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B. M. McCoy

Publications and source records attributed to B. M. McCoy.

At least 19 recordsLinked to original sources
arXiv2022

Factorization of Ising correlations C(M,N) for $ ν= \, -k$ and M+N odd, $M \le N$, $T < T_c$ and their lambda extensions

We study the factorizations of Ising low-temperature correlations C(M,N) for $ν=-k$ and M+N odd, $M \le N$, for both the cases $M\neq 0$ where there are two factors, and $M=0$ where there are four factors. We find that the two factors for $ M \neq 0$ satisfy the same non-linear differential equation and, similarly, for M=0 the four factors each satisfy Okamoto sigma-form of Painlevé VI equations with the same Okamoto parameters. Using a Landen transformation we show, for $M\neq 0$, that the previous non-linear differential equation can actually be reduced to an Okamoto sigma-form of Painlevé VI equation. For both the two and four factor case, we find that there is a one parameter family of boundary conditions on the Okamoto sigma-form of Painlevé VI equations which generalizes the factorization of the correlations C(M,N) to an additive decomposition of the corresponding sigma's solutions of the Okamoto sigma-form of Painlevé VI equation which we call lambda extensions. At a special value of the parameter, the lambda-extensions of the factors of C(M,N) reduce to homogeneous polynomials in the complete elliptic functions of the first and second kind. We also generalize some Tracy-Widom (Painlevé V) relations between the sum and difference of sigma's to this Painlevé VI framework.

math-ph↗
arXiv2020

The Ising correlation $C(M,N)$ for $ν=-k$

We present Painlev{é} VI sigma form equations for the general Ising low and high temperature two-point correlation functions $ C(M,N)$ with $M \leq N $ in the special case $ν= -k$ where $ν= \, \sinh 2E_h/k_BT/\sinh 2E_v/k_BT$. More specifically four different non-linear ODEs depending explicitly on the two integers $M $ and $N$ emerge: these four non-linear ODEs correspond to distinguish respectively low and high temperature, together with $ M+N$ even or odd. These four different non-linear ODEs are also valid for $M \ge N$ when $ ν= -1/k$. For the low-temperature row correlation functions $ C(0,N)$ with $ N$ odd, we exhibit again for this selected $ν= \, -k$ condition, a remarkable phenomenon of a Painlevé VI sigma function being the sum of four Painlevé VI sigma functions having the same Okamoto parameters. We show in this $ν= \, -k$ case for $ T < T_c $ and also $ T > T_c$, that $ C(M,N)$ with $ M \leq N $ is given as an $ N \times N$ Toeplitz determinant.

math-ph↗
arXiv2017

Analyticity of the Ising susceptibility: An interpretation

We discuss the implications of studies of partition function zeros and equimodular curves for the analytic properties of the Ising model on a square lattice in a magnetic field. In particular we consider the dense set of singularities in the susceptibility of the Ising model at $H=0$ found by Nickel and its relation to the analyticity of the field theory computations of Fonseca and Zamolodchikov.

math-ph↗
arXiv2016

The anisotropic Ising correlations as elliptic integrals: duality and differential equations

We present the reduction of the correlation functions of the Ising model on the anisotropic square lattice to complete elliptic integrals of the first, second and third kind, the extension of Kramers-Wannier duality to anisotropic correlation functions, and the linear differential equations for these anisotropic correlations. More precisely, we show that the anisotropic correlation functions are homogeneous polynomials of the complete elliptic integrals of the first, second and third kind. We give the exact dual transformation matching the correlation functions and the dual correlation functions. We show that the linear differential operators annihilating the general two-point correlation functions are factorised in a very simple way, in operators of decreasing orders.

