On the absence of simultaneous reflection and transmission in integrable impurity systems

TL;DR

This paper proves that integrable impurity systems cannot have simultaneous reflection and transmission unless specific commutation conditions are met, with exceptions only in certain free or diagonal scattering theories.

Contribution

It establishes fundamental constraints on the scattering amplitudes in integrable impurity systems, identifying when simultaneous reflection and transmission are possible.

Findings

Simultaneous reflection and transmission require commuting amplitudes with rapidity exchange.

Only theories with rapidity-independent bulk scattering matrices allow such solutions.

Free Boson, Fermion, and Federbush models are the main exceptions.

Abstract

We establish that the Yang-Baxter equations in the presence of an impurity can in general only admit solutions of simultaneous tranmission and reflection when the transmission and reflection amplitudes commute in the defect degrees of freedom with an additional exchange of the corresponding rapidities. In the absence of defect degrees of freedom we show in complete generality, that the only exceptions to this are theories which possess rapidity independent bulk scattering matrices. In particular bulk theories with diagonal scattering matrices, can only be the free Boson and Fermion, the Federbush model and their generalizations. These anyonic solutions do not admit the possibility of excited impurity states.