26 research outputs found
The 3-d Random Field Ising Model at zero temperature
We study numerically the zero temperature Random Field Ising Model on cubic
lattices of various linear sizes in three dimensions. For each random field
configuration we vary the ferromagnetic coupling strength . We find that in
the infinite volume limit the magnetization is discontinuous in . The energy
and its first derivative are continuous. The approch to the thermodynamic
limit is slow, behaving like with for the gaussian
distribution of the random field. We also study the bimodal distribution , and we find similar results for the magnetization but with a
different value of the exponent . This raises the question of the
validity of universality for the random field problem.Comment: 8 pages, 3 PostScript Figure
Disorder driven phase transitions of the large q-state Potts model in 3d
Phase transitions induced by varying the strength of disorder in the large-q
state Potts model in 3d are studied by analytical and numerical methods. By
switching on the disorder the transition stays of first order, but different
thermodynamical quantities display essential singularities. Only for strong
enough disorder the transition will be soften into a second-order one, in which
case the ordered phase becomes non-homogeneous at large scales, while the
non-correlated sites percolate the sample. In the critical regime the critical
exponents are found universal: \beta/\nu=0.60(2) and \nu=0.73(1).Comment: 4 pages; 3 figure
On the complexity of some birational transformations
Using three different approaches, we analyze the complexity of various
birational maps constructed from simple operations (inversions) on square
matrices of arbitrary size. The first approach consists in the study of the
images of lines, and relies mainly on univariate polynomial algebra, the second
approach is a singularity analysis, and the third method is more numerical,
using integer arithmetics. Each method has its own domain of application, but
they give corroborating results, and lead us to a conjecture on the complexity
of a class of maps constructed from matrix inversions
Spectral Properties of Statistical Mechanics Models
The full spectrum of transfer matrices of the general eight-vertex model on a
square lattice is obtained by numerical diagonalization. The eigenvalue spacing
distribution and the spectral rigidity are analyzed. In non-integrable regimes
we have found eigenvalue repulsion as for the Gaussian orthogonal ensemble in
random matrix theory. By contrast, in integrable regimes we have found
eigenvalue independence leading to a Poissonian behavior, and, for some points,
level clustering. These first examples from classical statistical mechanics
suggest that the conjecture of integrability successfully applied to quantum
spin systems also holds for classical systems.Comment: 4 pages, 1 Revtex file and 4 postscript figures tarred, gzipped and
uuencode
Symmetry, complexity and multicritical point of the two-dimensional spin glass
We analyze models of spin glasses on the two-dimensional square lattice by
exploiting symmetry arguments. The replicated partition functions of the Ising
and related spin glasses are shown to have many remarkable symmetry properties
as functions of the edge Boltzmann factors. It is shown that the applications
of homogeneous and Hadamard inverses to the edge Boltzmann matrix indicate
reduced complexities when the elements of the matrix satisfy certain
conditions, suggesting that the system has special simplicities under such
conditions. Using these duality and symmetry arguments we present a conjecture
on the exact location of the multicritical point in the phase diagram.Comment: 32 pages, 6 figures; a few typos corrected. To be published in J.
Phys.
A classification of four-state spin edge Potts models
We classify four-state spin models with interactions along the edges
according to their behavior under a specific group of symmetry transformations.
This analysis uses the measure of complexity of the action of the symmetries,
in the spirit of the study of discrete dynamical systems on the space of
parameters of the models, and aims at uncovering solvable ones. We find that
the action of these symmetries has low complexity (polynomial growth, zero
entropy). We obtain natural parametrizations of various models, among which an
unexpected elliptic parametrization of the four-state chiral Potts model, which
we use to localize possible integrability conditions associated with high genus
curves.Comment: 5 figure
Post-critical set and non existence of preserved meromorphic two-forms
We present a family of birational transformations in depending on
two, or three, parameters which does not, generically, preserve meromorphic
two-forms. With the introduction of the orbit of the critical set (vanishing
condition of the Jacobian), also called ``post-critical set'', we get some new
structures, some "non-analytic" two-form which reduce to meromorphic two-forms
for particular subvarieties in the parameter space. On these subvarieties, the
iterates of the critical set have a polynomial growth in the \emph{degrees of
the parameters}, while one has an exponential growth out of these subspaces.
The analysis of our birational transformation in is first carried out
using Diller-Favre criterion in order to find the complexity reduction of the
mapping. The integrable cases are found. The identification between the
complexity growth and the topological entropy is, one more time, verified. We
perform plots of the post-critical set, as well as calculations of Lyapunov
exponents for many orbits, confirming that generically no meromorphic two-form
can be preserved for this mapping. These birational transformations in ,
which, generically, do not preserve any meromorphic two-form, are extremely
similar to other birational transformations we previously studied, which do
preserve meromorphic two-forms. We note that these two sets of birational
transformations exhibit totally similar results as far as topological
complexity is concerned, but drastically different results as far as a more
``probabilistic'' approach of dynamical systems is concerned (Lyapunov
exponents). With these examples we see that the existence of a preserved
meromorphic two-form explains most of the (numerical) discrepancy between the
topological and probabilistic approach of dynamical systems.Comment: 34 pages, 7 figure
Minimum spanning trees on random networks
We show that the geometry of minimum spanning trees (MST) on random graphs is
universal. Due to this geometric universality, we are able to characterise the
energy of MST using a scaling distribution () found using uniform
disorder. We show that the MST energy for other disorder distributions is
simply related to . We discuss the relationship to invasion
percolation (IP), to the directed polymer in a random media (DPRM) and the
implications for the broader issue of universality in disordered systems.Comment: 4 pages, 3 figure
Level Statistics of XXZ Spin Chains with Discrete Symmetries: Analysis through Finite-size Effects
Level statistics is discussed for XXZ spin chains with discrete symmetries
for some values of the next-nearest-neighbor (NNN) coupling parameter. We show
how the level statistics of the finite-size systems depends on the NNN coupling
and the XXZ anisotropy, which should reflect competition among quantum chaos,
integrability and finite-size effects. Here discrete symmetries play a central
role in our analysis. Evaluating the level-spacing distribution, the spectral
rigidity and the number variance, we confirm the correspondence between
non-integrability and Wigner behavior in the spectrum. We also show that
non-Wigner behavior appears due to mixed symmetries and finite-size effects in
some nonintegrable cases.Comment: 19 pages, 6 figure
Random Matrix Theory and higher genus integrability: the quantum chiral Potts model
We perform a Random Matrix Theory (RMT) analysis of the quantum four-state
chiral Potts chain for different sizes of the chain up to size L=8. Our
analysis gives clear evidence of a Gaussian Orthogonal Ensemble statistics,
suggesting the existence of a generalized time-reversal invariance.
Furthermore a change from the (generic) GOE distribution to a Poisson
distribution occurs when the integrability conditions are met. The chiral Potts
model is known to correspond to a (star-triangle) integrability associated with
curves of genus higher than zero or one. Therefore, the RMT analysis can also
be seen as a detector of ``higher genus integrability''.Comment: 23 pages and 10 figure