Title: Localization of Extended Quantum Objects,
Abstract: A quantum system of particles can exist in a localized phase, exhibiting
ergodicity breaking and maintaining forever a local memory of its initial
conditions. We generalize this concept to a system of extended objects, such as
strings and membranes, arguing that such a system can also exhibit localization
in the presence of sufficiently strong disorder (randomness) in the
Hamiltonian. We show that localization of large extended objects can be mapped
to a lower-dimensional many-body localization problem. For example, motion of a
string involves propagation of point-like signals down its length to keep the
different segments in causal contact. For sufficiently strong disorder, all
such internal modes will exhibit many-body localization, resulting in the
localization of the entire string. The eigenstates of the system can then be
constructed perturbatively through a convergent 'string locator expansion.' We
propose a type of out-of-time-order string correlator as a diagnostic of such a
string localized phase. Localization of other higher-dimensional objects, such
as membranes, can also be studied through a hierarchical construction by
mapping onto localization of lower-dimensional objects. Our arguments are
'asymptotic' ($i.e.$ valid up to rare regions) but they extend the notion of
localization (and localization protected order) to a host of settings where
such ideas previously did not apply. These include high-dimensional
ferromagnets with domain wall excitations, three-dimensional topological phases
with loop-like excitations, and three-dimensional type-II superconductors with
flux line excitations. In type-II superconductors, localization of flux lines
could stabilize superconductivity at energy densities where a normal state
would arise in thermal equilibrium.