Many-body localization and transition by density matrix renormalization group and exact diagonalization studies

Lim, S. P.; Sheng, D. N.

doi:10.1103/physrevb.94.045111

Cited by 94 publications

(103 citation statements)

References 75 publications

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“…Furthermore, we mapped out the mobility edge in this model, demonstrating that the method has a good energy resolution. Our criterion requires a sample of adjacent eigenstates in an energy window, and hence it can be straightforwardly evaluated by "spectral transformation" methods applicable to much larger system sizes, either via iterative diagonalizations [24] or DMRG [48][49][50][51].…”

Section: Discussionmentioning

confidence: 99%

Criterion for Many-Body Localization-Delocalization Phase Transition

2015

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We propose a new approach to probing ergodicity and its breakdown in one-dimensional quantum manybody systems based on their response to a local perturbation. We study the distribution of matrix elements of a local operator between the system's eigenstates, finding a qualitatively different behavior in the manybody localized (MBL) and ergodic phases. To characterize how strongly a local perturbation modifies the eigenstates, we introduce the parameter GðLÞ ¼ hlnðV nm =δÞi, which represents the disorder-averaged ratio of a typical matrix element of a local operator V to energy level spacing δ; this parameter is reminiscent of the Thouless conductance in the single-particle localization. We show that the parameter GðLÞ decreases with system size L in the MBL phase and grows in the ergodic phase. We surmise that the delocalization transition occurs when GðLÞ is independent of system size, GðLÞ ¼ G c ∼ 1. We illustrate our approach by studying the many-body localization transition and resolving the many-body mobility edge in a disordered one-dimensional XXZ spin-1=2 chain using exact diagonalization and time-evolving block-decimation methods. Our criterion for the MBL transition gives insights into microscopic details of transition. Its direct physical consequences, in particular, logarithmically slow transport at the transition and extensive entanglement entropy of the eigenstates, are consistent with recent renormalization-group predictions.

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Section: Discussionmentioning

confidence: 99%

Criterion for Many-Body Localization-Delocalization Phase Transition

2015

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“…However, since the number of eigenstates is exponential in the system size, for large N one can only tackle the eigenstates in a certain energy window using MPS (see, e.g., Refs. [49][50][51]). On the other hand, spectral tensor networks are meant to encode an approximation to all eigenstates at once, which is a desirable property if one aims to calculate dynamical properties of local observables in MBL systems.…”

Section: Tensor Network Ansatzmentioning

confidence: 99%

“…It was suggested that the accuracy of the approximation for a given chain length can be increased by increasing the number of layers (the depth of the quantum circuit). Compared to the methods targeting eigenstates within an energy window [49][50][51][52], this procedure is constructed to efficiently represent all eigenstates with sufficient accuracy, providing access to dynamical properties of local observables.…”

Section: Introductionmentioning

confidence: 99%

Efficient Representation of Fully Many-Body Localized Systems Using Tensor Networks

Wahl¹,

Pal²,

Simon³

2017

Phys. Rev. X

125

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We propose a tensor network encoding the set of all eigenstates of a fully many-body localized system in one dimension. Our construction, conceptually based on the ansatz introduced in Phys. Rev. B 94, 041116(R) (2016), is built from two layers of unitary matrices which act on blocks of l contiguous sites. We argue that this yields an exponential reduction in computational time and memory requirement as compared to all previous approaches for finding a representation of the complete eigenspectrum of large many-body localized systems with a given accuracy. Concretely, we optimize the unitaries by minimizing the magnitude of the commutator of the approximate integrals of motion and the Hamiltonian, which can be done in a local fashion. This further reduces the computational complexity of the tensor networks arising in the minimization process compared to previous work. We test the accuracy of our method by comparing the approximate energy spectrum to exact diagonalization results for the random-field Heisenberg model on 16 sites. We find that the technique is highly accurate deep in the localized regime and maintains a surprising degree of accuracy in predicting certain local quantities even in the vicinity of the predicted dynamical phase transition. To demonstrate the power of our technique, we study a system of 72 sites, and we are able to see clear signatures of the phase transition. Our work opens a new avenue to study properties of the many-body localization transition in large systems.

