25 research outputs found
Beyond Blobs in Percolation Cluster Structure: The Distribution of 3-Blocks at the Percolation Threshold
The incipient infinite cluster appearing at the bond percolation threshold
can be decomposed into singly-connected ``links'' and multiply-connected
``blobs.'' Here we decompose blobs into objects known in graph theory as
3-blocks. A 3-block is a graph that cannot be separated into disconnected
subgraphs by cutting the graph at 2 or fewer vertices. Clusters, blobs, and
3-blocks are special cases of -blocks with , 2, and 3, respectively. We
study bond percolation clusters at the percolation threshold on 2-dimensional
square lattices and 3-dimensional cubic lattices and, using Monte-Carlo
simulations, determine the distribution of the sizes of the 3-blocks into which
the blobs are decomposed. We find that the 3-blocks have fractal dimension
in 2D and in 3D. These fractal dimensions are
significantly smaller than the fractal dimensions of the blobs, making possible
more efficient calculation of percolation properties. Additionally, the
closeness of the estimated values for in 2D and 3D is consistent with the
possibility that is dimension independent. Generalizing the concept of
the backbone, we introduce the concept of a ``-bone'', which is the set of
all points in a percolation system connected to disjoint terminal points
(or sets of disjoint terminal points) by disjoint paths. We argue that the
fractal dimension of a -bone is equal to the fractal dimension of
-blocks, allowing us to discuss the relation between the fractal dimension
of -blocks and recent work on path crossing probabilities.Comment: All but first 2 figs. are low resolution and are best viewed when
printe
A Simple Model of Liquid-liquid Phase Transitions
In recent years, a second fluid-fluid phase transition has been reported in
several materials at pressures far above the usual liquid-gas phase transition.
In this paper, we introduce a new model of this behavior based on the
Lennard-Jones interaction with a modification to mimic the different kinds of
short-range orientational order in complex materials. We have done Monte Carlo
studies of this model that clearly demonstrate the existence of a second
first-order fluid-fluid phase transition between high- and low-density liquid
phases
String order in spin liquid phases of spin ladders
Two-leg spin ladders have a rich phase diagram if rung, diagonal and
plaquette couplings are allowed for. Among the possible phases there are two
Haldane-type spin liquid phases without local order parameter, which differ,
however, in the topology of the short range valence bonds. We show that these
phases can be distinguished numerically by two different string order
parameters. We also point out that long range string- and dimer orders can
coexist
Tricritical Behavior of Two-Dimensional Scalar Field Theories
We compute by Monte Carlo numerical simulations the critical exponents of
two-dimensional scalar field theories at the tricritical point.
The results are in agreement with the Zamolodchikov conjecture based on
conformal invariance.Comment: 13 pages, uuencode tar-compressed Postscript file, preprint numbers:
IF/UFRJ/25/94, DFTUZ 94.06 and NYU--TH--94/10/0
New Criticality of 1D Fermions
One-dimensional massive quantum particles (or 1+1-dimensional random walks)
with short-ranged multi-particle interactions are studied by exact
renormalization group methods. With repulsive pair forces, such particles are
known to scale as free fermions. With finite -body forces (m = 3,4,...), a
critical instability is found, indicating the transition to a fermionic bound
state. These unbinding transitions represent new universality classes of
interacting fermions relevant to polymer and membrane systems. Implications for
massless fermions, e.g. in the Hubbard model, are also noted. (to appear in
Phys. Rev. Lett.)Comment: 10 pages (latex), with 2 figures (not included
Correlation decay and conformal anomaly in the two-dimensional random-bond Ising ferromagnet
The two-dimensional random-bond Ising model is numerically studied on long
strips by transfer-matrix methods. It is shown that the rate of decay of
correlations at criticality, as derived from averages of the two largest
Lyapunov exponents plus conformal invariance arguments, differs from that
obtained through direct evaluation of correlation functions. The latter is
found to be, within error bars, the same as in pure systems. Our results
confirm field-theoretical predictions. The conformal anomaly is calculated
from the leading finite-width correction to the averaged free energy on strips.
Estimates thus obtained are consistent with , the same as for the pure
Ising model.Comment: RevTeX 3, 11 pages +2 figures, uuencoded, IF/UFF preprin
Finite-size investigation of scaling corrections in the square-lattice three-state Potts antiferromagnet square-lattice three-state Potts antiferromagnet
We investigate the finite-temperature corrections to scaling in the
three-state square-lattice Potts antiferromagnet, close to the critical point
at T=0. Numerical diagonalization of the transfer matrix on semi-infinite
strips of width sites, , yields finite-size estimates of
the corresponding scaled gaps, which are extrapolated to . Owing to
the characteristics of the quantities under study, we argue that the natural
variable to consider is x\eta_3=2.00(1)\eta_{{\bf P}_{\rm
stagg}}=3$, corresponding to the staggered polarization.Comment: RevTex4, 5 pages, 2 .eps figures include
Magnetoresistance of Three-Constituent Composites: Percolation Near a Critical Line
Scaling theory, duality symmetry, and numerical simulations of a random
network model are used to study the magnetoresistance of a
metal/insulator/perfect conductor composite with a disordered columnar
microstructure. The phase diagram is found to have a critical line which
separates regions of saturating and non-saturating magnetoresistance. The
percolation problem which describes this line is a generalization of
anisotropic percolation. We locate the percolation threshold and determine the
t = s = 1.30 +- 0.02, nu = 4/3 +- 0.02, which are the same as in
two-constituent 2D isotropic percolation. We also determine the exponents which
characterize the critical dependence on magnetic field, and confirm numerically
that nu is independent of anisotropy. We propose and test a complete scaling
description of the magnetoresistance in the vicinity of the critical line.Comment: Substantially revised version; description of behavior in finite
magnetic fields added. 7 pages, 7 figures, submitted to PR
Extended scaling relations for planar lattice models
It is widely believed that the critical properties of several planar lattice
models, like the Eight Vertex or the Ashkin-Teller models, are well described
by an effective Quantum Field Theory obtained as formal scaling limit. On the
basis of this assumption several extended scaling relations among their indices
were conjectured. We prove the validity of some of them, among which the ones
by Kadanoff, [K], and by Luther and Peschel, [LP].Comment: 32 pages, 7 fi
Phase Transitions Between Topologically Distinct Gapped Phases in Isotropic Spin Ladders
We consider various two-leg ladder models exhibiting gapped phases. All of
these phases have short-ranged valence bond ground states, and they all exhibit
string order. However, we show that short-ranged valence bond ground states
divide into two topologically distinct classes, and as a consequence, there
exist two topologically distinct types of string order. Therefore, not all
gapped phases belong to the same universality class. We show that phase
transitions occur when we interpolate between models belonging to different
topological classes, and we study the nature of these transitions.Comment: 11 pages, 16 postscript figure