Akhilesh Pandey

Gaussian ensembles of random Hermitian matrices intermediate between orthogonal and unitary ones

A Gaussian ensemble of Hermitian matrices depending on a parameter α is considered. When α=0, the ensemble is Gaussian Orthogonal, and when α=1, it is Gaussian Unitary. An analytic expression for then-level correlation and cluster functions is given for anyn and 0≦α≦1. This ensemble is of relevance in the study of time reversal symmetry breaking of nuclear interactions.

Statistical properties of many-particle spectra: III. Ergodic behavior in random-matrix ensembles

Abstract The ergodic problem is defined for random-matrix ensembles and some conditions for ergodicity given. Ergodic properties are demonstrated for the orthogonal, unitary and symplectic cases of the Gaussian and circular ensembles, and also for the Poisson ensemble. The one-point measures, viz., the eigenvalue density, the number statistic and the k thnearest-neighbor spacings are shown to be ergodic and the ensemble variances of the corresponding spectral averages are explicitly calculated. It is moreover shown, by using Dysons cluster functions, that all the k -point correlation functions are themselves ergodic as are therefore the fluctuation measures which follow from them. It is proved also that the local fluctuation properties of the Gaussian ensembles are stationary over the spectrum.

Statistical properties of many-particle spectra V. Fluctuations and symmetries

Abstract The manner in which energy-level and strength fluctuations change as a good symmetry is gradually broken is studied for a wide variety of chaotic systems describable in terms of random-matrix models. The local parameter Λ which governs the transition is identified as the mean-square symmetry-breaking matrix element in units of the local average spacing; its basic significance is clarified by exhibiting it as the “time” variable in a hierarchic set of diffusion equations. While many transitions, including Poisson → GOE, are considered, the transition GOE → GUE, from Gaussian orthogonal to Gaussian unitary ensembles, is studied in particular detail because of its connection with the breaking of time-reversal invariance (TRI), especially in complex nuclei for which the relevance of the GOE has been experimentally confirmed (GOE → GUE enters also into the study of chaotic magnetic systems). It is shown that the ensemble theory is directly applicable to real systems so that the transition equations derived for various spectral and strength measures can be applied to fluctuation data to give multiparticle bounds on the TRI-breaking part of the Hamiltonian. Analyses of spectral and strength fluctuations in the nuclear neutron-resonance and proton-resonance regions give bounds on the symmetry-breaking matrix elements as approximately one-tenth of the local average spacing. The analysis however must not stop with determining a bound on Λ; although the mere existence of a nonzero Λ could be of interest, indicating the occurrence of a symmetry breaking, the significance, if any, of an upper bound will usually not be obvious. It is therefore essential to determine, from the local multibody parameter Λ, the global (two-body) quantity α, the relative norm of the time-reversal noninvariant (TRNI) part of the nucleon-nucleon interaction. We do that in the following paper.

Statistical properties of many-particle spectra. IV. New ensembles by Stieltjes transform methods☆

Abstract New Gaussian matrix ensembles, with arbitrary centroids and variances for the matrix elements, are defined as modifications of the three standard ones—GOE, GUE and GSE. The average density and two-point correlation function are given in the general case in terms of the corresponding Stieltjes transforms, first used by Pastur for the density. It is shown for the centroid-modified ensemble K + αH that when the operator K preserves the underlying symmetries of the standard ensemble H, then, as the magnitude of α grows, the transition of the fluctuations to those of H is very rapid and discontinuous in the limit of asymptotic dimensionality. Corresponding results are found for other ensembles. A similar Dyson result for the effects of the breaking of a model symmetry on the fluctuations is generalized to any model symmetry, as well as to the fundamental symmetries such as time-reversal invariance.

Statistical properties of many-particle spectra VI. Fluctuation bounds on N-NT-noninvariance

