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Many-body localization landscape

Shankar Balasubramanian, Yunxiang Liao, and Victor Galitski
Phys. Rev. B 101, 014201 – Published 6 January 2020

Abstract

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 Schrödinger 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.

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  • Received 24 September 2019
  • Revised 7 December 2019

DOI:https://doi.org/10.1103/PhysRevB.101.014201

©2020 American Physical Society

Physics Subject Headings (PhySH)

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Shankar Balasubramanian1,2, Yunxiang Liao3, and Victor Galitski3

  • 1Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
  • 2Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
  • 3Joint Quantum Institute and Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA

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Issue

Vol. 101, Iss. 1 — 1 January 2020

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