1+1D Bosonization and Zero Modes

With Tomonaga-Luttinger we are usually interested in largish systems and so the zero modes are hard to distinguish from any other low $k$, low $E$ mode. Saying this another way we remember that for large systems the distant boundary conditions have no effects that can be distingushed by local measurements --- but it is these boundary conditions that determine the existance (or not) of exact zero-energy modes. Thus the zero modes cannot be detected locally.

For small systems, and when we worry about the effects of boundary conditions, degenerate ground states, etc, the zero modes are important and need special treatment. They are especially important when we are interested in topological effects, such as different sectors that are intertwined by vortex/vertex opertors.

The zero mode in the case of Tomonaga-Luttinger model is not considered in most of the places because the q = 0 part in the Coulomb interaction cancels out the positive background. For details, see Bruus and Flensberg.

Regarding the boundary conditions, in the case of 1+1 dimensions, the edge consists of 2 points and thus considered not very interesting (Not pretty sure about this but I have not seen much treatment of edge modes in 1+1 dimensions).