We investigate one-dimensional charge conserving, spin-singlet (SSS) and spin-triplet (STS) superconductors in the presence of boundary fields. In systems with Open Boundary Conditions (OBC) it has been demonstrated that STS display a four-fold topol
ogical degeneracy, protected by the $mathbb{Z}_2$ symmetry which reverses the spins of all fermions, whereas SSS are topologically trivial. In this work we show that it is not only the type of the bulk superconducting instability that determines the eventual topological nature of a phase, but rather the interplay between bulk and boundary properties. In particular we show by means of the Bethe Ansatz technique that SSS may as well be in a $mathbb{Z}_2$-protected topological phase provided suitable twisted open boundary conditions ${widehat{OBC}}$ are imposed. More generally, we find that depending on the boundary fields, a given superconductor, either SSS or STS, may exhibits several types of phases such as topological, mid-gap and trivial phases; each phase being characterized by a boundary fixed point which which we determine. Of particular interest are the mid-gap phases which are stabilized close to the topological fixed point. They include both fractionalized phases where spin-$frac{1}{4}$ bound-states are localized at the two edges of the system and un-fractionalized phases where a spin-$frac{1}{2}$ bound-state is localized at either the left or the right edge.
Charge conserving spin singlet and spin triplet superconductors in one dimension are described by the $U(1)$ symmetric Thirring Hamiltonian. We solve the model with open boundary conditions on the a finite line segment by means of the Bethe Ansatz. W
e show that the ground state displays a fourfold degeneracy when the bulk is in the spin triplet superconducting phase. This degeneracy corresponds to the existence of zero energy boundary bound states localized at the edges which may be interpreted, in the light of the previous semi-classical analysis due to Kesselman and Berg cite{Keselman2015}, as resulting from the existence of fractional spin $pm 1/4$ localized at the two edges of the system.
Quantum impurity models are prevalent throughout many body physics, providing some prime examples of strongly correlated systems. Aside from being of great interest in themselves they can provide deep insight into the effects of strong correlations i
n general. The classic example is the Kondo model wherein a magnetic impurity is screened at low energies by a non interacting metallic bath. Here we consider a magnetic impurity coupled to a quantum wire with pairing interaction which dynamically generates a mass gap. Using Bethe Ansatz we solve the system exactly finding that it exhibits both screened and unscreened phases for an antiferromagnetic impurity. We determine the ground state density of states and magnetization in both phases as well as the excitations. In contrast to the well studied case of magnetic impurities in superconductors we find that there are no intragap bound states in the spectrum. The phase transition is not associated to a level crossing but with quantum fluctuations.
