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 ${\varphi }_{L,R}$ 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 $m_0$ and shifts the values of the boundary fields by ${\varphi }_{L,R}\to {\varphi }_{L,R}+\pi$. When the mass parameter $m_0<0$ and the boundary fields ${\varphi }_{L,R}=0$ or, equivalently, due to duality symmetry, when the mass parameter $m_0>0$ and the boundary fields ${\varphi }_{L,R}=\pi$, 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 $m_0<0$ and the boundary fields ${\varphi }_{L,R}=\pi$ and, equivalently, due to the duality symmetry, when the mass parameter $m_0>0$ and the boundary fields ${\varphi }_{L,R}=0$, the system is at a topological point where it exhibits a symmetry protected topological phase, which is characterized by the existence of zero energy bound states at both boundaries. As a result, the system exhibits a twofold degenerate ground state in each fermionic parity sector and an emergent boundary Hilbert space. For a given value of the mass parameter $m_0$, we find that the bound state structure and the structure of the boundary Hilbert space remains intact even if the boundary fields take values way from the special values corresponding to the topological and trivial points, provided these values are less than certain critical values. We refer to these regions around the trivial and topological points as the trivial and midgap phases, respectively. We find that the critical values depend on the strength of the interactions in the bulk, which is characterized by the Luttinger parameter $\beta$. 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.

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• Anyons
• Bosonization
• Conformal field theory
• Effective field theory
• Fractionalization
• Holon
• Quantum field theory
• Spin-singlet pairing
• Spin-triplet pairing
• Spinon
• Superconducting gap
• Symmetry protected topological states
• Topological field theories
• Topological superconductors