A review of sigma models for quantum chaotic dynamics

Altland, Alexander; Gnutzmann, Sven; Haake, Fritz; Micklitz, Tobias

doi:10.1088/0034-4885/78/8/086001

Cited by 20 publications

(24 citation statements)

References 48 publications

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“…If one wants to strictly confront individual chaotic systems, without any mild averaging over a small set of parameters whatsoever, the EFT can be stabilized by an average over energy [14][15][16]. A subsequent expansion in smooth field fluctuations then yields an action which, by design, disposes with the information on high frequency noise.…”

Section: Ensemble Vs Individual Systemmentioning

confidence: 99%

Late time physics of holographic quantum chaos

Altland

Sonner

2021

SciPost Phys.

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Quantum chaotic systems are often defined via the assertion that their spectral statistics coincides with, or is well approximated by, random matrix theory. In this paper we explain how the universal content of random matrix theory emerges as the consequence of a simple symmetry-breaking principle and its associated Goldstone modes. This allows us to write down an effective-field theory (EFT) description of quantum chaotic systems, which is able to control the level statistics up to an accuracy {O} \left(e^{-S} \right)O(e−S) with SS the entropy. We explain how the EFT description emerges from explicit ensembles, using the example of a matrix model with arbitrary invariant potential, but also when and how it applies to individual quantum systems, without reference to an ensemble. Within AdS/CFT this gives a general framework to express correlations between ``different universes’’ and we explicitly demonstrate the bulk realization of the EFT in minimal string theory where the Goldstone modes are bound states of strings stretching between bulk spectral branes. We discuss the construction of the EFT of quantum chaos also in higher dimensional field theories, as applicable for example for higher-dimensional AdS/CFT dual pairs.

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Section: Ensemble Vs Individual Systemmentioning

confidence: 99%

Late time physics of holographic quantum chaos

Altland

Sonner

2021

SciPost Phys.

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“…If η 1 then the latter correctly interpolates between the two analytically known expressions for the ground state |0 in the limit x ∼ 1 and |x| 1, resp. 71 . The excited states |k ≡ Φ k (z) of Ĥ with energies E k > 0 can be labeled by a set of quantum numbers k = (n, l, λ, λ), where n and l are integers and λ, λ are Grassmanns.…”

Section: Appendix B: Transfer Matrix Methodsmentioning

confidence: 99%

“…At the heart of the color-flavor transformation lies the identity 70,71 2π 0 dφ 0 2π e q (ψ T 1,q e iφ 0 ψ 2,q +ψ T 2,q e −iφ 0 ψ 1,q ) (A12)…”

Section: Appendix A: Effective Field Theorymentioning

confidence: 99%

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Probing the topological Anderson transition with quantum walks

Bagrets,

Kim,

Barkhofen

et al. 2021

Preprint

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We consider one-dimensional quantum walks in optical linear networks with synthetically introduced disorder and tunable system parameters allowing for the engineered realization of distinct topological phases. The option to directly monitor the walker's probability distribution makes this optical platform ideally suited for the experimental observation of the unique signatures of the onedimensional topological Anderson transition. We analytically calculate the probability distribution describing the quantum critical walk in terms of a (time staggered) spin polarization signal and propose a concrete experimental protocol for its measurement. Numerical simulations back the realizability of our blueprint with current date experimental hardware.

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“…We unfold the spectrum by expressing each eigenvalue in units of the mean level spacing e n = En ∆E . The average microscopic density of states is then defined by d(e) = n δ(e − e n ) (24) where · is an appropriate averaging procedure. In the standard Wigner-Dyson classes this procedure should lead to d(e) = 1 up to numerical noise.…”

Section: Quantum Chaos and Numerical Analysis Of Quantum Spectramentioning

confidence: 99%

Quantum chaos for nonstandard symmetry classes in the Feingold-Peres model of coupled tops

2017

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We consider two coupled quantum tops with angular momentum vectors L and M. The coupling Hamiltonian defines the Feingold-Peres model, which is a known paradigm of quantum chaos. We show that this model has a nonstandard symmetry with respect to the Altland-Zirnbauer tenfold symmetry classification of quantum systems, which extends the well-known threefold way of Wigner and Dyson (referred to as "standard" symmetry classes here). We identify the nonstandard symmetry classes BDI_{0} (chiral orthogonal class with no zero modes), BDI_{1} (chiral orthogonal class with one zero mode), and CI (antichiral orthogonal class) as well as the standard symmetry class AI (orthogonal class). We numerically analyze the specific spectral quantum signatures of chaos related to the nonstandard symmetries. In the microscopic density of states and in the distribution of the lowest positive energy eigenvalue, we show that the Feingold-Peres model follows the predictions of the Gaussian ensembles of random-matrix theory in the appropriate symmetry class if the corresponding classical dynamics is chaotic. In a crossover to mixed and near-integrable classical dynamics, we show that these signatures disappear or strongly change.

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A review of sigma models for quantum chaotic dynamics

Cited by 20 publications

References 48 publications

Late time physics of holographic quantum chaos

Late time physics of holographic quantum chaos

Probing the topological Anderson transition with quantum walks

Quantum chaos for nonstandard symmetry classes in the Feingold-Peres model of coupled tops

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