Theory of a Planckian metal

We present a lattice model of fermions with $N$ flavors and random interactions which describes a Planckian metal at low temperatures, $T rightarrow 0$, in the solvable limit of large $N$. We begin with quasiparticles around a Fermi surface with effective mass $m^ast$, and then include random interactions which lead to fermion spectral functions with frequency scaling with $k_B T/hbar$. The resistivity, $rho$, obeys the Drude formula $rho = m^ast/(n e^2 tau_{textrm{tr}})$, where $n$ is the density of fermions, and the transport scattering rate is $1/tau_{textrm{tr}} = f , k_B T/hbar$; we find $f$ of order unity, and essentially independent of the strength and form of the interactions. The random interactions are a generalization of the Sachdev-Ye-Kitaev models; it is assumed that processes non-resonant in the bare quasiparticle energies only renormalize $m^ast$, while resonant processes are shown to produce the Planckian behavior.