Statistical localization: From strong fragmentation to strong edge modes

Abstract

Certain disorder-free Hamiltonians are nonergodic due to a $s\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}g$ $f\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}g\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}n$ of the Hilbert space. Here, the authors introduce the notion of ``statistically localized integrals of motion'' (SLIOM) to characterize these systems. Despite SLIOMs being nonlocal operators, they become spatially localized to subextensive regions when their expectation value is taken in typical states. These can also result in statistically localized $s\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}g$ $z\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}o$ $m\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}d\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}s$, leading to topological string order for certain highly excited eigenstates as well as infinitely long-lived edge magnetization along with a thermalizing bulk.