1,843 research outputs found
The spin contribution to the form factor of quantum graphs
Following the quantisation of a graph with the Dirac operator (spin-1/2) we
explain how additional weights in the spectral form factor K(\tau) due to spin
propagation around orbits produce higher order terms in the small-\tau
asymptotics in agreement with symplectic random matrix ensembles. We determine
conditions on the group of spin rotations sufficient to generate CSE
statistics.Comment: 9 page
Generic identifiability and second-order sufficiency in tame convex optimization
We consider linear optimization over a fixed compact convex feasible region
that is semi-algebraic (or, more generally, "tame"). Generically, we prove that
the optimal solution is unique and lies on a unique manifold, around which the
feasible region is "partly smooth", ensuring finite identification of the
manifold by many optimization algorithms. Furthermore, second-order optimality
conditions hold, guaranteeing smooth behavior of the optimal solution under
small perturbations to the objective
Clarke subgradients of stratifiable functions
We establish the following result: if the graph of a (nonsmooth)
real-extended-valued function
is closed and admits a Whitney stratification, then the norm of the gradient of
at relative to the stratum containing bounds from below
all norms of Clarke subgradients of at . As a consequence, we obtain
some Morse-Sard type theorems as well as a nonsmooth Kurdyka-\L ojasiewicz
inequality for functions definable in an arbitrary o-minimal structure
Semiclassical Approach to Parametric Spectral Correlation with Spin 1/2
The spectral correlation of a chaotic system with spin 1/2 is universally
described by the GSE (Gaussian Symplectic Ensemble) of random matrices in the
semiclassical limit. In semiclassical theory, the spectral form factor is
expressed in terms of the periodic orbits and the spin state is simulated by
the uniform distribution on a sphere. In this paper, instead of the uniform
distribution, we introduce Brownian motion on a sphere to yield the parametric
motion of the energy levels. As a result, the small time expansion of the form
factor is obtained and found to be in agreement with the prediction of
parametric random matrices in the transition within the GSE universality class.
Moreover, by starting the Brownian motion from a point distribution on the
sphere, we gradually increase the effect of the spin and calculate the form
factor describing the transition from the GOE (Gaussian Orthogonal Ensemble)
class to the GSE class.Comment: 25 pages, 2 figure
Spectral Statistics for the Dirac Operator on Graphs
We determine conditions for the quantisation of graphs using the Dirac
operator for both two and four component spinors. According to the
Bohigas-Giannoni-Schmit conjecture for such systems with time-reversal symmetry
the energy level statistics are expected, in the semiclassical limit, to
correspond to those of random matrices from the Gaussian symplectic ensemble.
This is confirmed by numerical investigation. The scattering matrix used to
formulate the quantisation condition is found to be independent of the type of
spinor. We derive an exact trace formula for the spectrum and use this to
investigate the form factor in the diagonal approximation
Semiclassical Approach to Chaotic Quantum Transport
We describe a semiclassical method to calculate universal transport
properties of chaotic cavities. While the energy-averaged conductance turns out
governed by pairs of entrance-to-exit trajectories, the conductance variance,
shot noise and other related quantities require trajectory quadruplets; simple
diagrammatic rules allow to find the contributions of these pairs and
quadruplets. Both pure symmetry classes and the crossover due to an external
magnetic field are considered.Comment: 33 pages, 11 figures (appendices B-D not included in journal version
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
Intermediate statistics for a system with symplectic symmetry: the Dirac rose graph
We study the spectral statistics of the Dirac operator on a rose-shaped
graph---a graph with a single vertex and all bonds connected at both ends to
the vertex. We formulate a secular equation that generically determines the
eigenvalues of the Dirac rose graph, which is seen to generalise the secular
equation for a star graph with Neumann boundary conditions. We derive
approximations to the spectral pair correlation function at large and small
values of spectral spacings, in the limit as the number of bonds approaches
infinity, and compare these predictions with results of numerical calculations.
Our results represent the first example of intermediate statistics from the
symplectic symmetry class.Comment: 26 pages, references adde
- …