6 research outputs found
Beyond the Heisenberg time: Semiclassical treatment of spectral correlations in chaotic systems with spin 1/2
The two-point correlation function of chaotic systems with spin 1/2 is
evaluated using periodic orbits. The spectral form factor for all times thus
becomes accessible. Equivalence with the predictions of random matrix theory
for the Gaussian symplectic ensemble is demonstrated. A duality between the
underlying generating functions of the orthogonal and symplectic symmetry
classes is semiclassically established
Periodic-orbit theory of universal level correlations in quantum chaos
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.Comment: 44 pages, 11 figures, changed title; final version published in New
J. Phys. + additional appendices B-F not included in the journal versio
On determination of statistical properties of spectra from parametric level dynamics
We analyze an approach aiming at determining statistical properties of
spectra of time-periodic quantum chaotic system based on the parameter dynamics
of their quasienergies. In particular we show that application of the methods
of statistical physics, proposed previously in the literature, taking into
account appropriate integrals of motion of the parametric dynamics is fully
justified, even if the used integrals of motion do not determine the invariant
manifold in a unique way. The indetermination of the manifold is removed by
applying Dirac's theory of constrained Hamiltonian systems and imposing
appropriate primary, first-class constraints and a gauge transformation
generated by them in the standard way. The obtained results close the gap in
the whole reasoning aiming at understanding statistical properties of spectra
in terms of parametric dynamics.Comment: 9 pages without figure
Semiclassical Theory for Universality in Quantum Chaos with Symmetry Crossover
We address the quantum-classical correspondence for chaotic systems with a
crossover between symmetry classes. We consider the energy level statistics of
a classically chaotic system in a weak magnetic field. The generating function
of spectral correlations is calculated by using the semiclassical
periodic-orbit theory. An explicit calculation up to the second order,
including the non-oscillatory and oscillatory terms, agrees with the prediction
of random matrix theory. Formal expressions of the higher order terms are also
presented. The nonlinear sigma (NLS) model of random matrix theory, in the
variant of the Bosonic replica trick, is also analyzed for the crossover
between the Gaussian orthogonal ensemble and Gaussian unitary ensemble. The
diagrammatic expansion of the NLS model is interpreted in terms of the periodic
orbit theory.Comment: 25 pages, 4 figures, 1 tabl
Semiclassical approach to discrete symmetries in quantum chaos
We use semiclassical methods to evaluate the spectral two-point correlation
function of quantum chaotic systems with discrete geometrical symmetries. The
energy spectra of these systems can be divided into subspectra that are
associated to irreducible representations of the corresponding symmetry group.
We show that for (spinless) time reversal invariant systems the statistics
inside these subspectra depend on the type of irreducible representation. For
real representations the spectral statistics agree with those of the Gaussian
Orthogonal Ensemble (GOE) of Random Matrix Theory (RMT), whereas complex
representations correspond to the Gaussian Unitary Ensemble (GUE). For systems
without time reversal invariance all subspectra show GUE statistics. There are
no correlations between non-degenerate subspectra. Our techniques generalize
recent developments in the semiclassical approach to quantum chaos allowing one
to obtain full agreement with the two-point correlation function predicted by
RMT, including oscillatory contributions.Comment: 26 pages, 8 Figure