**Parameshwar R. Pasnoori (University of Maryland)**

### Spin fractionalization 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. The 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, the systems with SPT exhibit non-local order parameter and robust ground state degeneracy associated with protected fractionalized gapless excitations at the edges. In the first part of the talk, I will consider an example of an SPT system with strong electron-electron correlations which is called ”one dimensional charge conserving superconductor”. By applying magnetic fields at the edges, I shall show that due to the interplay between the bulk and the boundary, the system gives rise to an SPT phase protected by the Z2 spin flip symmetry. This phase is characterized by the existence of exponentially localized fractional spin 41 at the edges satisfying an emergent Clifford algebra. This leads to the existence of two zero energy Majorana modes (ZEM) at each edge resulting in four fold topological degeneracy of the ground states. In the second part of the talk I will consider dynamical boundary conditions by coupling each edge to a Kondo impurity. This allows one to study the Kondo effect in the presence of strong correlations among electrons. I will show that in contrast to the BCS case, each impurity exhibits three regimes: renormalized Kondo regime, YSR regime and unscreened regime, and the well known Shiba (YSR) states exist only in the narrow YSR regime. The full phase diagram of the system with two Kondo impurities, one at each edge, exhibits an emergent boundary supersymmetry (SUSY). When the quantum fluctuations associated with the impurities are suppressed, the SUSY algebra is reduced down to Clifford algebra associated with the Majorana modes. In the third part of the talk I will consider the 1/2 XXZ chain which exhibits SSB. I will show that this system also exhibits exponentially localized spin 1/4 at the edges associated with strong Majorana zero modes which map different symmetry broken sectors. All the systems described above undergo Hilbert space fragmentation, where the Hilbert space is comprised of a certain number of towers of excited states. As the parameters are varied, they undergo Boundary eigenstate phase transition where the number of towers of the Hilbert space changes.