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Many-body localization landscape

TitleMany-body localization landscape
Publication TypeJournal Article
Year of Publication2020
AuthorsS. Balasubramanian, Y. Liao, and V. Galitski
JournalPhys. Rev. B
Date PublishedJAN 6
Type of ArticleArticle

We generalize the notion of ``localization landscape,{''} introduced by M. Filoche and S. Mayboroda {[}Proc. Natl. Acad. Sci. USA 109, 14761 (2012)] for the single-particle Schrodinger operator, to a wide class of interacting many-body Hamiltonians. The many-body localization landscape (MBLL) is defined on a graph in the Fock space, whose nodes represent the basis vectors in the Fock space and edges correspond to transitions between the nodes connected by the hopping term in the Hamiltonian. It is shown that in analogy to the single-particle case, the inverse MBLL plays the role of an effective potential in the Fock space. We construct a generalized discrete Agmon metric and prove Agmon inequalities on the Fock-state graph to obtain bounds on the exponential decay of the many-body wave functions in the Fock space. The corresponding construction is motivated by the semiclassical WKB approximation, but the bounds are exact and fully quantum mechanical. We then prove a series of locality theorems which establish where in the Fock space we expect eigenstates to localize. Using these results as well as the locator expansion, we establish evidence for the existence of many-body localized states for a wide class of lattice models in any physical dimension in at least a part of their Hilbert space. The key to this argument is the observation that in sharp contrast to the conventional locator expansion for the Green's function, the locator expansion for the landscape function contains no resonances. For short-range hopping, which limits the connectivity of the Fock-state graph, the locator series is proven to be convergent and bounded by a simple geometric series. This, in combination with the discrete Agmon-type inequalities and the locality theorems, shows that localization for a fraction of the Hilbert space survives weak interactions and weak hopping at least for some realizations of disorder, but cannot prove or rule out localization of the entire Hilbert space. We qualitatively discuss potential breakdown of the locator expansion in the MBLL for long-range hopping and the appearance of a mobility edge in higher-dimensional theories.