  Heat rectifiers would facilitate energy management operations such as
cooling, or energy harvesting, but devices of practical interest are still
missing. Understanding heat rectification at a fundamental level is key to help
us find or design such devices. The match or mismatch of the phonon band
spectrum of device segments for forward or reverse temperature bias of the
thermal baths at device boundaries, was proposed as the mechanism behind
rectification. However no explicit, theoretical relation derived from first
principles had been found so far between heat fluxes and spectral matching. We
study heat rectification in a minimalistic chain of two coupled ions. The
fluxes and rectification can be calculated analytically. We propose a
definition of the matching that sets an upper bound for the heat flux. In a
regime where the device rectifies optimally, matching and flux ratios for
forward and reverse configurations are found to be proportional. The results
can be extended to a system of N particles in arbitrary traps with
nearest-neighbor linear interactions.
