Beyond the Heisenberg time: semiclassical treatment of spectral correlations in chaotic systems with spin 1/2

Braun, P. A.

doi:10.1088/1751-8113/45/4/045102

Cited by 4 publications

(11 citation statements)

References 31 publications

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“…Again, for the unitary symmetry class the diagonal approximation yields the exact finite-N CUE generating function and, with N → ∞, the exact GUE result. We recover the previously obtained pseudo-orbit expansions [1][2][3] for higher-order corrections in the limit N → ∞, for all Wigner-Dyson symmetry classes.…”

Section: Summary and Discussionsupporting

confidence: 76%

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Chaotic maps and flows: exact Riemann–Siegel lookalike for spectral fluctuations

Braun¹,

Haake²

2012

J. Phys. A: Math. Theor.

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Abstract. To treat the spectral statistics of quantum maps and flows that are fully chaotic classically, we use the rigorous Riemann-Siegel lookalike available for the spectral determinant of unitary time evolution operators F . Concentrating on dynamics without time reversal invariance we get the exact two-point correlator of the spectral density for finite dimension N of the matrix representative of F , as phenomenologically given by random matrix theory. In the limit N → ∞ the correlator of the Gaussian unitary ensemble is recovered. Previously conjectured cancelations of contributions of pseudo-orbits with periods beyond half the Heisenberg time are shown to be implied by the Riemann-Siegel lookalike.

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Section: Summary and Discussionsupporting

confidence: 76%

“…The phenomenological description previously given by random-matrix theory has been fully recovered for individual (rather than ensemble averages of) systems from the unitary, orthogonal [1,2], and symplectic [3] symmetry classes.…”

Section: Introductionmentioning

confidence: 99%

Chaotic maps and flows: exact Riemann–Siegel lookalike for spectral fluctuations

Braun¹,

Haake²

2012

J. Phys. A: Math. Theor.

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“…Considering the exponential factors in Z ζ the sum Z ∼ Z I + Z II (15) will smoothly interpolate between Z I and Z II as we change Im e C,D from positive to negative values, providing thus the high-energy asymptotics of the generating function for real e C,D . It coincides with the result obtained differently in [34].…”

Section: The Generating Functionmentioning

confidence: 99%

“…Its understanding took many investigations, to mention but a few relevant here: the diagonal approximation [6], a heuristic derivation of the oscillatory part of the correlation function for the unitary unversality class [7], the orbit partnership role in spectral correlations [8], the semiclassical treatment of spin in the level statistics [9,10,11], the periodic orbit theory of the small time form factor [12]. The complete semiclassical derivation of the two-point correlation function for the unitary and orthogonal universality classes is given in [13,14] and for the symplectic class in [15]. Three-and higher-level correlation functions are considered in [16].…”

Section: Introductionmentioning

confidence: 99%

New approach to periodic orbit theory of spectral correlations

Braun¹,

Waltner²

2019

J. Phys. A: Math. Theor.

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The existing periodic orbit theory of spectral correlations for classically chaotic systems relies on the Riemann-Siegel-like representation of the spectral determinants which is still largely hypothetical. We suggest a simpler derivation using analytic continuation of the periodic-orbit expansion of the pertinent generating function from the convergence border to physically important real values of its arguments. As examples we consider chaotic systems without time reversal as well as the Riemann zeta function and Dirichlet L-functions zeros.New approach to periodic orbit theory of spectral correlations

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“…For such systems GSE statistics have been predicted and/or observed numerically in examples such as quantum billiards [7], maps [8] and quantum graphs [9], and explained using periodic-orbit theory [10,11]. However to date there has been no experimental observation.…”

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confidence: 99%

GSE statistics without spin

Joyner

Müller

Sieber

2014

EPL

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Energy levels statistics following the Gaussian Symplectic Ensemble (GSE) of Random Matrix Theory have been predicted theoretically and observed numerically in numerous quantum chaotic systems. However in all these systems there has been one unifying feature: the combination of halfinteger spin and time-reversal invariance. Here we provide an alternative mechanism for obtaining GSE statistics that is based on geometric symmetries of a quantum system which alleviates the need for spin. As an example, we construct a quantum graph with a particular discrete symmetry given by the quaternion group Q8. GSE statistics is then observed within one of its subspectra.In the 1950s and 1960s Wigner and Dyson pioneered the use of random matrices in modelling the statistical properties of the energy eigenvalues belonging to complicated quantum systems [1,2]. The techniques they developed spawned a new field of mathematics which has since become known as Random Matrix Theory (RMT) and its application has spread far and wide to many areas of Mathematics and Physics [3]. In particular it was later conjectured [4] that the high-lying quantum energy levels of classically chaotic systems are faithful to random matrix averages.One of the cornerstones of RMT is Dyson's three-fold way [2], which groups quantum systems without geometric symmetries into three distinct types. The first occurs if time-reversal invariance is broken, for example by a magnetic field, meaning the quantum Hamiltonian H is inherently complex. The remaining two appear if there is an antiunitary time-reversal operator T which leaves H invariant, i.e. [T , H] = 0. They are then distinguished by either T 2 = 1 or T 2 = −1, in which case H is real symmetric or quaternion-real respectively. For chaotic systems, RMT makes predictions in all three instances by averaging over an ensemble of Hermitian matrices with the appropriate internal structure and Gaussian weighted elements. These are referred to as the Gaussian Unitary, Orthogonal and Symplectic ensembles (GUE, GOE and GSE). We note that the number of symmetry classes can be extended to ten if additional anti-commuting symmetries are present [5,6] but this is beyond the scope of this letter.In systems without geometrical symmetries timereversal invariance with T 2 = −1, and hence GSE statistics, can only arise if the wavefunctions have an even number of components, commonly associated with halfinteger spin. For such systems GSE statistics have been predicted and/or observed numerically in examples such as quantum billiards [7], maps [8] and quantum graphs [9], and explained using periodic-orbit theory [10,11]. However to date there has been no experimental observation.For systems with geometric symmetries the situation becomes more involved. Here the Hilbert space decomposes into subspaces invariant under symmetry transformations, and the spectral statistics inside these subspaces depends both on the system's behaviour under time reversal and on the nature of the subspace. For example 3-fold rotationally invariant...

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Beyond the Heisenberg time: semiclassical treatment of spectral correlations in chaotic systems with spin 1/2

Cited by 4 publications

References 31 publications

Chaotic maps and flows: exact Riemann–Siegel lookalike for spectral fluctuations

Chaotic maps and flows: exact Riemann–Siegel lookalike for spectral fluctuations

New approach to periodic orbit theory of spectral correlations

GSE statistics without spin

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