Upper Bound on Diffusivity

The linear growth of operators in local quantum systems leads to an effective light cone even if the system is nonrelativistic. We show that the consistency of diffusive transport with this light cone places an upper bound on the diffusivity: $D\lesssim v^2{\tau }_{\text{eq}}$. The operator growth velocity $v$ defines the light cone, and ${\tau }_{\text{eq}}$ is the local equilibration time scale, beyond which the dynamics of conserved densities is diffusive. We verify that the bound is obeyed in various weakly and strongly interacting theories. In holographic models, this bound establishes a relation between the hydrodynamic and leading nonhydrodynamic quasinormal modes of planar black holes. Our bound relates transport data—including the electrical resistivity and the shear viscosity—to the local equilibration time, even in the absence of a quasiparticle description. In this way, the bound sheds light on the observed $T$-linear resistivity of many unconventional metals, the shear viscosity of the quark-gluon plasma, and the spin transport of unitary fermions.

Figure 1

The diffusive Green’s function is large outside the operator growth light cone at early times. It follows that the true Green’s function must not be diffusive over that regime.