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arXiv:1605.03003 · Physical Review Letters

Diagonalization and Many-Body Localization for a Disordered Quantum Spin Chain

John Z Imbrie

Published Jul 5, 2016• 289 citations
arXivHTMLPDF

Abstract

We consider a weakly interacting quantum spin chain with random local interactions. We prove that many-body localization follows from a physically reasonable assumption that limits the extent of level attraction in the statistics of eigenvalues. In a Kolmogorov-Arnold-Moser-style construction, a sequence of local unitary transformations is used to diagonalize the Hamiltonian by deforming the initial tensor-product basis into a complete set of exact many-body eigenfunctions.

DOI
10.1103/physrevlett.117.027201

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We consider a weakly interacting quantum spin chain with random local interactions. We prove that many-body localization follows from a physically reasonable assumption that limits the extent of level attraction in the statistics of eigenvalues. In a Kolmogorov-Arnold-Moser-style construction, a sequence of local unitary transformations is used to diagonalize the Hamiltonian by deforming the initial tensor-product basis into a complete set of exact many-body eigenfunctions.
Abstract We study the many-body localization (MBL) properties of the Heisenberg XXZ spin- 12\frac 1221​ chain in a random magnetic field. We prove that the system exhibits localization in any given energy interval at the bottom of the spectrum in a nontrivial region of the parameter space. This region, which includes weak interaction and strong disorder regimes, is independent of the size of the system and depends only on the energy interval. Our approach is based on the reformulation of the localization problem as an expression of quasi-locality for functions of the random many-body XXZ Hamiltonian. This allows us to extend the fractional moment method for proving localization, previously derived in a single-particle localization context, to the many-body setting.
Strong long-range interaction leads to localization in a closed quantum system without disorders. Employing the exact diagonalization method, I numerically investigate thermalization and many-body localization in translational invariant quantum chains with finite Coulomb interactions. In the computational basis, excluding all trivial degeneracies, the interaction-induced localization is well demonstrated in aspects of level statistics, eigenstate expectation values, and Anderson localization on graphs constructed of many-body bases. The nature of localization for generic eigenstates is attributed to the quasidisorder from the power-law interactions, and the full localization in the Hilbert space is similar to that in the disorder case. However, due to real-space symmetries, the long-time dynamics is dominated by the degenerated eigenstates and eventually reaches homogeneity in real space. On the other hand, the entanglement entropy exhibits size dependence beyond the area law for the same reason, even deep in the localized state, indicating an incomplete localization in real space.
Rare regions with weak disorder (Griffiths regions) have the potential to spoil localization. We describe a non-perturbative construction of local integrals of motion (LIOMs) for a weakly interacting spin chain in one dimension, under a physically reasonable assumption on the statistics of eigenvalues. We discuss ideas about the situation in higher dimensions, where one can no longer ensure that interactions involving the Griffiths regions are much smaller than the typical energy-level spacing for such regions. We argue that ergodicity is restored in dimension d >1, although equilibration should be extremely slow, similar to the dynamics of glasses. This article is part of the themed issue ‘Breakdown of ergodicity in quantum systems: from solids to synthetic matter’.
We present the study of the structure of many-body eigenfunctions in a one-dimensional disordered spin chain.We discuss the choice of an appropriate basis in the Hilbert space, where the problem can be seen as an Anderson model defined on a high-dimensional non-trivial graph, determined by the many-body Hamiltonian.The comparison with the usual behavior of wave-functions in finite dimensional Anderson localization allows us to put in light the main differences of the many-body case.At high disorder, the typical eigenfunctions do not seem to localize though they occupy a infinitesimal portion of the Hilbert space in the thermodynamic limit.We perform a detailed analysis of the distribution of the wave-function coefficients and their peculiar scaling in the small and large disorder phase.We propose a criterion to identify the position of the transition by looking at the long tails of these distributions.The results coming from exact diagonalization show signs of breaking of ergodicity when the disorder reaches a critical value that agrees with the estimation of the many-body localization transition in the same model.
Despite a very good understanding of single-particle Anderson localization in one-dimensional (1D) disordered systems, many-body effects are still full of surprises; a famous example is the interaction-driven many-body localization (MBL) problem, about which much has been written, and perhaps the best is yet to come. Interestingly enough the noninteracting limit provides a natural playground to study nontrivial multiparticle physics, offering the possibility to test some general mechanisms with very large-scale exact diagonalization simulations. In this paper, we first revisit the 1D many-body Anderson insulator through the lens of extreme value theory, focusing on the extreme polarizations of the equivalent spin chain model in a random magnetic field. A many-body-induced chain breaking mechanism is explored numerically, and compared to an analytically solvable toy model. A unified description, from weak to large disorder strengths WWW emerges, where the disorder-dependent average localization length $\ensuremath{\xi}(W)$ governs the extreme events leading to chain breaks. In particular, tails of the local magnetization distributions are controlled by $\ensuremath{\xi}(W)$. Remarkably, we also obtain a quantitative understanding of the full distribution of the extreme polarizations, which is given by a Fr\'echet-type law. In a second part, we explore finite interaction physics and the MBL question. For the available system sizes, we numerically quantify the difference in the extreme value distributions between the interacting problem and the noninteracting Anderson case. Strikingly, we observe a sharp ``extreme-statistics transition'' as WWW changes, which may coincide with the MBL transition.