The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Uzy Smilansky - One of the best experts on this subject based on the ideXlab platform.
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Periodic Orbit theory of anderson localization on graphs
Physical Review Letters, 2000Co-Authors: Holger Schanz, Uzy SmilanskyAbstract:We present the first quantum system where Anderson localization is completely described within Periodic-Orbit theory. The model is a quantum graph analogous to an aPeriodic Kronig-Penney model in one dimension. The exact expression for the probability to return to an initially localized state is computed in terms of classical trajectories. It saturates to a finite value due to localization, while the diagonal approximation decays diffusively. Our theory is based on the identification of families of isometric Orbits. The coherent Periodic-Orbit sums within these families, and the summation over all families, are performed analytically using advanced combinatorial methods.
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Periodic Orbit theory and spectral statistics for quantum graphs
Annals of Physics, 1999Co-Authors: Tsampikos Kottos, Uzy SmilanskyAbstract:Abstract We quantize graphs (networks) which consist of a finite number of bonds and vertices. We show that the spectral statistics of fully connected graphs is well reproduced by random matrix theory. We also define a classical phase space for the graphs, where the dynamics is mixing and the Periodic Orbits proliferate exponentially. An exact trace formula for the quantum spectrum is developed in terms of the same Periodic Orbits, and it is used to investigate the origin of the connection between random matrix theory and the underlying chaotic classical dynamics. Being an exact theory, and due to its relative simplicity, it offers new insights into this problem which is at the forefront of the research in quantum chaos and related fields.
Petr Braun - One of the best experts on this subject based on the ideXlab platform.
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Periodic Orbit theory of universal level correlations in quantum chaos
arXiv: Chaotic Dynamics, 2009Co-Authors: Sebastian Muller, Stefan Heusler, Alexander Altland, Petr Braun, Fritz HaakeAbstract:Using Gutzwiller's semiclassical Periodic-Orbit theory we demonstrate universal behaviour of the two-point correlator of the density of levels for quantum systems whose classical limit is fully chaotic. We go beyond previous work in establishing the full correlator such that its Fourier transform, the spectral form factor, is determined for all times, below and above the Heisenberg time. We cover dynamics with and without time reversal invariance (from the orthogonal and unitary symmetry classes). A key step in our reasoning is to sum the Periodic-Orbit expansion in terms of a matrix integral, like the one known from the sigma model of random-matrix theory.
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Periodic Orbit theory of level correlations
Physical Review Letters, 2007Co-Authors: Stefan Heusler, Sebastian Muller, Alexander Altland, Petr Braun, Fritz HaakeAbstract:We present a semiclassical explanation of the so-called Bohigas-Giannoni-Schmit conjecture which asserts universality of spectral fluctuations in chaotic dynamics. We work with a generating function whose semiclassical limit is determined by quadruplets of sets of Periodic Orbits. The asymptotic expansions of both the nonoscillatory and the oscillatory part of the universal spectral correlator are obtained. Borel summation of the series reproduces the exact correlator of random-matrix theory.
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Periodic Orbit theory of universality in quantum chaos
Physical Review E, 2005Co-Authors: Sebastian Muller, Stefan Heusler, Petr Braun, Fritz Haake, Alexander AltlandAbstract: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 $K(\ensuremath{\tau})$ as power series in the time $\ensuremath{\tau}$. Each term ${\ensuremath{\tau}}^{n}$ 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 nontrivial properties of permutations. We show our series to be equivalent to perturbative implementations of the nonlinear $\ensuremath{\sigma}$ models for the Wigner-Dyson ensembles of random matrices and for disordered systems; our families of Orbit pairs have a one-to-one relationship with Feynman diagrams known from the $\ensuremath{\sigma}$ model.
Xiao-song Yang - One of the best experts on this subject based on the ideXlab platform.
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Singular cycles connecting saddle Periodic Orbit and saddle equilibrium in piecewise smooth systems
Nonlinear Dynamics, 2019Co-Authors: Lei Wang, Xiao-song YangAbstract:For flows, the singular cycles connecting saddle Periodic Orbit and saddle equilibrium can potentially result in the so-called singular horseshoe, which means the existence of a non-uniformly hyperbolic chaotic invariant set. However, it is very hard to find a specific dynamical system that exhibits such singular cycles in general. In this paper, the existence of the singular cycles involving saddle Periodic Orbits is studied by two types of piecewise smooth systems: One is the piecewise smooth systems having an admissible saddle point with only real eigenvalues and an admissible saddle Periodic Orbit, and the other is the piecewise smooth systems having an admissible saddle-focus and an admissible saddle Periodic Orbit. Several kinds of sufficient conditions are obtained for the existence of only one heteroclinic cycle or only two heteroclinic cycles in the two types of piecewise smooth systems, respectively. In addition, some examples are presented to illustrate the results.
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Singular cycles connecting saddle Periodic Orbit and saddle equilibrium in piecewise smooth systems
arXiv: Dynamical Systems, 2018Co-Authors: Lei Wang, Xiao-song YangAbstract:For flows, the singular cycles connecting saddle Periodic Orbit and saddle equilibrium can poten- tially result in the so-called singular horseshoe, which means the existence of a non-uniformly hyperbolic chaotic invariant set. However, it is very hard to find a specific dynamical system that exhibits such singular cycles in general. In this paper, the existence of the singular cycles involved in saddle Periodic Orbits is studied by two types of piecewise affine systems: one is the piecewise affine system having an admissible saddle point with only real eigenvalues and an admissible saddle Periodic Orbit, and the other is the piecewise affine system having an admissible saddle- focus and an admissible saddle Periodic Orbit. Precisely, several kinds of sufficient conditions are obtained for the existence of only one heteroclinic cycle or only two heteroclinic cycles in the two types of piecewise affine systems, respectively. In addition, some examples are presented to illustrate the results.
