2,171 research outputs found
Random Matrix Theory and Classical Statistical Mechanics. I. Vertex Models
A connection between integrability properties and general statistical
properties of the spectra of symmetric transfer matrices of the asymmetric
eight-vertex model is studied using random matrix theory (eigenvalue spacing
distribution and spectral rigidity). For Yang-Baxter integrable cases,
including free-fermion solutions, we have found a Poissonian behavior, whereas
level repulsion close to the Wigner distribution is found for non-integrable
models. For the asymmetric eight-vertex model, however, the level repulsion can
also disappearand the Poisson distribution be recovered on (non Yang--Baxter
integrable) algebraic varieties, the so-called disorder varieties. We also
present an infinite set of algebraic varieties which are stable under the
action of an infinite discrete symmetry group of the parameter space. These
varieties are possible loci for free parafermions. Using our numerical
criterion we have tested the generic calculability of the model on these
algebraic varieties.Comment: 25 pages, 7 PostScript Figure
Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals
Lattice statistical mechanics, often provides a natural (holonomic) framework
to perform singularity analysis with several complex variables that would, in a
general mathematical framework, be too complex, or could not be defined.
Considering several Picard-Fuchs systems of two-variables "above" Calabi-Yau
ODEs, associated with double hypergeometric series, we show that holonomic
functions are actually a good framework for actually finding the singular
manifolds. We, then, analyse the singular algebraic varieties of the n-fold
integrals , corresponding to the decomposition of the magnetic
susceptibility of the anisotropic square Ising model. We revisit a set of
Nickelian singularities that turns out to be a two-parameter family of elliptic
curves. We then find a first set of non-Nickelian singularities for and , that also turns out to be rational or ellipic
curves. We underline the fact that these singular curves depend on the
anisotropy of the Ising model. We address, from a birational viewpoint, the
emergence of families of elliptic curves, and of Calabi-Yau manifolds on such
problems. We discuss the accumulation of these singular curves for the
non-holonomic anisotropic full susceptibility.Comment: 36 page
Multicritical Points of Potts Spin Glasses on the Triangular Lattice
We predict the locations of several multicritical points of the Potts spin
glass model on the triangular lattice. In particular, continuous multicritical
lines, which consist of multicritical points, are obtained for two types of
two-state Potts (i.e., Ising) spin glasses with two- and three-body
interactions on the triangular lattice. These results provide us with numerous
examples to further verify the validity of the conjecture, which has succeeded
in deriving highly precise locations of multicritical points for several spin
glass models. The technique, called the direct triangular duality, a variant of
the ordinary duality transformation, directly relates the triangular lattice
with its dual triangular lattice in conjunction with the replica method.Comment: 18 pages, 2, figure
Location of the Multicritical Point for the Ising Spin Glass on the Triangular and Hexagonal Lattices
A conjecture is given for the exact location of the multicritical point in
the phase diagram of the +/- J Ising model on the triangular lattice. The
result p_c=0.8358058 agrees well with a recent numerical estimate. From this
value, it is possible to derive a comparable conjecture for the exact location
of the multicritical point for the hexagonal lattice, p_c=0.9327041, again in
excellent agreement with a numerical study. The method is a variant of duality
transformation to relate the triangular lattice directly with its dual
triangular lattice without recourse to the hexagonal lattice, in conjunction
with the replica method.Comment: 9 pages, 1 figure; Minor corrections in notatio
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
On the Symmetries of Integrability
We show that the Yang-Baxter equations for two dimensional models admit as a
group of symmetry the infinite discrete group . The existence of
this symmetry explains the presence of a spectral parameter in the solutions of
the equations. We show that similarly, for three-dimensional vertex models and
the associated tetrahedron equations, there also exists an infinite discrete
group of symmetry. Although generalizing naturally the previous one, it is a
much bigger hyperbolic Coxeter group. We indicate how this symmetry can help to
resolve the Yang-Baxter equations and their higher-dimensional generalizations
and initiate the study of three-dimensional vertex models. These symmetries are
naturally represented as birational projective transformations. They may
preserve non trivial algebraic varieties, and lead to proper parametrizations
of the models, be they integrable or not. We mention the relation existing
between spin models and the Bose-Messner algebras of algebraic combinatorics.
Our results also yield the generalization of the condition so often
mentioned in the theory of quantum groups, when no parameter is available.Comment: 23 page
The diagonal Ising susceptibility
We use the recently derived form factor expansions of the diagonal two-point
correlation function of the square Ising model to study the susceptibility for
a magnetic field applied only to one diagonal of the lattice, for the isotropic
Ising model.
We exactly evaluate the one and two particle contributions
and of the corresponding susceptibility, and obtain linear
differential equations for the three and four particle contributions, as well
as the five particle contribution , but only modulo a given
prime. We use these exact linear differential equations to show that, not only
the russian-doll structure, but also the direct sum structure on the linear
differential operators for the -particle contributions are
quite directly inherited from the direct sum structure on the form factors .
We show that the particle contributions have their
singularities at roots of unity. These singularities become dense on the unit
circle as .Comment: 18 page
Analytical evidence for the absence of spin glass transition on self-dual lattices
We show strong evidence for the absence of a finite-temperature spin glass
transition for the random-bond Ising model on self-dual lattices. The analysis
is performed by an application of duality relations, which enables us to derive
a precise but approximate location of the multicritical point on the Nishimori
line. This method can be systematically improved to presumably give the exact
result asymptotically. The duality analysis, in conjunction with the
relationship between the multicritical point and the spin glass transition
point for the symmetric distribution function of randomness, leads to the
conclusion of the absence of a finite-temperature spin glass transition for the
case of symmetric distribution. The result is applicable to the random bond
Ising model with or Gaussian distribution and the Potts gauge glass on
the square, triangular and hexagonal lattices as well as the random three-body
Ising model on the triangular and the Union-Jack lattices and the four
dimensional random plaquette gauge model. This conclusion is exact provided
that the replica method is valid and the asymptotic limit of the duality
analysis yields the exact location of the multicritical pointComment: 11 Pages, 4 figures, 1 table. submitted to J. Phys. A Math. Theo
Exact location of the multicritical point for finite-dimensional spin glasses: A conjecture
We present a conjecture on the exact location of the multicritical point in
the phase diagram of spin glass models in finite dimensions. By generalizing
our previous work, we combine duality and gauge symmetry for replicated random
systems to derive formulas which make it possible to understand all the
relevant available numerical results in a unified way. The method applies to
non-self-dual lattices as well as to self dual cases, in the former case of
which we derive a relation for a pair of values of multicritical points for
mutually dual lattices. The examples include the +-J and Gaussian Ising spin
glasses on the square, hexagonal and triangular lattices, the Potts and Z_q
models with chiral randomness on these lattices, and the three-dimensional +-J
Ising spin glass and the random plaquette gauge model.Comment: 27 pages, 3 figure
The Fuchsian differential equation of the square lattice Ising model susceptibility
Using an expansion method in the variables that appear in the
-dimensional integrals representing the -particle contribution to the
Ising square lattice model susceptibility , we generate a long series of
coefficients for the 3-particle contribution , using a
polynomial time algorithm. We give the Fuchsian differential equation of order
seven for that reproduces all the terms of our long series. An
analysis of the properties of this Fuchsian differential equation is performed.Comment: 15 pages, no figures, submitted to J. Phys.
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