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arXiv:1411.0660 · Physical Review B

Many-body localization edge in the random-field Heisenberg chain

David J. Luitz, Nicolas Laflorencie, Fabien Alet

Published Feb 9, 2015• 1,022 citations
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Abstract

The authors study the phenomena of many-body localization in a random field Heisenberg chain. In this paper the authors use a shift-inverse exact diagonalization approach that allows them to study the mid-spectrum spectral properties of the model for system sizes of up to N=22. This has allow the authors to identify the many-body localization edge.

DOI
10.1103/physrevb.91.081103

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The authors study the phenomena of many-body localization in a random field Heisenberg chain. In this paper the authors use a shift-inverse exact diagonalization approach that allows them to study the mid-spectrum spectral properties of the model for system sizes of up to N=22. This has allow the authors to identify the many-body localization edge.
We provide a pedagogical review on the calculation of highly excited eigenstates of disordered interacting quantum systems which can undergo a many-body localization (MBL) transition, using shift-invert exact diagonalization. We also provide an example code at https://bitbucket.org/dluitz/sinvert_mbl. Through a detailed analysis of the simulational parameters of the random field Heisenberg spin chain, we provide a practical guide on how to perform efficient computations. We present data for mid-spectrum eigenstates of spin chains of sizes up to L = 26. This work is also geared towards readers with interest in efficiency of parallel sparse linear algebra techniques that will find a challenging application in the MBL problem.
We provide a pedagogical review on the calculation of highly excited eigenstates of disordered interacting quantum systems which can undergo a many-body localization (MBL) transition, using shift-invert exact diagonalization. We also provide an example code at https://bitbucket.org/dluitz/sinvert_mbl. Through a detailed analysis of the simulational parameters of the random field Heisenberg spin chain, we provide a practical guide on how to perform efficient computations. We present data for mid-spectrum eigenstates of spin chains of sizes up to L=26 L = 26 . This work is also geared towards readers with interest in efficiency of parallel sparse linear algebra techniques that will find a challenging application in the MBL problem.
We study the many-body localization of the random-field Heisenberg chain using the nested shift-invert Lanczos method with an iterative linear solver. We use the minimum residual method (MINRES) inside each Lanczos iteration. The memory consumption of the proposed method is only proportional to the dimension of the Hilbert space. We also introduce a preconditioner that takes into account the effects of disorder and interaction in the random-field Heisenberg chain. As a probe of many-body localization transition, we propose a unitary operator called the twist operator, which has a clear interpretation in the real space. We discuss its behavior for thermal and localized eigenstates. We demonstrate the efficiency of the nested iterative shift-invert diagonalization method with the proposed preconditioner for the many-body localization problem and estimate the transition point of the random-field Heisenberg chain more precisely based on the finite-size analysis of the expectation value of the twist operator.
Abstract We study the one-dimensional spin-1/2 Heisenberg chain with Dzyaloshinskii-Moriya interaction in a random magnetic field using exact diagonalization. In order to obtain many-body mobility edge at infinite temperature, we employ a polynomial filtered Lanczos method that can avoid the fill-in problem when implementing the commonly used shift-and-invert transformation. In stark contrast to the original Heisenberg model, although the localized phase always conforms to Poisson statistics, the ergodic phase exhibits the Gaussian unitary ensemble rather than the Gaussian orthogonal ensemble statistics due to the lack of complex conjugation symmetry. The boundary between the ergodic and localized phases is determined by carefully performing finite-size scalings for the level statistics, entanglement entropy and its standard deviation, as well as fluctuations of the total spin of the system. The two phases are also well distinguished by the full delocalization or localization in the Hilbert space wherein the participation entropies present. To indicate the localized phase in experiment, we propose a scheme for realizing the out-of-time-order correlator on a modern nuclear magnetic resonance quantum simulator.
We study the many-body localization problem in the non-Abelian SU(2)-invariant random antiferromagnetic exchange model in 1D. Exact and sparse matrix diagonalization methods are used to calculate eigenvalues and eigenvectors of the Hamiltonian matrix. We investigate the behavior of the energy level gap-ratio statistic, participation ratio, entanglement entropy and the entanglement spectral parameter as a function of disorder strength. Different distributions of random couplings are considered. We find, up to L = 24 , a clear distinction between our non-Abelian model and the more often studied random field Heisenberg model: the regime of seemingly localized behavior is much less pronounced in the random exchange model than in the field model case.
We construct a solvable spin chain model of many-body localization (MBL) with a tunable mobility edge. This simple model not only demonstrates analytically the existence of mobility edges in interacting one-dimensional (1D) disordered systems, but also allows us to study their physics. By establishing a connection between MBL and a quantum central limit theorem (QCLT), we show that many-body localization-delocalization transitions can be visualized as tuning a mobility edge in the energy spectrum. Since the effective disorder strength for individual eigenstates depends on energy density, we identify "energy-resolved disorder strength" as a physical mechanism for the appearance of mobility edges, and support the universality of this mechanism by arguing its presence in a large class of models including the random-field Heisenberg chain. We also construct models with multiple mobility edges. All our constructions can be made translationally invariant.