Localization of Extended Quantum Objects

TL;DR

This paper extends the concept of many-body localization to systems with extended objects like strings and membranes, showing they can also exhibit localization under strong disorder, with implications for various high-dimensional quantum phases.

Contribution

It introduces a framework for understanding localization of extended objects, mapping their dynamics to lower-dimensional localized problems, and proposes diagnostic tools like string correlators.

Findings

Localization of strings and membranes is possible with strong disorder.

Eigenstates can be constructed via a string locator expansion.

Localization can stabilize phases like superconductivity in high dimensions.

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.