Statistical localization: from strong fragmentation to strong edge modes

Certain disorder-free Hamiltonians can be non-ergodic due to a strong fragmentation of the Hilbert space into disconnected sectors. We show how to characterize such systems by introducing the notion of 'statistically localized integrals of motion’ (SLIOM), whose eigenvalues label the connected components of the Hilbert space. SLIOMs are not spatially localized in the operator sense, but appear localized to sub-extensive regions when their expectation value is taken in typical states with a finite density of particles. We will illustrate this general concept on several Hamiltonians, both with and without dipole conservation. For the former we uncover additional SLIOMs due to dipole moment conservation on finite regions of the chain. Moreover, we explain that there exist perturbations which destroy these integrals of motion in the bulk of the system, while keeping them on the boundary. This results in statistically localized strong zero modes, leading to infinitely long-lived edge magnetizations along with a thermalizing bulk. We also show that these edge modes can lead to the appearance of topological string order in a certain subset of highly excited eigenstates, and conclude providing experimental realizations of these models using existing experimental platforms.