307 research outputs found
The spectral form factor is not self-averaging
The spectral form factor, k(t), is the Fourier transform of the two level
correlation function C(x), which is the averaged probability for finding two
energy levels spaced x mean level spacings apart. The average is over a piece
of the spectrum of width W in the neighborhood of energy E0. An additional
ensemble average is traditionally carried out, as in random matrix theory.
Recently a theoretical calculation of k(t) for a single system, with an energy
average only, found interesting nonuniversal semiclassical effects at times t
approximately unity in units of {Planck's constant) /(mean level spacing). This
is of great interest if k(t) is self-averaging, i.e, if the properties of a
typical member of the ensemble are the same as the ensemble average properties.
We here argue that this is not always the case, and that for many important
systems an ensemble average is essential to see detailed properties of k(t). In
other systems, notably the Riemann zeta function, it is likely possible to see
the properties by an analysis of the spectrum.Comment: 4 pages, RevTex, no figures, submitted to Phys. Rev. Lett., permanent
e-mail address, [email protected]
Spectral correlations in systems undergoing a transition from periodicity to disorder
We study the spectral statistics for extended yet finite quasi 1-d systems
which undergo a transition from periodicity to disorder. In particular we
compute the spectral two-point form factor, and the resulting expression
depends on the degree of disorder. It interpolates smoothly between the two
extreme limits -- the approach to Poissonian statistics in the (weakly)
disordered case, and the universal expressions derived for the periodic case.
The theoretical results agree very well with the spectral statistics obtained
numerically for chains of chaotic billiards and graphs.Comment: 16 pages, Late
First-Principles Electronic Structure of Solid Picene
To explore the electronic structure of the first aromatic superconductor,
potassium-doped solid picene which has been recently discovered by Mitsuhashi
et al with the transition temperatures K, we have obtained a
first-principles electronic structure of solid picene as a first step toward
the elucidation of the mechanism of the superconductivity. The undoped crystal
is found to have four conduction bands, which are characterized in terms of the
maximally localized Wannier orbitals. We have revealed how the band structure
reflects the stacked arrangement of molecular orbitals for both undoped and
doped (Kpicene) cases, where the bands are not rigid. The Fermi surface for
Kpicene is a curious composite of a warped two-dimensional surface and a
three-dimensional one.Comment: 5 pages, 4 figure
Real roots of Random Polynomials: Universality close to accumulation points
We identify the scaling region of a width O(n^{-1}) in the vicinity of the
accumulation points of the real roots of a random Kac-like polynomial
of large degree n. We argue that the density of the real roots in this region
tends to a universal form shared by all polynomials with independent,
identically distributed coefficients c_i, as long as the second moment
\sigma=E(c_i^2) is finite. In particular, we reveal a gradual (in contrast to
the previously reported abrupt) and quite nontrivial suppression of the number
of real roots for coefficients with a nonzero mean value \mu_n = E(c_i) scaled
as \mu_n\sim n^{-1/2}.Comment: Some minor mistakes that crept through into publication have been
removed. 10 pages, 12 eps figures. This version contains all updates, clearer
pictures and some more thorough explanation
Semiclassical spectral correlator in quasi one-dimensional systems
We investigate the spectral statistics of chaotic quasi one dimensional
systems such as long wires. To do so we represent the spectral correlation
function through derivatives of a generating function and
semiclassically approximate the latter in terms of periodic orbits. In contrast
to previous work we obtain both non-oscillatory and oscillatory contributions
to the correlation function. Both types of contributions are evaluated to
leading order in for systems with and without time-reversal
invariance. Our results agree with expressions from the theory of disordered
systems.Comment: 10 pages, no figure
Characterization of Quantum Chaos by the Autocorrelation Function of Spectral Determinants
The autocorrelation function of spectral determinants is proposed as a
convenient tool for the characterization of spectral statistics in general, and
for the study of the intimate link between quantum chaos and random matrix
theory, in particular. For this purpose, the correlation functions of spectral
determinants are evaluated for various random matrix ensembles, and are
compared with the corresponding semiclassical expressions. The method is
demonstrated by applying it to the spectra of the quantized Sinai billiards in
two and three dimensions.Comment: LaTeX, 32 pages, 6 figure
Correlations between spectra with different symmetry: any chance to be observed?
A standard assumption in quantum chaology is the absence of correlation
between spectra pertaining to different symmetries. Doubts were raised about
this statement for several reasons, in particular, because in semiclassics
spectra of different symmetry are expressed in terms of the same set of
periodic orbits. We reexamine this question and find absence of correlation in
the universal regime. In the case of continuous symmetry the problem is reduced
to parametric correlation, and we expect correlations to be present up to a
certain time which is essentially classical but larger than the ballistic time
Correlations between zeros of a random polynomial
We obtain exact analytical expressions for correlations between real zeros of
the Kac random polynomial. We show that the zeros in the interval are
asymptotically independent of the zeros outside of this interval, and that the
straightened zeros have the same limit translation invariant correlations. Then
we calculate the correlations between the straightened zeros of the SO(2)
random polynomial.Comment: 31 pages, 2 figures; a revised version of the J. Stat. Phys. pape
QUANTIZATION OF A CLASS OF PIECEWISE AFFINE TRANSFORMATIONS ON THE TORUS
We present a unified framework for the quantization of a family of discrete
dynamical systems of varying degrees of "chaoticity". The systems to be
quantized are piecewise affine maps on the two-torus, viewed as phase space,
and include the automorphisms, translations and skew translations. We then
treat some discontinuous transformations such as the Baker map and the
sawtooth-like maps. Our approach extends some ideas from geometric quantization
and it is both conceptually and calculationally simple.Comment: no. 28 pages in AMSTE
Spectral statistics in chaotic systems with a point interaction
We consider quantum systems with a chaotic classical limit that are perturbed
by a point-like scatterer. The spectral form factor K(tau) for these systems is
evaluated semiclassically in terms of periodic and diffractive orbits. It is
shown for order tau^2 and tau^3 that off-diagonal contributions to the form
factor which involve diffractive orbits cancel exactly the diagonal
contributions from diffractive orbits, implying that the perturbation by the
scatterer does not change the spectral statistic. We further show that
parametric spectral statistics for these systems are universal for small
changes of the strength of the scatterer.Comment: LaTeX, 21 pages, 7 figures, small corrections, new references adde
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