Theory of a Planckian Metal

We present a lattice model of fermions with N flavors and random interactions that describes a Planckian metal at low temperatures T→0 in the solvable limit of large N. We begin with quasiparticles around a Fermi surface with effective mass m* and then include random interactions that lead to fermion spectral functions with frequency scaling with kBT/ℏ. The resistivity ρ obeys the Drude formula ρ=m*/(ne²τₜᵣ), where n is the density of fermions, and the transport scattering rate is 1/τₜᵣ=fkBT/ℏ; 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 nonresonant in the bare quasiparticle energies only renormalize m*, while resonant processes are shown to produce the Planckian behavior.