arXiv subjects

A. D. Jackson

Publications and source records attributed to A. D. Jackson.

At least 55 records · Page 3Linked to original sources

arXiv2000

Spectral ergodicity and normal modes in ensembles of sparse matrices

We investigate the properties of sparse matrix ensembles with particular regard for the spectral ergodicity hypothesis, which claims the identity of ensemble and spectral averages of spectral correlators. An apparent violation of the spectral ergodicity is observed. This effect is studied with the aid of the normal modes of the random matrix spectrum, which describe fluctuations of the eigenvalues around their average positions. This analysis reveals that spectral ergodicity is not broken, but that different energy scales of the spectra are examined by the two averaging techniques. Normal modes are shown to provide a useful complement to traditional spectral analysis with possible applications to a wide range of physical systems.

cond-mat↗

arXiv2000

Apparent Superluminal Behavior

The apparent superluminal propagation of electromagnetic signals seen in recent experiments is shown to be the result of simple and robust properties of relativistic field equations. Although the wave front of a signal passing through a classically forbidden region can never move faster than light, an attenuated replica of the signal is reproduced ``instantaneously'' on the other side of the barrier. The reconstructed signal, causally connected to the forerunner rather than the bulk of the input signal, appears to move through the barrier faster than light.

physics.class-ph↗

arXiv2000

Analytical results for a trapped, weakly-interacting Bose-Einstein condensate under rotation

We examine the problem of a repulsive, weakly-interacting and harmonically trapped Bose-Einstein condensate under rotation. We derive a simple analytic expression for the energy incorporating the interactions when the angular momentum per particle is between zero and one and find that the interaction energy decreases linearly as a function of the angular momentum in agreement with previous numerical and limiting analytical studies.

cond-mat↗

arXiv2000

Weakly Interacting Bose-Einstein Condensates Under Rotation: Mean-field versus Exact Solutions

We consider a weakly-interacting, harmonically-trapped Bose-Einstein condensed gas under rotation and investigate the connection between the energies obtained from mean-field calculations and from exact diagonalizations in a subspace of degenerate states. From the latter we derive an approximation scheme valid in the thermodynamic limit of many particles. Mean-field results are shown to emerge as the correct leading-order approximation to exact calculations in the same subspace.

cond-mat↗

arXiv2000

Random matrix model for chiral symmetry breaking and color superconductivity in QCD at finite density

We consider a random matrix model which describes the competition between chiral symmetry breaking and the formation of quark Cooper pairs in QCD at finite density. We study the evolution of the phase structure in temperature and chemical potential with variations of the strength of the interaction in the quark-quark channel and demonstrate that the phase diagram can realize a total of six different topologies. A vector interaction representing single-gluon exchange reproduces a topology commonly encountered in previous QCD models, in which a low-density chiral broken phase is separated from a high-density diquark phase by a first-order line. The other five topologies either do not possess a diquark phase or display a new phase and new critical points. Since these five cases require large variations of the coupling constants away from the values expected for a vector interaction, we conclude that the phase diagram of finite density QCD has the topology suggested by single-gluon exchange and that this topology is robust.

hep-ph↗

arXiv1999

Partitioning Composite Finite Systems

We compare different analytical and numerical methods for studying the partitions of a finite system into fragments. We propose a new numerical method of exploring the partition space by generating the Markov chains of partitions based on the Metropolis algorithm. The advantages of the new method for the problems where partitions are sampled with non-trivial weights are demonstrated.

math-ph↗

arXiv1999

A Random Matrix Model for Color Superconductivity at Zero Chemical Potential

We discuss random matrix models for the spontaneous breaking of both chiral and color symmetries at zero chemical potential and finite temperature. Exploring different Lorentz and gauge symmetric color structures of the random matrix interactions, we find that spontaneous chiral symmetry breaking is always thermodynamically preferred over diquark condensation. Stable diquark condensates appear only as SU(2) rotated chiral condensates, which do not represent an independent thermodynamic phase. Our analysis is based on general symmetry arguments and hence suggests that no stable and independent diquark phase can form in QCD with two flavors at zero quark chemical potential.

hep-ph↗

arXiv1999

Finding the Pion in the Chiral Random Matrix Vacuum

The existence of a Goldstone boson is demonstrated in chiral random matrix theory. After determining the effective coupling and calculating the scalar and pseudoscalar propagators, a random phase approximation summation reveals the massless pion and massive sigma modes expected whenever chiral symmetry is spontaneously broken.

hep-ph↗

arXiv1998

Rigidity and Normal Modes in Random Matrix Spectra

We consider the Gaussian ensembles of random matrices and describe the normal modes of the eigenvalue spectrum, i.e., the correlated fluctuations of eigenvalues about their most probable values. The associated normal mode spectrum is linear, and for large matrices, the normal modes are found to be Chebyshev polynomials of the second kind. We contrast this with the behaviour of a sequence of uncorrelated levels, which has a quadratic normal mode spectrum. The difference in the rigidity of random matrix spectra and sequences of uncorrelated levels can be attributed to this difference in the normal mode spectra. We illustrate this by calculating the number variance in the two cases.

nucl-th↗

arXiv1998

Crossover to Non-universal Microscopic Spectral Fluctuations in Lattice Gauge Theory

