Boundary phenomena and phase transitions in strongly correlated one dimensional systems.

One dimensional quantum systems exhibit many interesting physical phenomena as a result of strong correlations. The gapped systems with symmetries exhibit exotic phases and are categorized as spontaneous symmetry breaking (SSB) or symmetry protected topological (SPT) phases. Systems with SSB have non vanishing local order parameter and a discrete symmetry is spontaneously broken leading to degenerate pairing in the spectrum. In contrast, systems with SPT exhibit non-local order parameter and robust ground state degeneracy associated with protected fractionalized gapless excitations at the edges. In this work we consider systems belonging to both classes: the spin 1/2 XXZ chain which exhibits SSB and the charge conserving superconductor which exhibits SPT, and study them under the effect of symmetry breaking fields at the edges. We find that these systems exhibit a rich phase diagram and show that certain phases display spin fractionalization associated with strong Majorana zero modes at the edges. We then consider spin 1/2 Heisenberg XXX chain, a gapless system that does not fall into either the SPT or SSB classes. We show that it exhibits a rich phase diagram and contains zero modes which arise at high energies. We show that in all three systems described above, the Hilbert space is comprised of a certain number of towers of excited states, and that they exhibit a new type of phase transition named 'Hilbert space' or 'eigenstate' phase transition where the number of towers of the Hilbert space changes as result of the application of the edge fields.