Orbit bifurcations and spectral statistics

Berry, Michael; Keating, Jon P; Prado, Sandra Denise

doi:10.1088/0305-4470/31/13/001

Cited by 33 publications

(49 citation statements)

References 28 publications

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“…This goes far beyond the already established fact that bifurcations contribute to the spectral statistics when regular and chaotic orbits coexist (Berry et al 1998), because these new semiclassical ®uctuation phenomena involve many competing bifurcations, not just one.…”

Section: Discussionmentioning

confidence: 94%

Universal twinkling exponents for spectral fluctuations associated with mixed chaology

Berry

Keating

Schomerus

2000

Proc. R. Soc. Lond. A

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For systems that are neither fully integrable nor fully chaotic, bifurcations of periodic orbits give rise to semiclassically emergent singularities in the ®uctuating part N ® of the energy-level counting function. The bifurcations dominate the spectral momentsin the limit~! 0. We argue that M m (~) ¹ const:=~¸m , and calculate the twinkling exponents¸m as the result of a competition between bifurcations with di¬erent codimensions and repetition numbers.

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Section: Discussionmentioning

confidence: 94%

Universal twinkling exponents for spectral fluctuations associated with mixed chaology

Berry

Keating

Schomerus

2000

Proc. R. Soc. Lond. A

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“…Similar results have been obtained for moments of spectral counting functions and wave functions in closed mixed systems. [24][25][26][27] In the following we briefly review the semiclassical theory of transport through antidot lattices and discuss modifications in the presence of bifurcations. We give an overview of different types of bifurcations and discuss their influence on moments of conductivity fluctuations.…”

Section: Introductionmentioning

confidence: 99%

Universal quantum signature of mixed dynamics in antidot lattices

2005

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We investigate phase coherent ballistic transport through antidot lattices in the generic case where the classical phase space has both regular and chaotic components. It is shown that the conductivity fluctuations have a non-Gaussian distribution, and that their moments have a power-law dependence on a semiclassical parameter, with fractional exponents. These exponents are obtained from bifurcating periodic orbits in the semiclassical approximation. They are universal in situations where sufficiently long orbits contribute.

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“…Such phenomena are characteristic of dynamics in systems with a mixed phase space, where both regular and chaotic motion occurs. The contribution from a periodic orbit to the Gutzwiller trace formula and to the Bogomolny scar formula diverges when the orbit bifurcates, hinting that bifurcations are likely to give rise to fundamentally new behaviour in quantum fluctuation statistics in the semiclassical limit.That individual orbit bifurcations can have an important and sometimes dominant influence on spectral statistics was first demonstrated by Berry et al (1998). Let N (E) denote the spectral counting function (i.e.…”

mentioning

confidence: 99%

“…That individual orbit bifurcations can have an important and sometimes dominant influence on spectral statistics was first demonstrated by Berry et al (1998). Let N (E) denote the spectral counting function (i.e.…”

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confidence: 99%

Orbit Bifurcations and Quantum Fluctuation Statistics

Keating

2003

Ann. Henri Poincaré

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Recent results concerning the semiclassical influence of bifurcating classical periodic orbits on quantum fluctuation statistics are reviewed. These point to the appearance of a new class of universal scaling exponents that characterize the divergence of spectral and wavefunction statistics in systems where chaotic and regular dynamics coexist (i.e. in systems with a mixed phase space) in the limit as → 0.According to a conjecture of Berry (Berry 1977), statistical fluctuations on the scale of the de Broglie wavelength in the quantum wavefunctions of classically chaotic systems can be modeled in the semiclassical limit by those of gaussian random functions (e.g. random superpositions of plane waves). This in turn prompts the conjecture that spectral statistics on the scale of the mean level spacing should, in generic chaotic systems, coincide with those of the eigenvalues of random matrices (Bohigas et al. 1983).On scales larger than a de Broglie wavelength, wavefunctions can be scarred by classical periodic orbits (Heller 1984, Kaplan 1999. According to a theory developed by Bogomolny (Bogomolny 1988) and Berry (Berry 1989), in completely chaotic systems with two degrees of freedom, where all of the periodic orbits are isolated and unstable, these scars have a width of the order of 1/2 , and, in the square of the modulus of the wavefunctions, an amplitude also of the order of 1/2 . It follows from the Gutzwiller trace formula (Gutzwiller 1971), which relates the quantum density of states of a system to a sum over its classical periodic orbits, that the periodic orbits also strongly influence spectral statistics on scales larger than the mean level spacing in the semiclassical limit (Berry 1985).The question arises as to what happens when, as one changes system parameters, periodic orbits bifurcate; that is when combinations of stable and unstable orbits collide and transmute, or annihilate. Such phenomena are characteristic of dynamics in systems with a mixed phase space, where both regular and chaotic motion occurs. The contribution from a periodic orbit to the Gutzwiller trace formula and to the Bogomolny scar formula diverges when the orbit bifurcates, hinting that bifurcations are likely to give rise to fundamentally new behaviour in quantum fluctuation statistics in the semiclassical limit.That individual orbit bifurcations can have an important and sometimes dominant influence on spectral statistics was first demonstrated by Berry et al. (1998). Let N (E) denote the spectral counting function (i.e. the number of energy levels E n < E) and letN (E) = N (E) denote the local average of N over an

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Orbit bifurcations and spectral statistics

Cited by 33 publications

References 28 publications

Universal twinkling exponents for spectral fluctuations associated with mixed chaology

Universal twinkling exponents for spectral fluctuations associated with mixed chaology

Universal quantum signature of mixed dynamics in antidot lattices

Orbit Bifurcations and Quantum Fluctuation Statistics

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