  Simple intersections between one-dimensional channels can act as coherent
beam splitters for non-interacting electrons. Here we examine how coherent
splitting at such intersections is affected by inter-particle interactions, in
the special case of an intersection of topological edge states. We derive an
effective impurity model which represents the edge-state intersection within
Luttinger liquid theory at low energy. For Luttinger K = 1 / 2 , we compute the
exact time-dependent expectation values of the charge density as well as the
density-density correlation functions. In general a single incoming charge
density wave packet will split into four outgoing wave packets with
transmission and reflection coefficients depending on the strengths of the
tunnelling processes between the wires at the junction. We find that when
multiple charge density wave packets from different directions pass through the
intersection at the same time, reflection and splitting of the packets depend
on the relative phases of the waves. Active use of this phase-dependent
splitting of wave packets may make Luttinger interferometry possible. We also
find that coherent incident packets generally suffer partial decoherence from
the intersection, with some of their initially coherent signal being
transferred into correlated quantum noise. In an extreme case four incident
coherent wave packets can be transformed entirely into density-density
correlations, with the charge density itself having zero expectation value
everywhere in the final state.