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“…We have proposed and analyzed the presumably simplest possible protocol for quantum magnets to exhibit the absence of ergodic dynamics, and thus Many-Body Localization in the form of remanent magnetization in initially ferromagnetically polarized antiferromagnets. The present calculation illustrates how the perturbative construction of conserved quantities allows one to make analytic predictions for quantities of experimental relevance.Our explicit recipe for constructing the conserved quantities is an analytical alternative to several recent numerical schemes based on DMRG [41][42][43][44] or quantum Monte Carlo [45] that allow one to study properties of specific MBL eigenstates. Since the simple formula (6) is derived under the sole assumption that the conserved operators have spectrum ±1, it could be applied to the conserved pseudo-spins constructed numerically in Refs.…”

mentioning

confidence: 99%

“…Our explicit recipe for constructing the conserved quantities is an analytical alternative to several recent numerical schemes based on DMRG [41][42][43][44] or quantum Monte Carlo [45] that allow one to study properties of specific MBL eigenstates. Since the simple formula (6) is derived under the sole assumption that the conserved operators have spectrum ±1, it could be applied to the conserved pseudo-spins constructed numerically in Refs.…”

mentioning

confidence: 99%

Remanent Magnetization: Signature of Many-Body Localization in Quantum Antiferromagnets

Müller

2017

Phys. Rev. Lett.

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We study the remanent magnetization in antiferromagnetic, many-body localized quantum spin chains, initialized in a fully magnetized state. Its long time limit is an order parameter for the localization transition, which is readily accessible by standard experimental probes in magnets. We analytically calculate its value in the strong-disorder regime exploiting the explicit construction of quasi-local conserved quantities of the localized phase. We discuss analogies in cold atomic systems.Introduction. The non-equilibrium dynamics in disordered, isolated quantum systems have been subject to theoretical investigations ever since the notion of localization was introduced in [1]. Spin systems in random fields are prototypical models to analyze the disorderinduced breakdown of thermalization: a large number of studies on disordered spin chains [2][3][4][5][6][7][8][9][10][11][12][13] has provided evidence for a dynamical phase transition between a weakdisorder phase which thermalizes, and a Many-Body Localized (MBL) phase in which excitations do not diffuse, ergodicity is broken and local memory of the initial conditions persists for infinite time [14][15][16].Signatures of MBL are found in the properties of individual many-body eigenstates. Even highly excited eigenstates exhibit area-law scaling of the bipartite entanglement entropy [4,7,8,17] and Poissonian level statistics [2,18,19], both being incompatible with thermalization [20][21][22]. Novel dynamical properties such as the logarithmic spreading of entanglement have been observed in direct simulations of the time evolution [3,23]. The non-equilibrium physics of MBL systems has been probed experimentally in artificial quantum systems made of cold atomic gases [24,25] and trapped ion systems [26], while an indirect signature in the from of strongly suppressed absorption of radiation was found in electron-glasses [27]. However, direct observations of MBL in solid-state materials are still lacking.It has been argued [28][29][30] that the properties of MBL systems are related to the existence of extensively many quasi-local conserved operators that strongly constrain the quantum dynamics, preventing both transport and thermalization. Their existence also follows as a corollary from Imbrie's rigorous arguments in favor of MBL [31,32].In this work, we propose a experimentally readily observable consequence of MBL in quantum magnets: the out-of-equilibrium remanent magnetization that persists after ferromagnetically polarizing an antiferromagnet whose total magnetization is not a conserved. The remanence implies non-ergodicity, since ergodic dynamics would relax the magnetization completely (cf. Fig. 1 for a schematic sketch of the protocol). As an example, we consider an antiferromagnetic, anisotropic Heisenberg spin-1/2 chain(1) subject to random fields h k along the Ising axis. We assume J z < 0, as well as J x = J y to ensure the nonconservation of the total magnetization. Such Hamiltonians can be realized, e.g., in Ising compounds with both exchange and dipola...

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Many-body localization and transition by density matrix renormalization group and exact diagonalization studies

Cited by 94 publications

References 75 publications

Criterion for Many-Body Localization-Delocalization Phase Transition

Criterion for Many-Body Localization-Delocalization Phase Transition

Efficient Representation of Fully Many-Body Localized Systems Using Tensor Networks

Remanent Magnetization: Signature of Many-Body Localization in Quantum Antiferromagnets

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