Statistical localization: From strong fragmentation to strong edge modes

Abstract

Certain disorder-free Hamiltonians can be non-ergodic due to a strong fragmentation of the Hilbert space into disconnected sectors. Here, we 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 illustrate this general concept on several Hamiltonians, both with and without dipole conservation. Furthermore, we demonstrate 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, constituting the first example of such strong edge modes in a non-integrable model. We also show that in a particular example, these edge modes lead to the appearance of topological string order in a certain subset of highly excited eigenstates. Some of our suggested models can be realized in Rydberg quantum simulators.CONTENTS * These authors contributed equally to this work.

Observing non-ergodicity due to kinetic constraints in tilted Fermi-Hubbard chains

“…By tuning the direction of the tilt in a 2D lattice, dipole- and higher-moment conserving models could be realized 20 , 57 enabling studies beyond the hydrodynamic regime 58 . Moreover, it will be interesting to explore the connection between lattice gauge theories and the phenomenon of Hilbert-space fragmentation 21 , 27 , 29 , 59 – 61 , which could be addressed experimentally in a similar model 62 .…”

Fracton phases of matter

“…In turn, Reference [110] showed how these localized states are intricately connected with the many-body scar paradigm, in which a small number of non-thermal states exist in an otherwise thermal spectrum [111]. All of this unexpected behavior, arising as a consequence of two simple conservation laws, has led to numerous recent investigations into ergodicity breaking in fracton systems [112][113][114][115][116][117].…”

Long-Lived Interacting Phases of Matter Protected by Multiple Time-Translation Symmetries in Quasiperiodically Driven Systems

“…The ability to engineer such conserved quantities in a robust manner can lead to a variety of interesting dynamical phenomena, including the recently discussed idea of "Hilbert space shattering" of Refs. [135][136][137][138], where the Hilbert space fractures into exponentially many subsectors.…”