The spectrum of the Dirac operator near zero virtuality obtained in lattice gauge simulations is known to be universally described by chiral random matrix theory. We address the question of the maximum energy for which this universality persists. For this purpose, we analyze large ensembles of complete spectra of the Euclidean Dirac operator for staggered fermions. We calculate the disconnected scalar susceptibility and the microscopic number variance for the chiral symplectic ensemble of random matrices and compare the results with lattice Dirac spectra for quenched SU(2). The crossover to a non-universal regime is clearly identified and found to scale with the square of the linear lattice size and with $f_π^2$, in agreement with theoretical expectations.

hep-ph↗

arXiv1998

On the Phase Diagram of QCD

We analyze the phase diagram of QCD with two massless quark flavors in the space of temperature, T, and chemical potential of the baryon charge, mu, using available experimental knowledge of QCD, insights gained from various models, as well as general and model independent arguments including continuity, universality, and thermodynamic relations. A random matrix model is used to describe the chiral symmetry restoration phase transition at finite T and mu. In agreement with general arguments, this model predicts a tricritical point in the T mu plane. Certain critical properties at such a point are universal and can be relevant to heavy ion collision experiments.

hep-ph↗

arXiv1998

The Ginsparg-Wilson relation and local chiral random matrix theory

A chiral random matrix model with locality is constructed and examined. The Nielsen-Ninomiya no-go theorem is circumvented by the use of a generally applicable modified Dirac operator which respects the Ginsparg-Wilson relation. We observe the expected universal behaviour of the eigenvalue density in the microscopic limit.

hep-lat↗

arXiv1998

Solitary waves in clouds of Bose-Einstein condensed atoms

We consider the conditions under which solitary waves can exist in elongated clouds of Bose-Einstein condensed atoms. General expressions are derived for the velocity, characteristic size, and spatial profile of solitary waves, and the low- and high-density limits are examined.

cond-mat↗

arXiv1997

Random-matrix universality in the small-eigenvalue spectrum of the lattice Dirac operator

We analyze complete spectra of the lattice Dirac operator in SU(2) gauge theory and demonstrate that the distribution of low-lying eigenvalues is described by random matrix theory. We present possible practical applications of this random-matrix universality. In particular, reliable extrapolations of lattice gauge data to the thermodynamic limit are discussed.

hep-lat↗

arXiv1997

Yang-Lee zeros of a random matrix model for QCD at finite density

We study the Yang-Lee zeros of a random matrix partition function with the global symmetries of the QCD partition function. We consider both zeros in the complex chemical potential plane and in the complex mass plane. In both cases we find that the zeros are located on a curve. In the thermodynamic limit, the zeros appear to merge to form a cut. The shape of this limiting curve can be obtained from a saddle-point analysis of the partition function. An explicit solution for the line of zeros in the complex chemical potential plane at zero mass is given in the form of a transcendental equation.

hep-lat↗

arXiv1997

Random Quantum Billiards

We present a random matrix model suitable for the quantum mechanical description of a particle confined to move inside a two-dimensional domain. Here, the ensemble average corresponds to an average over domain shapes. Although this approach is formally distinct from that of the familiar Gaussian orthogonal ensemble, there is remarkable agreement in the resulting distributions of level spacings and in the spectral rigidity. We discuss possible applications of this ensemble to the analysis of real data sets.

chao-dyn↗

arXiv1997

Universality of Correlation Functions in Random Matrix Models of QCD

We demonstrate the universality of the spectral correlation functions of a QCD inspired random matrix model that consists of a random part having the chiral structure of the QCD Dirac operator and a deterministic part which describes a schematic temperature dependence. We calculate the correlation functions analytically using the technique of Itzykson-Zuber integrals for arbitrary complex super-matrices. An alternative exact calculation for arbitrary matrix size is given for the special case of zero temperature, and we reproduce the well-known Laguerre kernel. At finite temperature, the microscopic limit of the correlation functions are calculated in the saddle point approximation. The main result of this paper is that the microscopic universality of correlation functions is maintained even though unitary invariance is broken by the addition of a deterministic matrix to the ensemble.

hep-th↗

arXiv1997

Fermion determinants in matrix models of QCD at nonzero chemical potential

The presence of a chemical potential completely changes the analytical structure of the QCD partition function. In particular, the eigenvalues of the Dirac operator are distributed over a finite area in the complex plane, whereas the zeros of the partition function in the complex mass plane remain on a curve. In this paper we study the effects of the fermion determinant at nonzero chemical potential on the Dirac spectrum by means of the resolvent, G(z), of the QCD Dirac operator. The resolvent is studied both in a one-dimensional U(1) model (Gibbs model) and in a random matrix model with the global symmetries of the QCD partition function. In both cases we find that, if the argument z of the resolvent is not equal to the mass m in the fermion determinant, the resolvent diverges in the thermodynamic limit. However, for z =m the resolvent in both models is well defined. In particular, the nature of the limit $z \rightarrow m$ is illuminated in the Gibbs model. The phase structure of the random matrix model in the complex m and μ-planes is investigated both by a saddle point approximation and via the distribution of Yang-Lee zeros. Both methods are in complete agreement and lead to a well-defined chiral condensate and quark number density.

hep-lat↗