We present a lattice model of fermions with $N$ flavors and random interactions that describes a Planckian metal at low temperatures $T\to 0$ 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 $k_{B}T/\hslash$. The resistivity $\rho$ obeys the Drude formula $\rho =m^{*}/(n{e}^2{\tau }_{\mathrm{tr}})$, where $n$ is the density of fermions, and the transport scattering rate is $1/{\tau }_{\mathrm{tr}}=f{k}_{B}T/\hslash$; 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.

• Received 17 June 2019

DOI:https://doi.org/10.1103/PhysRevLett.123.066601