We solve the one dimensional massive Thirring model or equivalently the sine-Gordon model in the repulsive regime with general Dirichlet boundary conditions, which are characterized by two boundary fields ϕ₋, ₑ associated with the left and right boundaries respectively. In the presence of these boundary fields, which explicitly break the charge conjugation symmetry, the system exhibits a duality symmetry which changes the sign of the mass parameter m₀ and shifts the values of the boundary fields by ϕ₋, ₑ ϕ₋, ₑ+π. When the mass parameter m₀0 and the boundary fields ϕ₋, ₑ=π, the system is at a trivial point. Here, the ground state is unique just as in the case of periodic boundary conditions. In contrast, when the mass parameter m₀0 and the boundary fields ϕ₋, ₑ=0, the system is at a topological point where it exhibits a symmetry protected topological (SPT) phase, which is characterized by the existence of zero energy bound states at both the boundaries. For a given value of the mass parameter m₀, we find that these phases remain stable in the presence of symmetry breaking fields at the boundary, provided they are smaller than certain critical values which depend on the strength of the interactions in the bulk. Hence, we show that the stability of the SPT and trivial phases depends on the interplay of the symmetry breaking boundary field values and the bulk interaction strength.
Pasnoori et al. (Tue,) studied this question.