Reasonable and robust Hamiltonians violating the Third Law

It has recently been shown that the third law of thermodynamics is violated by an entire class of classical Hamiltonians in one dimension, over a finite volume of coupling-constant space, assuming only that certain elementary symmetries are exact, and that the interactions are finite-ranged. However, until now, only the existence of such Hamiltonians was known, while almost nothing was known of the nature of the couplings. Here we show how to define the subvolume of these Hamiltonians---a `wedge' W in a d-dimensional space---in terms of simple properties of a directed graph. We then give a simple expression for a specific Hamiltonian H* in this wedge, and show that H* is a physically reasonable Hamiltonian, in the sense that its coupling constants lie within an envelope which decreases smoothly, as a function of the range l, to zero at l=r+1, where r is the range of the interaction.