16 research outputs found
A Trace Formula for Products of Diagonal Matrix Elements in Chaotic Systems
We derive a trace formula for , where
is the diagonal matrix element of the operator in the energy basis
of a chaotic system. The result takes the form of a smooth term plus
periodic-orbit corrections; each orbit is weighted by the usual Gutzwiller
factor times , where is the average of the classical
observable along the periodic orbit . This structure for the orbit
corrections was previously proposed by Main and Wunner (chao-dyn/9904040) on
the basis of numerical evidence.Comment: 8 pages; analysis made more rigorous in the revised versio
Wave function correlations on the ballistic scale: Exploring quantum chaos by quantum disorder
We study the statistics of wave functions in a ballistic chaotic system. The
statistical ensemble is generated by adding weak smooth disorder. The
conjecture of Gaussian fluctuations of wave functions put forward by Berry and
generalized by Hortikar and Srednicki is proven to hold on sufficiently short
distances, while it is found to be strongly violated on larger scales. This
also resolves the conflict between the above conjecture and the wave function
normalization. The method is further used to study ballistic correlations of
wave functions in a random magnetic field.Comment: 4 page
Semiclassical spatial correlations in chaotic wave functions
We study the spatial autocorrelation of energy eigenfunctions corresponding to classically chaotic systems in the semiclassical regime.
Our analysis is based on the Weyl-Wigner formalism for the spectral average
of , defined as the average over eigenstates within an energy window
centered at . In this framework is the Fourier
transform in momentum space of the spectral Wigner function . Our study reveals the chord structure that
inherits from the spectral Wigner function showing the interplay between the
size of the spectral average window, and the spatial separation scale. We
discuss under which conditions is it possible to define a local system
independent regime for . In doing so, we derive an expression
that bridges the existing formulae in the literature and find expressions for
valid for any separation size .Comment: 24 pages, 3 figures, submitted to PR
Eigenfunction Statistics on Quantum Graphs
We investigate the spatial statistics of the energy eigenfunctions on large
quantum graphs. It has previously been conjectured that these should be
described by a Gaussian Random Wave Model, by analogy with quantum chaotic
systems, for which such a model was proposed by Berry in 1977. The
autocorrelation functions we calculate for an individual quantum graph exhibit
a universal component, which completely determines a Gaussian Random Wave
Model, and a system-dependent deviation. This deviation depends on the graph
only through its underlying classical dynamics. Classical criteria for quantum
universality to be met asymptotically in the large graph limit (i.e. for the
non-universal deviation to vanish) are then extracted. We use an exact field
theoretic expression in terms of a variant of a supersymmetric sigma model. A
saddle-point analysis of this expression leads to the estimates. In particular,
intensity correlations are used to discuss the possible equidistribution of the
energy eigenfunctions in the large graph limit. When equidistribution is
asymptotically realized, our theory predicts a rate of convergence that is a
significant refinement of previous estimates. The universal and
system-dependent components of intensity correlation functions are recovered by
means of an exact trace formula which we analyse in the diagonal approximation,
drawing in this way a parallel between the field theory and semiclassics. Our
results provide the first instance where an asymptotic Gaussian Random Wave
Model has been established microscopically for eigenfunctions in a system with
no disorder.Comment: 59 pages, 3 figure
Semiclassical Construction of Random Wave Functions for Confined Systems
We develop a statistical description of chaotic wavefunctions in closed
systems obeying arbitrary boundary conditions by combining a semiclassical
expression for the spatial two-point correlation function with a treatment of
eigenfunctions as Gaussian random fields. Thereby we generalize Berry's
isotropic random wave model by incorporating confinement effects through
classical paths reflected at the boundaries. Our approach allows to explicitly
calculate highly non-trivial statistics, such as intensity distributions, in
terms of usually few short orbits, depending on the energy window considered.
