Spectral statistics of a pseudo-integrable map: the general case

Bogomolny, E.; Dubertrand, Rémy; Schmit, C.

doi:10.1088/0951-7715/22/9/003

Cited by 13 publications

(31 citation statements)

References 24 publications

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“…Whether the high energy entanglement spectrum deep in the MBL phase retains a memory of any other properties of the critical point (such as other critical exponents), and if so how this information could be extracted, is an intriguing question that we leave to future work. We note that semi-Poisson statistics are also a diagnostic of pseudo-integrability [32,33], and our results thus suggest that the entanglement Hamiltonian in the many body localized phase should be pseudointegrable (i.e. not integrable but also not chaotic), an observation that may open new lines of attack on the localized phase.…”

Section: Discussionmentioning

confidence: 62%

“…We note that semi-Poisson statistics are a known diagnostic of pseudo-integrability [32,33], and our results this suggest that while the entanglement Hamiltonian of a non-interacting Anderson insulator will be integrable, the entanglement Hamiltonian of a many body localized system will be only pseudo-integrable (i.e. not chaotic but also not integrable).…”

Section: Introductionmentioning

confidence: 60%

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Many-body localization and thermalization: Insights from the entanglement spectrum

Geraedts¹,

Nandkishore²,

Regnault³

2016

Phys. Rev. B

137

125

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We study the entanglement spectrum in the many body localizing and thermalizing phases of one and two dimensional Hamiltonian systems, and periodically driven 'Floquet' systems. We focus on the level statistics of the entanglement spectrum as obtained through numerical diagonalization, finding structure beyond that revealed by more limited measures such as entanglement entropy. In the thermalizing phase the entanglement spectrum obeys level statistics governed by an appropriate random matrix ensemble. For Hamiltonian systems this can be viewed as evidence in favor of a strong version of the eigenstate thermalization hypothesis (ETH). Similar results are also obtained for Floquet systems, where they constitute a result 'beyond ETH', and show that the corrections to ETH governing the Floquet entanglement spectrum have statistical properties governed by a random matrix ensemble. The particular random matrix ensemble governing the Floquet entanglement spectrum depends on the symmetries of the Floquet drive, and therefore can depend on the choice of origin of time. In the many body localized phase the entanglement spectrum is also found to show level repulsion, following a semi-Poisson distribution (in contrast to the energy spectrum, which follows a Poisson distribution). This semi-Poisson distribution is found to come mainly from states at high entanglement energies. The observed level repulsion only occurs for interacting localized phases. We also demonstrate that equivalent results can be obtained by calculating with a single typical eigenstate, or by averaging over a microcanonical energy window -a surprising result in the localized phase. This discovery of new structure in the pattern of entanglement of localized and thermalizing phases may open up new lines of attack on many body localization, thermalization, and the localization transition.

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Section: Discussionmentioning

confidence: 62%

Section: Introductionmentioning

confidence: 60%

Many-body localization and thermalization: Insights from the entanglement spectrum

Geraedts¹,

Nandkishore²,

Regnault³

2016

Phys. Rev. B

137

125

View full text Add to dashboard Cite

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“…This ensemble is identical to RS ensemble, but with a constant g = aN which now depends on the size of the matrix. Spectral statistics of this model turn out to be very different from those of the RS ensemble [20]. For rational a = m/b it displays intermediate spectral properties which depend on the remainder of mN modulo b.…”

Section: Intermediate Mapmentioning

confidence: 83%

“…using the result of (20). Putting together equations (9), (19), (31) and (38), the first-order correction to moments I q is…”

Section: First-order Termmentioning

confidence: 99%

“…These quantum maps have exactly the form ( 6), but with g now depending on the matrix size, namely g = aN with a fixed. Spectral statistics of these ensembles have been shown to be of intermediate type [20]. We will also consider the well-studied CrBRM ensemble, which is the set of N ×N symmetric or Hermitian matrices whose elements M mn are independently distributed Gaussian variables with zero mean and variance…”

Section: The Modelsmentioning

confidence: 99%

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Multifractal dimensions for all moments for certain critical random-matrix ensembles in the strong multifractality regime

Bogomolny

Giraud²

2012

Phys. Rev. E

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We construct perturbation series for the qth moment of eigenfunctions of various critical random-matrix ensembles in the strong multifractality regime close to localization. Contrary to previous investigations, our results are valid in the region q<1/2. Our findings allow one to verify, at first leading orders in the strong multifractality limit, the symmetry relation for anomalous fractal dimensions Δ(q)=Δ(1-q), recently conjectured for critical models where an analog of the metal-insulator transition takes place. It is known that this relation is verified at leading order in the weak multifractality regime. Our results thus indicate that this symmetry holds in both limits of small and large coupling constant. For general values of the coupling constant we present careful numerical verifications of this symmetry relation for different critical random-matrix ensembles. We also present an example of a system closely related to one of these critical ensembles, but where the symmetry relation, at least numerically, is not fulfilled.

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Integrable random matrix ensembles

Bogomolny¹,

Giraud²,

Schmit³

2011

Nonlinearity

Self Cite

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We propose new classes of random matrix ensembles whose statistical properties are intermediate between statistics of Wigner-Dyson random matrices and Poisson statistics. The construction is based on integrable N -body classical systems with a random distribution of momenta and coordinates of the particles. The Lax matrices of these systems yield random matrix ensembles whose joint distribution of eigenvalues can be calculated analytically thanks to integrability of the underlying system. Formulas for spacing distributions and level compressibility are obtained for various instances of such ensembles.

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Spectral statistics of a pseudo-integrable map: the general case

Cited by 13 publications

References 24 publications

Many-body localization and thermalization: Insights from the entanglement spectrum

Many-body localization and thermalization: Insights from the entanglement spectrum

Multifractal dimensions for all moments for certain critical random-matrix ensembles in the strong multifractality regime

Integrable random matrix ensembles

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