Quantum chaos induced by scaled disorder
J. A. Verge
´
s
Instituto de Ciencia de Materiales de Madrid, Consejo Superior de Investigaciones Cientı ´ficas, Cantoblanco, E-28049 Madrid, Spain
E. Louis
Departamento de Fı ´sica Aplicada, Universidad de Alicante, Apartado 99, E-03080 Alicante, Spain
Received 23 November 1998
Quantum chaos is obtained for a two-dimensional square lattice with a number of vacancies that scales with
the linear size of the cluster L. The appearance of quantum chaos is signaled by both level and wave function
statistics. Since states are extended, ballistic transport behavior is expected. In particular, we show that the
static conductance increases linearly with L. S1063-651X9951104-5
PACS numbers: 05.45.Mt, 73.20.Dx, 03.65.Sq
The statistical properties of measurable magnitudes of
mesoscopic systems play an important role in the physics of
mesoscopic phenomena 1,2. Random matrix theory RMT
3 has been successfully used to explain most of the experi-
mentally known statistical results. The nonlinear supersym-
metric -model demonstrates the relevance of RMT in
slightly disordered systems 4 and makes detailed predic-
tions for some deviations 5. However, generalization of
these results to chaotic ballistic systems brings technical
complications, since average over disorder should be substi-
tuted by energy averaging of an action in which the Liouville
operator replaces the diffusion operator. Alternatively, one
can study disordered systems that are nevertheless ballistic
from the point of view of their transport properties. A billiard
having a rough surface is the model of choice 6,7. Other
possible models are distorted integrable billiards 8. Follow-
ing this idea, Blanter, Mirlin, and Muzykantskii have pre-
sented a detailed supersymmetric study of the statistical
properties of rough circular billiards 9. The level statistics
for the same problem was studied by Tripathi and
Khmelnitskii 10. Motivated by the important differences
between systems having surface or bulk disorder, we have
further analyzed our original model 7 in order to unravel
the relevant parameters. It happens that the crucial character-
istic is not the physical placement of defects but their num-
ber, or more precisely, the scaling of the number of defects
as the size of the system grows. If the ratio between the
number of defects and the billiard area, i.e., the defect den-
sity, is constant, transport properties of the system scale from
the diffusive regime towards localization at large enough
size scales. At the same time, statistical properties scale from
Wigner-Dyson behavior to Poisson statistics. On the other
hand, if the number of defects is proportional to the number
of surface sites, i.e., defect density is inversely proportional
to linear size, transport properties are ballistic at all size
scales see below. Diffusive or localized transport behavior
is never reached. Statistical properties are well described by
RMT at any system size. Moreover, the detailed distribution
of defects over the billiard does not matter. In this way, we
arrive at the simplest model showing chaotic statistics and
ballistic transport properties: a square cluster of side L with a
number of vacancies of order L placed at random positions.
The model of a quantum chaotic billiard presented in this
Rapid Communication is not only the more general one pos-
sible but also simpler than the original one because the sub-
stitution of defects by vacancies eliminates one unnecessary
technical complication. Nondiagonal or topological disorder
occupies the place of diagonal disorder, eliminating one ir-
relevant parameter from our model, the width of the distri-
bution of diagonal energies. Only one energy scale remains,
the one defined by the hopping integral. The other superflu-
ous characteristic of our former billiard model was the place-
ment of all the defects on the surface of the system. We were
modeling roughness in a practical implementation but here
we show that bulk roughness in the form of forbidden places
is also valid. In other words, what matters is just the rela-
tionship between forbidden and allowed sites but not their
relative spatial distribution.
Our model of a quantum billiard is described by means of
a tight-binding Hamiltonian with a single atomic level per
lattice site,
H
ˆ
=-
i , j
i
c
ˆ
i
†
c
ˆ
j
i
, 1
where the operator c
ˆ
i
destroys an electron on site i, all the
hopping integrals are taken equal to -1 and restricted to
nearest neighboring sites. j
i
gives just the labels of the ex-
isting nearest neighbors of site i. Periodic boundary condi-
tions are used for the study of spectral properties in order to
minimize finite size effects. Therefore, the difference be-
tween our Hamiltonian H
ˆ
and the one corresponding to an
ideal L L cluster of the square lattice is the absence of
hopping to and from L sites chosen at random among the L
2
sites defining the lattice. Spectral calculations have been car-
ried out on clusters of linear sizes up to L =100, whereas
conductance has been measured up to L =500.
The classical analog of our model shares some features
with the pinball game. Certainly, a classical L L table in-
cluding about L / a circular scatterers of linear size a centered
at random positions shows classical hard chaos. Notice that
our model is characterized by two length scales: a micro-
scopic one equal to a and a mesoscopic one given by L.
Scaling towards chaos requires a number of defects scatter-
RAPID COMMUNICATIONS
PHYSICAL REVIEW E APRIL 1999 VOLUME 59, NUMBER 4
PRE 59 1063-651X/99/594/38034/$15.00 R3803 ©1999 The American Physical Society