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