  Chaotic lattice models at high temperature are generically expected to
exhibit diffusive transport of all local conserved charges. Such diffusive
transport is usually associated with overdamped relaxation of the associated
currents. Here we show that by appropriately tuning the inter-particle
interactions, lattice models of chaotic fermions at infinite temperature can be
made to cross over from an overdamped regime of diffusion to an underdamped
regime of "hot band sound". We study a family of one-dimensional spinless
fermion chains with long-range density-density interactions, in which the
damping time of sound waves can be made arbitrarily long even as an effective
interaction strength is held fixed. Our results demonstrate that underdamped
sound waves of charge density can arise within a single band, far from
integrability, and at very high temperature.