Abstract The preceding paper (V) shows that fluctuation analysis applied to experimental data in a chaotic region of the nuclear spectrum, determines values or bounds for a parameter Λ, proportional to the mean-squared near-diagonal matrix element of a symmetry-breaking interaction. The immediate interest is with time-reversal invariance (TRI) for which bounds, not values, are found. We show here how this information can be reduced to a bound on α, the relative magnitude of the time-reversal noninvariant (TRNI) part of the nucleon-nucleon interaction. This is done in terms of the level and transition-strength densities, both given as explicit functions of the Hamiltonian parameters. It is shown that, just as univariate Gaussian densities are essential ingredients of the state density, so also bivariate Gaussians enter naturally into a theory for the transition strengths. Calculations made for a subset, namely 167,169Er, 233Th, and 239U, of the nuclei considered in (V), give α≲2×10−3, a value which by some arguments is at the boundary of fundamental interest. It is found that ΛD α 2 is roughly constant over the periodic table and varies only slowly with energy. This implies that, for α-bounds better by perhaps a factor 4 (or α-values), we need more small strengths in nuclei with small spacings such as 236U (which presently yields no Λ-bound). The methods introduced here, which admit a number of simplifying approximations, are applicable to TRI studies via detailed-balance experiments proceeding through a compound nucleus, to other symmetries, and to a wide class of other problems.

Fluctuations in intramolecular line shapes—random matrix theory

Abstract Random matrix theory is used to develop a model for the distribution of energy levels and intensities in intramolecular line shapes. The effects of missing lines, either due to their weak intensity, or due to the finite spectral resolution, are quantitatively incorporated. It is shown how the information regarding spectral fluctuations in intermediate size molecules is eroded in the large molecule statistical limit. Our predictions are compared with recent experimental data on highly vibrationally excited acetylene, and the relevant statistical measures are calculated.

Brownian-motion model of discrete spectra

Abstract The transition in spectral fluctuations in a quantum chaotic system as a good symmetry of the system is gradually broken is describable in terms of Dysons Brownian-motion model. Here the magnitude of symmetry breaking plays the role of ‘time’ while the ‘temperature’ determines the universality class of random matrices to which the system goes in ‘equilibrium’. Recently we have shown that the transition itself has a universal classification and is useful in studying exact or weakly broken symmetries in complex many-body systems. In this paper we derive the exact two-level correlation functions for the class of transitions which eventually go to the Gaussian unitary ensemble. It appears that the transition to the Gaussian orthogonal and symplectic ensembles may also be exactly solvable.

Statistical properties of many-particle spectra. II. Two-point correlations and fluctuations☆

Abstract The two-point correlation function for complex spectra described by the Gaussian Orthogonal Ensemble (GOE) is calculated, and its essential simplicity displayed, by an elementary procedure which derives from orthogonal invariance and the dominance of intrinsic binary correlations. The resultant function is used for an approximate calculation of the standard fluctuation measures. Good agreements are found with exact results where these are available, this incidentally demonstrating that the measures are, for the most part, two-point measures. It is shown that they vary slowly over the spectrum, a result which is in agreement both with experiment and with Monte Carlo calculations. The same technique can be used for higher-order correlation functions, and possibly also for more complicated ensembles in which case the results would be relevant to the question why GOE fluctuations give a good account of experimental results.

Level repulsion in the spectrum of two-dimensional harmonic oscillators

The asymptotic-energy limit of the density P(S) of spacings between adjacent levels of the two-dimensional harmonic oscillator (TDHO) spectrum is studied. It is shown that in any integer segment (M, M+1), containing approximately M/α levels, of the TDHO spectrum m + αn, P(S) has the form Σ wi δ(S - Si) where i takes on at most three values. For large M, P(S) displays strong level repulsion for irrational α, but it does not settle on a stationary form nor does its average over M. This is in marked contrast with the behaviour of generic integrable systems for which the Poisson statistics, P(S)= exp(-S), is known to apply.

Eigenvalue correlations in the circular ensembles

Dyson (1972) introduced two types of Brownian-motion ensembles of random matrices for studying approximate symmetries in complex quantum systems. The magnitude of symmetry breaking plays the role of a fictitious time t>or=0. The authors study the eigenvalue correlations in the circular-type ensembles which serve as models for the evolution operators of quantum maps with chaotic classical limits. In two cases involving time-reversal symmetry breaking they evaluate explicitly the eigenangle-density correlation functions of all orders for all t and for all values of the matrix dimensionality N. The general case is described by a hierarchic set of relations among the correlation functions. As a function of t, the transition in the correlations is found to be rapid for large N, discontinuous for N to infinity . As a function of a local parameter Lambda , which measures the mean square symmetry-admixing matrix element in units of the local average spacing, the transition is found to be smooth. The same Lambda -dependent results were found earlier for the Gaussian-type ensembles which serve as models for the Hamiltonian operators of autonomous chaotic systems.