Duality symmetry, zero energy modes and boundary spectrum of the sine-Gordon/massive Thirring model

We solve the one-dimensional massive Thirring model, which is equivalent to the one-dimensional sine-Gordon model, with two types of Dirchlet boundary conditions: open boundary conditions (OBC) and twisted open boundary conditions (OBC). The system exhibits a duality symmetry which relates models with opposite bare mass parameters and boundary conditions, i.e: m₀ ↔ - m₀, OBC↔OBC. For m₀<0 and OBC, the system is in a trivial phase whose ground state is unique, as in the case of periodic boundary conditions. In contrast, for m₀<0 and OBC, the system is in a topological phase characterized by the existence of zero energy modes (ZEMs) localized at each boundary. As dictated by the duality symmetry, for m₀>0 and OBC, the trivial phase occurs, whereas the topological phase occurs for m₀>0 and OBC. In addition, we analyze the structure of the boundary excitations, finding significant differences between the attractive $(g>0)$ and the repulsive $(g<0)$ regimes.