Sebastian Muller - One of the best experts on this subject based on the ideXlab platform.
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Periodic Orbit theory of universal level correlations in quantum chaos
arXiv: Chaotic Dynamics, 2009Co-Authors: Sebastian Muller, Stefan Heusler, Alexander Altland, Petr Braun, Fritz HaakeAbstract:Using Gutzwiller's semiclassical Periodic-Orbit theory we demonstrate universal behaviour of the two-point correlator of the density of levels for quantum systems whose classical limit is fully chaotic. We go beyond previous work in establishing the full correlator such that its Fourier transform, the spectral form factor, is determined for all times, below and above the Heisenberg time. We cover dynamics with and without time reversal invariance (from the orthogonal and unitary symmetry classes). A key step in our reasoning is to sum the Periodic-Orbit expansion in terms of a matrix integral, like the one known from the sigma model of random-matrix theory.
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Periodic Orbit theory of level correlations
Physical Review Letters, 2007Co-Authors: Stefan Heusler, Sebastian Muller, Alexander Altland, Petr Braun, Fritz HaakeAbstract:We present a semiclassical explanation of the so-called Bohigas-Giannoni-Schmit conjecture which asserts universality of spectral fluctuations in chaotic dynamics. We work with a generating function whose semiclassical limit is determined by quadruplets of sets of Periodic Orbits. The asymptotic expansions of both the nonoscillatory and the oscillatory part of the universal spectral correlator are obtained. Borel summation of the series reproduces the exact correlator of random-matrix theory.
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Periodic Orbit approach to universality in quantum chaos
arXiv: Chaotic Dynamics, 2005Co-Authors: Sebastian MullerAbstract:We show that in the semiclassical limit, classically chaotic systems have universal spectral statistics. Concentrating on short-time statistics, we identify the pairs of classical Periodic Orbits determining the small-$\tau$ behavior of the spectral form factor $K(\tau)$ of fully chaotic systems. The two Orbits within each pair differ only by their connections inside close self-encounters in phase space. The frequency of occurrence of these self-encounters is determined by ergodicity. Permutation theory is used to systematically sum over all topologically different families of such Orbit pairs. The resulting expansions of the form factor in powers of $\tau$ coincide with the predictions of random-matrix theory, both for systems with and without time-reversal invariance, and to all orders in $\tau$. Our results are closely related to the zero-dimensional nonlinear $\sigma$ model of quantum field theory. The relevant families of Orbit pairs are in one-to-one correspondence to Feynman diagrams appearing in the perturbative treatment of the $\sigma$ model.
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Periodic Orbit theory of universality in quantum chaos
Physical Review E, 2005Co-Authors: Sebastian Muller, Stefan Heusler, Petr Braun, Fritz Haake, Alexander AltlandAbstract: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 $K(\ensuremath{\tau})$ as power series in the time $\ensuremath{\tau}$. Each term ${\ensuremath{\tau}}^{n}$ 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 nontrivial properties of permutations. We show our series to be equivalent to perturbative implementations of the nonlinear $\ensuremath{\sigma}$ models for the Wigner-Dyson ensembles of random matrices and for disordered systems; our families of Orbit pairs have a one-to-one relationship with Feynman diagrams known from the $\ensuremath{\sigma}$ model.
Fritz Haake - One of the best experts on this subject based on the ideXlab platform.
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Periodic Orbit theory of universal level correlations in quantum chaos
arXiv: Chaotic Dynamics, 2009Co-Authors: Sebastian Muller, Stefan Heusler, Alexander Altland, Petr Braun, Fritz HaakeAbstract:Using Gutzwiller's semiclassical Periodic-Orbit theory we demonstrate universal behaviour of the two-point correlator of the density of levels for quantum systems whose classical limit is fully chaotic. We go beyond previous work in establishing the full correlator such that its Fourier transform, the spectral form factor, is determined for all times, below and above the Heisenberg time. We cover dynamics with and without time reversal invariance (from the orthogonal and unitary symmetry classes). A key step in our reasoning is to sum the Periodic-Orbit expansion in terms of a matrix integral, like the one known from the sigma model of random-matrix theory.
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Periodic Orbit theory of level correlations
Physical Review Letters, 2007Co-Authors: Stefan Heusler, Sebastian Muller, Alexander Altland, Petr Braun, Fritz HaakeAbstract:We present a semiclassical explanation of the so-called Bohigas-Giannoni-Schmit conjecture which asserts universality of spectral fluctuations in chaotic dynamics. We work with a generating function whose semiclassical limit is determined by quadruplets of sets of Periodic Orbits. The asymptotic expansions of both the nonoscillatory and the oscillatory part of the universal spectral correlator are obtained. Borel summation of the series reproduces the exact correlator of random-matrix theory.
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Periodic Orbit theory of universality in quantum chaos
Physical Review E, 2005Co-Authors: Sebastian Muller, Stefan Heusler, Petr Braun, Fritz Haake, Alexander AltlandAbstract: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 $K(\ensuremath{\tau})$ as power series in the time $\ensuremath{\tau}$. Each term ${\ensuremath{\tau}}^{n}$ 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 nontrivial properties of permutations. We show our series to be equivalent to perturbative implementations of the nonlinear $\ensuremath{\sigma}$ models for the Wigner-Dyson ensembles of random matrices and for disordered systems; our families of Orbit pairs have a one-to-one relationship with Feynman diagrams known from the $\ensuremath{\sigma}$ model.
