  The speed of information propagation is finite in quantum systems with local
interactions. In many such systems, local operators spread ballistically in
time and can be characterized by a "butterfly velocity", which can be measured
via out-of-time-ordered correlation functions. In general, the butterfly
velocity can depend asymmetrically on the direction of information propagation.
In this work, we construct a family of simple 2-local Hamiltonians for
understanding the asymmetric hydrodynamics of operator spreading. Our models
live on a one dimensional lattice and exhibit asymmetric butterfly velocities
between the left and right spatial directions. This asymmetry is transparently
understood in a free (non-interacting) limit of our model Hamiltonians, where
the butterfly speed can be understood in terms of quasiparticle velocities.
