Efficient variational diagonalization of fully many-body localized Hamiltonians

Pollmann, Frank; Khemani, Vedika; Cirac, J. Ignacio; Sondhi, S. L.

doi:10.48550/arxiv.1506.07179

Cited by 7 publications

(11 citation statements)

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“…We used this distinct feature of MBL systems to perform highly sensitive benchmarks of our MPS algorithm. Using our algorithm, we obtained eigenstates of large systems at disorder W = 4, which is closer to the MBL transition than previously reported [29][30][31][32][33].…”

supporting

confidence: 71%

“…Of course, the CG scheme induces extra errors, but as we have shown they are not severe; for example, our obtained energy variance (Fig. S7) is comparable or lower than other approaches [29][30][31][32][33]. Finally, to reduce the errors even further, it is possible to increase the number of CG iterations N CG , which is far more efficient than increasing the bond dimension χ in methods relying on exact inversion.…”

Section: B2 Performance Of the Algorithm And Discussionmentioning

confidence: 51%

“…In addition to providing new insights into the properties of MBL systems, the ES is of crucial importance for the matrix-product state (MPS) optimization algorithms such as the "density matrix renormalization group" (DMRG) [28]. While in principle the MPS naturally encode the eigenstates of MBL phase due to the area-law entropy, practical realizations of the efficient optimization algorithms are an active area of research [29][30][31][32][33]. In this work we develop an MPS optimization to tar- get the highly-excited states of a disordered XXZ chain in 1D, and use the power-law ES as a sensitive benchmark of its accuracy in large systems up to L = 30 spins.…”

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confidence: 99%

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Power-Law Entanglement Spectrum in Many-Body Localized Phases

et al. 2016

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Entanglement spectrum of the reduced density matrix contains information beyond the von Neumann entropy and provides unique insights into exotic orders or critical behavior of quantum systems. Here we show that strongly-disordered systems in the many-body localized phase have universal power-law entanglement spectra, arising from the presence of extensively many local integrals of motion. The power-law entanglement spectrum distinguishes many-body localized systems from ergodic systems, as well as from ground states of gapped integrable models or systems in the vicinity of scale-invariant critical points. We confirm our results using large-scale exact diagonalization. In addition, we develop a matrix-product state algorithm which allows us to access the eigenstates of large systems close to the localization transition, and discuss general implications of our results for variational studies of highly excited eigenstates in many-body localized systems. Introduction.-Recently, much progress has been made towards understanding the mechanisms of ergodicity and its breakdown in an isolated quantum many-body system. Currently, two generic classes of many-body systems are known: ergodic (thermal) systems and manybody localized (MBL) systems [1][2][3][4]. An ergodic system is one that acts as a heat bath for its subsystems, and therefore thermalizes as a result of unitary evolution [5][6][7]. By contrast, in MBL systems transport of energy is quenched by disorder, via a mechanism akin to the single-particle Anderson localization [8]. Nevertheless, MBL systems do reach stationary states [9, 10], which are highly non-thermal due to the emergence of extensively many quasi-local integrals of motion (LIOMs) [11][12][13].In addition to distinct dynamical properties, ergodic and MBL systems are sharply distinguished by the microscopic nature of their eigenstates. This difference can be probed via quantum-information measures, such as entanglement entropy (EE). Given a pure quantum state ψ of a many-body system S = L ∪ R, consisting of two subsystems L and R, the EE is defined as S = − D i λ i ln λ i , where {λ i }, i = 1, ..., D, are the eigenvalues of the reduced density matrixρ R = Tr L |ψ ψ|, and D is the dimensionality of the Hilbert space of R. The EE of highlyexcited eigenstates of thermal systems, which obey the "Eigenstate Thermalization Hypothesis" [5][6][7], is known to generically scale as the number of degrees of freedom in R ("volume law"). On the other hand, in an MBL system the EE of nearly all eigenstates obeys the "area law" [11,12,14]. This weaker scaling of EE makes MBL systems reminiscent of ground states of gapped systems [15].EE, while providing a quantitative measure of entanglement in a many-body state, contains no information about how it is created or how different degrees of freedom are entangled with each other. Therefore, to gain

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supporting

confidence: 71%

Section: B2 Performance Of the Algorithm And Discussionmentioning

confidence: 51%

mentioning

confidence: 99%

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Power-Law Entanglement Spectrum in Many-Body Localized Phases

et al. 2016

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“…Shortly before the completion of our work, we became aware of three preprints [40][41][42] which present similar ideas to access excited states of MBL systems via DMRG algorithms (these algorithms are more elaborate than the one used here); another method to compute features of the entire spectrum rather than individual states was introduced in preprint [43]. We complement these studies in the following way: (a) Our data was obtained over the course of 9 months using large-scale numerics (900.000 core hours); this allows us to access systems which are larger than those investigated in Refs.…”

Section: Introductionmentioning

confidence: 99%

Entanglement scaling of excited states in large one-dimensional many-body localized systems

Kennes

Karrasch

2016

Phys. Rev. B

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We study the properties of excited states in one-dimensional many-body localized (MBL) systems using a matrix product state algorithm. First, the method is tested for a large disordered noninteracting system, where for comparison we compute a quasi-exact reference solution via a Monte Carlo sampling of the single-particle levels. Thereafter, we present extensive data obtained for large interacting systems of L ∼ 100 sites and large bond dimensions χ ∼ 1700, which allows us to quantitatively analyze the scaling behavior of the entanglement S in the system. The MBL phase is characterized by a logarithmic growth S(L) ∼ log(L) over a large scale separating the regimes where volume and area laws hold. We check the validity of the eigenstate thermalization hypothesis. Our results are consistent with the existence of a mobility edge.

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“…Here, we show that an appropriate (local) mapping of the energy values to the wave functions leads to (quasi) local parent Hamiltonians. This provides another example for construction and study of the fully manybody-localized systems [7][8][9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Characterizing the parent Hamiltonians for a complete set of orthogonal wave functions: An inverse quantum problem

Ramezanpour

2016

Phys. Rev. A

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We study the inverse problem of constructing an appropriate Hamiltonian from a physically reasonable set of orthogonal wave functions for a quantum spin system. Usually, we are given a local Hamiltonian and our goal is to characterize the relevant wave functions and energies (the spectrum) of the system. Here, we take the opposite approach; starting from a reasonable collection of orthogonal wave functions, we try to characterize the associated parent Hamiltonians, to see how the wave functions and the energy values affect the structure of the parent Hamiltonian.Specifically, we obtain (quasi) local Hamiltonians by a complete set of (multilayer) product states and a local mapping of the energy values to the wave functions. On the other hand, a complete set of tree wave functions (having a tree structure) results to nonlocal Hamiltonians and operators which flip simultaneously all the spins in a single branch of the tree graph. We observe that even for a given set of basis states, the energy spectrum can significantly change the nature of interactions in the Hamiltonian. These effects can be exploited in a quantum engineering problem optimizing an objective functional of the Hamiltonian.

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Efficient variational diagonalization of fully many-body localized Hamiltonians

Cited by 7 publications

References 0 publications

Power-Law Entanglement Spectrum in Many-Body Localized Phases

Power-Law Entanglement Spectrum in Many-Body Localized Phases

Entanglement scaling of excited states in large one-dimensional many-body localized systems

Characterizing the parent Hamiltonians for a complete set of orthogonal wave functions: An inverse quantum problem

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