58 research outputs found
Analyticity and integrability in the chiral Potts model
We study the perturbation theory for the general non-integrable chiral Potts model depending on two chiral angles and a strength parameter and show how the analyticity of the ground state energy and correlation functions dramatically increases when the angles and the strength parameter satisfy the integrability condition. We further specialize to the superintegrable case and verify that a sum rule is obeyed
Critical behaviour of the two-dimensional Ising susceptibility
We report computations of the short-distance and the long-distance (scaling)
contributions to the square-lattice Ising susceptibility in zero field close to
T_c. Both computations rely on the use of nonlinear partial difference
equations for the correlation functions. By summing the correlation functions,
we give an algorithm of complexity O(N^6) for the determination of the first N
series coefficients. Consequently, we have generated and analysed series of
length several hundred terms, generated in about 100 hours on an obsolete
workstation. In terms of a temperature variable, \tau, linear in T/T_c-1, the
short-distance terms are shown to have the form \tau^p(ln|\tau|)^q with p>=q^2.
To O(\tau^14) the long-distance part divided by the leading \tau^{-7/4}
singularity contains only integer powers of \tau. The presence of irrelevant
variables in the scaling function is clearly evident, with contributions of
distinct character at leading orders |\tau|^{9/4} and |\tau|^{17/4} being
identified.Comment: 11 pages, REVTex
The saga of the Ising susceptibility
We review developments made since 1959 in the search for a closed form for
the susceptibility of the Ising model. The expressions for the form factors in
terms of the nome and the modulus are compared and contrasted. The
generalized correlations are defined and explicitly
computed in terms of theta functions for .Comment: 19 pages, 1 figur
Correction induced by irrelevant operators in the correlators of the 2d Ising model in a magnetic field
We investigate the presence of irrelevant operators in the 2d Ising model
perturbed by a magnetic field, by studying the corrections induced by these
operators in the spin-spin correlator of the model. To this end we perform a
set of high precision simulations for the correlator both along the axes and
along the diagonal of the lattice. By comparing the numerical results with the
predictions of a perturbative expansion around the critical point we find
unambiguous evidences of the presence of such irrelevant operators. It turns
out that among the irrelevant operators the one which gives the largest
correction is the spin 4 operator T^2 + \bar T^2 which accounts for the
breaking of the rotational invariance due to the lattice. This result agrees
with what was already known for the correlator evaluated exactly at the
critical point and also with recent results obtained in the case of the thermal
perturbation of the model.Comment: 28 pages, no figure
On the magnetic perturbation of the Ising model on the sphere
In this letter we will extend the analysis given by Al. Zamolodchikov for the
scaling Yang-Lee model on the sphere to the Ising model in a magnetic field. A
numerical study of the partition function and of the vacuum expectation values
(VEV) is done by using the truncated conformal space (TCS) approach. Our
results strongly suggest that the partition function is an entire function of
the coupling constant.Comment: 8 pages, 1 figure, revised version, references adde
Universal ratios of critical amplitudes in the Potts model universality class
Monte Carlo (MC) simulations and series expansions (SE) data for the energy,
specific heat, magnetization, and susceptibility of the three-state and
four-state Potts model and Baxter-Wu model on the square lattice are analyzed
in the vicinity of the critical point in order to estimate universal
combinations of critical amplitudes. We also form effective ratios of the
observables close to the critical point and analyze how they approach the
universal critical-amplitude ratios. In particular, using the duality relation,
we show analytically that for the Potts model with a number of states ,
the effective ratio of the energy critical amplitudes always approaches unity
linearly with respect to the reduced temperature. This fact leads to the
prediction of relations among the amplitudes of correction-to-scaling terms of
the specific heat in the low- and high-temperature phases. It is a common
belief that the four-state Potts and the Baxter-Wu model belong to the same
universality class. At the same time, the critical behavior of the four-state
Potts model is modified by logarithmic corrections while that of the Baxter-Wu
model is not. Numerical analysis shows that critical amplitude ratios are very
close for both models and, therefore, gives support to the hypothesis that the
critical behavior of both systems is described by the same renormalization
group fixed point.Comment: Talk presented at CCP 2008, Ouro Preto, 5-9 August 200
Square lattice Ising model susceptibility: Series expansion method and differential equation for
In a previous paper (J. Phys. A {\bf 37} (2004) 9651-9668) we have given the
Fuchsian linear differential equation satisfied by , the
``three-particle'' contribution to the susceptibility of the isotropic square
lattice Ising model. This paper gives the details of the calculations (with
some useful tricks and tools) allowing one to obtain long series in polynomial
time. The method is based on series expansion in the variables that appear in
the -dimensional integrals representing the -particle contribution to
the isotropic square lattice Ising model susceptibility . The
integration rules are straightforward due to remarkable formulas we derived for
these variables. We obtain without any numerical approximation as
a fully integrated series in the variable , where , with the conventional Ising model coupling constant. We also
give some perspectives and comments on these results.Comment: 28 pages, no figur
Holonomy of the Ising model form factors
We study the Ising model two-point diagonal correlation function by
presenting an exponential and form factor expansion in an integral
representation which differs from the known expansion of Wu, McCoy, Tracy and
Barouch. We extend this expansion, weighting, by powers of a variable
, the -particle contributions, . The corresponding
extension of the two-point diagonal correlation function, , is shown, for arbitrary , to be a solution of the sigma
form of the Painlev{\'e} VI equation introduced by Jimbo and Miwa. Linear
differential equations for the form factors are obtained and
shown to have both a ``Russian doll'' nesting, and a decomposition of the
differential operators as a direct sum of operators equivalent to symmetric
powers of the differential operator of the elliptic integral . Each is expressed polynomially in terms of the elliptic integrals and . The scaling limit of these differential operators breaks the
direct sum structure but not the ``Russian doll'' structure. The previous -extensions, are, for singled-out values ( integers), also solutions of linear differential
equations. These solutions of Painlev\'e VI are actually algebraic functions,
being associated with modular curves.Comment: 39 page
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