Action correlations and random matrix theory

Smilansky, Uzy; Verdene, Basile

doi:10.1088/0305-4470/36/12/338

Cited by 18 publications

(26 citation statements)

References 27 publications

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“…The transfer matrix method is a standard tool in statistical mechanics for the calculation of the partition function of Ising lattices [37][38][39]. Due to the analogy between Ising partition functions (for imaginary temperature) and semiclassical periodic orbit sums, the method has also found applications in the quantum chaos domain [40][41][42][43].…”

Section: The Random-perturbation Limitmentioning

confidence: 99%

Semiclassical approach to fidelity amplitude

García-Mata¹,

Vallejos²,

Wisniacki³

2011

New J. Phys.

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The elusive nature of the Lyapunov regime Ignacio García-Mata and Diego A Wisniacki -Loschmidt echo for local perturbations: non-monotonic cross-over from the Fermigolden-rule to the escape-rate regime Arseni Goussev, Daniel Waltner, Klaus Richter et al. and Diego A Wisniacki Abstract. The fidelity amplitude (FA) is a quantity of paramount importance in echo-type experiments. We use semiclassical theory to study the average FA for quantum chaotic systems under external perturbation. We explain analytically two extreme cases: the random dynamics limit-attained approximately by strongly chaotic systems-and the random perturbation limit, which shows a Lyapunov decay. Numerical simulations help us to bridge the gap between both the extreme cases.

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Section: The Random-perturbation Limitmentioning

confidence: 99%

Semiclassical approach to fidelity amplitude

García-Mata¹,

Vallejos²,

Wisniacki³

2011

New J. Phys.

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“…The first periodicorbit approach to K(τ ) was taken by Berry [5], who derived the leading term in (3) using "diagonal" pairs of coinciding (γ ′ = γ) and, for time T -invariant dynamics, mutually time-reversed (γ ′ = T γ) orbits, which obviously are identical in action. Starting with Argaman et al [6], off-diagonal orbit pairs were studied in [9,10,11]. The potential importance of close self-encounters in orbit pairs was first spelled out in work on electronic transport [12] and qualitatively discussed for spectral fluctuations in [13].…”

Section: Introductionmentioning

confidence: 99%

Periodic-orbit theory of universality in quantum chaos

et al. 2005

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We argue semiclassically, on the basis of Gutzwiller's periodic-orbit theory, that full classical chaos is paralleled by quantum energy spectra with universal spectral statistics, in agreement with random-matrix theory. For dynamics from all three Wigner-Dyson symmetry classes, we calculate the small-time spectral form factor K(τ ) as power series in the time τ . Each term τ n of that series is provided by specific families of pairs of periodic orbits. The contributing pairs are classified in terms of close self-encounters in phase space. The frequency of occurrence of self-encounters is calculated by invoking ergodicity. Combinatorial rules for building pairs involve non-trivial properties of permutations. We show our series to be equivalent to perturbative implementations of the non-linear sigma models for the Wigner-Dyson ensembles of random matrices and for disordered systems; our families of orbit pairs are one-to-one with Feynman diagrams known from the sigma model.

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“…In spite of their success and significance these studies (and also random-matrix theory) cannot explain the deep quantum-to-classical correspondence between spectral correlations in a Hamiltonian systems and the dynamic properties of the underlying classical flow. This correspondence is the main focus in quantum chaos where semiclassical periodic-orbit theory has been used [21,50,52,55,73,228]. We will not go into the details of the semiclassical periodic orbit approach here but only summarise qualitatively some of the main results and ideas.…”

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confidence: 99%

Quantum graphs: Applications to quantum chaos and universal spectral statistics

Gnutzmann

Smilansky

2006

Advances in Physics

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450

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During the last years quantum graphs have become a paradigm of quantum chaos with applications from spectral statistics to chaotic scattering and wave function statistics. In the first part of this review we give a detailed introduction to the spectral theory of quantum graphs and discuss exact trace formulae for the spectrum and the quantum-to-classical correspondence. The second part of this review is devoted to the spectral statistics of quantum graphs as an application to quantum chaos. Especially, we summarise recent developments on the spectral statistics of generic large quantum graphs based on two approaches: the periodic-orbit approach and the supersymmetry approach. The latter provides a condition and a proof for universal spectral statistics as predicted by random-matrix theory. Chapter 1 IntroductionThe general mathematical concept of a graph (network) as a set of elements which are connected by some relation has found applications in many branches of science, engineering, and also social science. A street network of a traffic engineer, the network of neurons studied by a neuroscientist, the structure of databases in computer science can all be described by graphs.Recently the Laplacian on a metric graph has gained a lot of attention in physics and mathematics in terms of the diffusion equation or Schrödinger equation. They have now become known as quantum graphs but different aspects are studied under various names such as quantum networks or quantum wires. They have a long history in mathematics and physics.In physics, the first application has probably been in the context of free electron models for organic molecules about seventy years ago by Pauling [198], an approach which has been further developed in subsequent years [172,173,201,209,77,182,204]. Quantum graphs have also been applied successfully to superconductivity in granular and artificial materials [12], acoustic and electromagnetic waveguide networks [112,181], the Anderson transition in a disordered wire [18,222], quantum Hall systems [71,151,150], fracton excitations in fractal structures [22,190], and mesoscopic quantum systems [142,166,242,243,241]. Quantum graphs have also been simulated experimentally [141].The construction of self-adjoint operators, or wave equations with appropriate boundary conditions on graphs has first been addressed by Ruedenberg and Scherr [209] (see also [204]). They considered graphs as an idealisation of networks of wires or wave guides of finite cross-section in the limit where the diameter of the wire is much smaller than any other length scale. Similar approaches to graphs as networks of thin wires with a finite diameter, or fat quantum graphs as they are now called, have been a topic in mathematical physics recently [211,212,208,167,106,202].Another interesting aproach which has been discussed mainly by Exner and his coworkers is based on leaky graphs [101,100,102,103,105,104 , 108, 109]. Here, a finite attractive potential in the Schrödinger equation is centered on a metric graph. A leaky g...

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Action correlations and random matrix theory

Cited by 18 publications

References 27 publications

Semiclassical approach to fidelity amplitude

Semiclassical approach to fidelity amplitude

Periodic-orbit theory of universality in quantum chaos

Quantum graphs: Applications to quantum chaos and universal spectral statistics

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