pub69 - PHYSICAL REVIEW B 82, 104202 2010 Universality of a...

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Universality of a family of random matrix ensembles with logarithmic soft-conFnement potentials Jinmyung Choi and K. A. Muttalib Department of Physics, University of Florida, Gainesville, Florida 32611-8440, USA s Received 9 June 2010; revised manuscript received 23 July 2010; published 8 September 2010 d Recently we introduced a family of U s N d invariant random matrix ensembles which is characterized by a parameter l describing logarithmic soft-conFnement potentials V s H d,f ln H g s 1+ l d s l. 0 d . We showed that we can study eigenvalue correlations of these “ l ensembles” based on the numerical construction of the corre- sponding orthogonal polynomials with respect to the weight function exp f ± s ln x d 1+ l g . In this work, we expand our previous work and show that: s i d the eigenvalue density is given by a power law of the form r s x d ~ f ln x g l ±1 / x and s ii d the two-level kernel has an anomalous structure, which is characteristic of the critical ensembles. We further show that the anomalous part, or the so-called “ghost-correlation peak,” is controlled by the parameter l ; decreasing l increases the anomaly. We also identify the two-level kernel of the l ensembles in the semiclassical regime, which can be written in a sinh-kernel form with more general argument that reduces to that of the critical ensembles for l =1. ²inally, we discuss the universality of the l ensembles, which includes Wigner-Dyson universality s l ` limit d , the uncorrelated Poisson-type behavior s l 0 limit d , and a critical behavior for all the intermediate l s 0 ,l,` d in the semiclassical regime. We also comment on the implications of our results in the context of the localization-delocalization problems as well as the N depen- dence of the two-level kernel of the fat-tail random matrices. DOI: 10.1103/PhysRevB.82.104202 PACS number s s d : 72.15.Rn, 05.40. 2 a, 05.60. 2 k, 05.90. 1 m I. INTRODUCTION Random matrix theory s RMT d deals with the statistical properties of eigenvalues and eigenvectors of random matri- ces drawn from a certain probability measure. The theory was successfully applied by Wigner 1 in the 1950s to describe the spectral properties of complex many-body nuclei where the underlying Hamiltonians are so complicated that a useful way to study the system turned out to be through a statistical treatment of the Hamiltonians. 2 Since then, RMT has been applied to a wide variety of systems in diverse areas, includ- ing, e.g., many-body atoms and nuclei, quantum chaos, me- soscopic disordered conductors, two-dimensional quantum gravity, conformal Feld theory, chiral phase transitions as well as zeros of Riemann zeta function, scale-free networks, biological networks, communication systems, and Fnancial markets. 3 , 4 This broad range of applicability of RMT in seemingly unrelated areas highlights the universal features of the correlations of the eigenvalues in RMT. Within the clas- sical Gaussian model pioneered by Wigner, these correla- tions are known as the Wigner-Dyson s WD d statistics of the
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This note was uploaded on 12/15/2011 for the course PHY 3221 taught by Professor Chan during the Spring '08 term at University of Florida.

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pub69 - PHYSICAL REVIEW B 82, 104202 2010 Universality of a...

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