Transport and chaos in lattice Sachdev-Ye-Kitaev models

We compute the transport and chaos properties of lattices of quantum Sachdev-Ye-Kitaev islands coupled by single fermion hopping, and with the islands coupled to a large number of local, low energy phonons. We find two distinct regimes of linear-in-temperature (T) resistivity, and describe the crossover between them. When the electron-phonon coupling is weak, we obtain the "incoherent metal" regime, where there is near-maximal chaos with front propagation at a butterfly velocity v(B), and the associated diffusivity D-chaos = v(B)(2)/(2 pi T) closely tracks the energy diffusivity. On the other hand, when the electron-phonon coupling is strong, and the linear resistivity is largely due to near-elastic scattering of electrons off nearly free phonons, we find that the chaos is far from maximal and spreads diffusively. We also describe the crossovers to low T regimes where the electronic quasiparticles are well defined.