We compare with numerical quantum results for the Africa billiard and derive
non-isotropic random wave models for other prominent confinement geometries.Comment: To be submitted to Physical Review Letter
Coulomb blockade conductance peak fluctuations in quantum dots and the independent particle model
We study the combined effect of finite temperature, underlying classical
dynamics, and deformations on the statistical properties of Coulomb blockade
conductance peaks in quantum dots. These effects are considered in the context
of the single-particle plus constant-interaction theory of the Coulomb
blockade. We present numerical studies of two chaotic models, representative of
different mean-field potentials: a parametric random Hamiltonian and the smooth
stadium. In addition, we study conductance fluctuations for different
integrable confining potentials. For temperatures smaller than the mean level
spacing, our results indicate that the peak height distribution is nearly
always in good agreement with the available experimental data, irrespective of
the confining potential (integrable or chaotic). We find that the peak bunching
effect seen in the experiments is reproduced in the theoretical models under
certain special conditions. Although the independent particle model fails, in
general, to explain quantitatively the short-range part of the peak height
correlations observed experimentally, we argue that it allows for an
understanding of the long-range part.Comment: RevTex 3.1, 34 pages (including 13 EPS and PS figures), submitted to
Phys. Rev.
The approach to thermal equilibrium in quantized chaotic systems
We consider many-body quantum systems that exhibit quantum chaos, in the
sense that the observables of interest act on energy eigenstates like banded
random matrices. We study the time-dependent expectation values of these
observables, assuming that the system is in a definite (but arbitrary) pure
quantum state. We induce a probability distribution for the expectation values
by treating the zero of time as a uniformly distributed random variable. We
show explicitly that if an observable has a nonequilibrium expectation value at
some particular moment, then it is overwhelmingly likely to move towards
equilibrium, both forwards and backwards in time. For deviations from
equilibrium that are not much larger than a typical quantum or thermal
fluctuation, we find that the time dependence of the move towards equilibrium
is given by the Kubo correlation function, in agreement with Onsager's
postulate. These results are independent of the details of the system's quantum
state.Comment: 15 pages, no figures; some arguments are clarified in the revised
versio
Quantum Billiards with Surface Scattering: Ballistic Sigma-Model Approach
Statistical properties of energy levels and eigenfunctions in a ballistic
system with diffusive surface scattering are investigated. The two-level
correlation function, the level number variance, the correlation function of
wavefunction intensities, and the inverse participation ratio are calculated.Comment: 4 pages REVTEX, two figures included as eps file
Semiclassical Theory of Coulomb Blockade Peak Heights in Chaotic Quantum Dots
We develop a semiclassical theory of Coulomb blockade peak heights in chaotic
quantum dots. Using Berry's conjecture, we calculate the peak height
distributions and the correlation functions. We demonstrate that the
corrections to the corresponding results of the standard statistical theory are
non-universal and can be expressed in terms of the classical periodic orbits of
the dot that are well coupled to the leads. The main effect is an oscillatory
dependence of the peak heights on any parameter which is varied; it is
substantial for both symmetric and asymmetric lead placement. Surprisingly,
these dynamical effects do not influence the full distribution of peak heights,
but are clearly seen in the correlation function or power spectrum. For
non-zero temperature, the correlation function obtained theoretically is in
good agreement with that measured experimentally.Comment: 5 color eps figure
Level and Eigenfunction Statistics in Billiards with Surface Scattering
Statistical properties of billiards with diffusive boundary scattering are
investigated by means of the supersymmetric sigma-model in a formulation
appropriate for chaotic ballistic systems. We study level statistics,
parametric level statistics, and properties of electron wavefunctions. In the
universal regime, our results reproduce conclusions of the random matrix
theory, while beyond this regime we obtain a variety of system-specific results
determined by the classical dynamics in the billiard. Most notably, we find
that level correlations do not vanish at arbitrary separation between energy
levels, or if measured at arbitrarily large difference of magnetic fields.
Saturation of the level number variance indicates strong rigidity of the
spectrum. To study spatial correlations of wavefunction amplitudes, we
reanalyze and refine derivation of the ballistic version of the sigma-model.
This allows us to obtain a proper matching of universal short-scale
correlations with system-specific ones.Comment: 19 pages, 5 figures included. Minor corrections, references adde