Bound-state instability of the chiral Luttinger liquid in one dimension

We have developed a "bootstrap'' method for solving a class of interacting one-dimensional chiral fermions. The conventional model for interacting rig ht-moving electrons with spin has an SO(4) symmetry, and can be written as four interacting Majorana fermions, each with the same velocity. We have fo und a method for solving some cases when the velocities of these Majorana f ermions are no longer equal. We demonstrate in some detail the remarkable r esult that corrections to the skeleton self-energy identically vanish fur t hese models, and this enables us to solve them exactly. For the cases where the model can be solved by bosonization, our method can be explicitly chec ked. However, we are also able to solve some cases where the excitation spe ctrum differs qualitatively from a Luttinger liquid. Of particular interest is the so-called SO(3) model, where a triplet of Majorana fermions, moving at one velocity, interact with a single Majorana fermion moving at another velocity. Using our method we show, that a sharp bound (or antibound) stat e splits off from the original Luttinger-liquid continuum, cutting off the x-ray singularity to form a broad incoherent excitation with a lifetime tha t grows linearly with frequency.