov exponent, and/or specific heat.