# Theory of a Planckian metal

## TL;DR

This paper introduces a lattice model of fermions with random interactions that captures the behavior of a Planckian metal, exhibiting linear-in-temperature resistivity and frequency scaling consistent with experimental observations.

## Contribution

It presents a solvable large-N lattice model that generalizes SYK models to describe Planckian metal behavior with universal scattering rates.

## Key findings

- Resistivity obeys Drude formula with a universal scattering rate proportional to temperature.
- Spectral functions show frequency scaling with T/ħ, indicating Planckian dissipation.
- Random interactions produce temperature-linear resistivity independent of interaction details.

## Abstract

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.