Finite size scaling in crossover among different random matrix ensembles in microscopic lattice models

Modak, Ranjan; Mukerjee, Subroto

doi:10.1088/1367-2630/16/9/093016

Cited by 40 publications

(38 citation statements)

References 36 publications

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“…Only in the case of a linear variation of the covariance with Hamming distance the level spacing statistics show Poissonian statistics implying nonthermal behavior in the system. However that can imply either emergence of MBL phase which is believed to have emergent conservation laws 6,[19][20][21] or conventional integrability of integrable system with dynamical symmetries that inhibit ergodicity [22][23][24][25] . After the introduction of additional interactions which can break the integrability of the system, the system shows a thermal to MBL transition as a function of the slope of the linear variation of covariance ( Fig.…”

Section: Discussionmentioning

confidence: 99%

Many-body localization due to correlated disorder in Fock space

et al. 2019

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In presence of strong enough disorder one dimensional systems of interacting spinless fermions at non-zero filling factor are known to be in a many body localized phase. When represented in 'Fock space', the Hamiltonian of such a system looks like that of a single 'particle' hopping on a Fock lattice in the presence of a random disordered potential. The coordination number of the Fock lattice increases linearly with the system size L in one dimension. Thus in the thermodynamic limit L → ∞, the disordered interacting problem in one dimension maps on to an Anderson model with infinite coordination number. Despite this, this system displays localization which appears counterintuitive. A close observation of the on-site disorder potentials on the Fock lattice reveals a large degree of correlation among them as they are derived from an exponentially smaller number of on-site disorder potentials in real space. This indicates that the correlations between the on-site disorder potentials on a Fock lattice has a strong effect on the localization properties of the corresponding many-body system. This intuition is also consistent with studies of quantum random energy model where the typical mid-spectrum states are ergodic and the on-site potentials in Fock space are completely uncorrelated. In this work we perform a systematic quantitative exploration of the nature of correlations of the Fock space potential required for localization. We study different functional variations of the disorder correlation in Fock lattice by analyzing the eigenspectrum obtained through exact diagonalization. Without changing the typical strength of the on-site disorder potential in Fock lattice we show that changing the correlation strength can induce thermalization or localization in systems. From among the various forms of correlations we study, we find that only the linear variation of correlations with Hamming distance in Fock space is able to drive a thermal-MBL phase transition where the transition is driven by the correlation strength. Systems with the other forms of correlations we study are found to be ergodic.

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Section: Discussionmentioning

confidence: 99%

Many-body localization due to correlated disorder in Fock space

et al. 2019

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“…The H 1 term breaks the integrability. Physical arguments as well as numerics strongly suggest that when H 1 is local, the system will show a cross-over behavior from an integrable regime to a chaotic regime for ∼ 1/L β [16][17][18][19]22,23 . In fact, following arguments similar to Ref.…”

Section: The Nature Of Chaotic Eigenstatesmentioning

confidence: 99%

Renyi entropy of chaotic eigenstates

Grover

2019

Phys. Rev. E

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Using arguments built on ergodicity, we derive an analytical expression for the Renyi entanglement entropies corresponding to the finite-energy density eigenstates of chaotic many-body Hamiltonians. The expression is a universal function of the density of states and is valid even when the subsystem is a finite fraction of the total system -a regime in which the reduced density matrix is not thermal. We find that in the thermodynamic limit, only the von Neumann entropy density is independent of the subsystem to the total system ratio V A /V, while the Renyi entropy densities depend non-linearly on V A /V. Surprisingly, Renyi entropies S n for n > 1 are convex functions of the subsystem size, with a volume law coefficient that depends on V A /V, and exceeds that of a thermal mixed state at the same energy density. We provide two different arguments to support our results: the first one relies on a many-body version of Berry's formula for chaotic quantum mechanical systems, and is closely related to eigenstate thermalization hypothesis. The second argument relies on the assumption that for a fixed energy in a subsystem, all states in its complement allowed by the energy conservation are equally likely. We perform Exact Diagonalization study on quantum spin-chain Hamiltonians to test our analytical predictions, and find good agreement.

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“…They have been used to study the ETH-MBL transition in finite size numerical simulations, in particular for an extensive analysis of the Heisenberg model in a random field. These characterizing quantities include energy level statistics [30][31][32][33][34][35], level statistics [25,36] as well as density of states [37] analyses of the entanglement spectrum and studies of the distribution of the entanglement entropy over a region of energy eigenstates [18,[38][39][40][41][42][43]. Necessarily, these methods rely on a physical understanding of the nature of either regime or of the transition.…”

Section: Introductionmentioning

confidence: 99%

Probing many-body localization with neural networks

2017

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We show that a simple artificial neural network trained on entanglement spectra of individual states of a many-body quantum system can be used to determine the transition between a many-body localized and a thermalizing regime. Specifically, we study the Heisenberg spin-1/2 chain in a random external field. We employ a multilayer perceptron with a single hidden layer, which is trained on labeled entanglement spectra pertaining to the fully localized and fully thermal regimes. We then apply this network to classify spectra belonging to states in the transition region. For training, we use a cost function that contains, in addition to the usual error and regularization parts, a term that favors a confident classification of the transition region states. The resulting phase diagram is in good agreement with the one obtained by more conventional methods and can be computed for small systems. In particular, the neural network outperforms conventional methods in classifying individual eigenstates pertaining to a single disorder realization. It allows us to map out the structure of these eigenstates across the transition with spatial resolution. Furthermore, we analyze the network operation using the dreaming technique to show that the neural network correctly learns by itself the power-law structure of the entanglement spectra in the many-body localized regime.

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Finite size scaling in crossover among different random matrix ensembles in microscopic lattice models

Cited by 40 publications

References 36 publications

Many-body localization due to correlated disorder in Fock space

Many-body localization due to correlated disorder in Fock space

Renyi entropy of chaotic eigenstates

Probing many-body localization with neural networks

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