Emergent boundary supersymmetry in a one dimensional superconductor

The interplay between bulk properties and boundary conditions in one-dimensional quantum systems, gives rise to many intriguing phenomena. These include the emergence of zero energy modes which are of significant interest to a variety of fields. In this work we investigate the presence of such zero modes in cases where the boundary conditions are dynamical and arise due to the coupling to some quantum degrees of freedom. In particular, we study a one-dimensional spin-singlet superconductor, modeled by the Gross-Neveu field theory, coupled to spin $\frac{1}{2}$ magnetic impurities at its boundaries via a spin-exchange interaction. We solve the model exactly for arbitrary values of the bulk and the impurity coupling strengths using nested coordinate Bethe ansatz and show that the system exhibits a rich boundary phase structure. For a range of couplings, the low energy degrees of freedom form irreducible representations of the supersymmetric $spl(2,1)\otimes spl(2,1)$ algebra which become degenerate at a specific point, indicating the emergence of supersymmetry in the low energy boundary degrees of freedom. We show that at the supersymmetric point there exist exact zero energy modes that map one ground state with the other. We express these in terms of the generators of the algebra.