math-ph↗
arXiv2014

Integrability vs non-integrability: Hard hexagons and hard squares compared

In this paper we compare the integrable hard hexagon model with the non-integrable hard squares model by means of partition function roots and transfer matrix eigenvalues. We consider partition functions for toroidal, cylindrical, and free-free boundary conditions up to sizes $40\times40$ and transfer matrices up to 30 sites. For all boundary conditions the hard squares roots are seen to lie in a bounded area of the complex fugacity plane along with the universal hard core line segment on the negative real fugacity axis. The density of roots on this line segment matches the derivative of the phase difference between the eigenvalues of largest (and equal) moduli and exhibits much greater structure than the corresponding density of hard hexagons. We also study the special point $z=-1$ of hard squares where all eigenvalues have unit modulus, and we give several conjectures for the value at $z=-1$ of the partition functions.

math-ph↗
arXiv2013

Hard hexagon partition function for complex fugacity

We study the analyticity of the partition function of the hard hexagon model in the complex fugacity plane by computing zeros and transfer matrix eigenvalues for large finite size systems. We find that the partition function per site computed by Baxter in the thermodynamic limit for positive real values of the fugacity is not sufficient to describe the analyticity in the full complex fugacity plane. We also obtain a new algebraic equation for the low density partition function per site.

math-ph↗
arXiv2012

The importance of the Ising model

Understanding the relationship which integrable (solvable) models, all of which possess very special symmetry properties, have with the generic non-integrable models that are used to describe real experiments, which do not have the symmetry properties, is one of the most fundamental open questions in both statistical mechanics and quantum field theory. The importance of the two-dimensional Ising model in a magnetic field is that it is the simplest system where this relationship may be concretely studied. We here review the advances made in this study, and concentrate on the magnetic susceptibility which has revealed an unexpected natural boundary phenomenon. When this is combined with the Fermionic representations of conformal characters, it is suggested that the scaling theory, which smoothly connects the lattice with the correlation length scale, may be incomplete for $H \neq 0$.

math-ph↗
arXiv2011

Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi-Yau equations

We give the exact expressions of the partial susceptibilities $χ^{(3)}_d$ and $χ^{(4)}_d$ for the diagonal susceptibility of the Ising model in terms of modular forms and Calabi-Yau ODEs, and more specifically, $_3F_2([1/3,2/3,3/2],\, [1,1];\, z)$ and $_4F_3([1/2,1/2,1/2,1/2],\, [1,1,1]; \, z)$ hypergeometric functions. By solving the connection problems we analytically compute the behavior at all finite singular points for $χ^{(3)}_d$ and $χ^{(4)}_d$. We also give new results for $χ^{(5)}_d$. We see in particular, the emergence of a remarkable order-six operator, which is such that its symmetric square has a rational solution. These new exact results indicate that the linear differential operators occurring in the $n$-fold integrals of the Ising model are not only "Derived from Geometry" (globally nilpotent), but actually correspond to "Special Geometry" (homomorphic to their formal adjoint). This raises the question of seeing if these "special geometry" Ising-operators, are "special" ones, reducing, in fact systematically, to (selected, k-balanced, ...) $_{q+1}F_q$ hypergeometric functions, or correspond to the more general solutions of Calabi-Yau equations.

math-ph↗
arXiv2011

Factorization of the Ising model form factors

We present a general method for analytically factorizing the n-fold form factor integrals $f^{(n)}_{N,N}(t)$ for the correlation functions of the Ising model on the diagonal in terms of the hypergeometric functions $_2F_1([1/2,N+1/2];[N+1];t)$ which appear in the form factor $f^{(1)}_{N,N}(t)$. New quadratic recursion and quartic identities are obtained for the form factors for n=2,3. For n= 2,3,4 explicit results are given for the form factors. These factorizations are proved for all N for n= 2,3. These results yield the emergence of palindromic polynomials canonically associated with elliptic curves. As a consequence, understanding the form factors amounts to describing and understanding an infinite set of palindromic polynomials, canonically associated with elliptic curves. From an analytical viewpoint the relation of these palindromic polynomials with hypergeometric functions associated with elliptic curves is made very explicitly, and from a differential algebra viewpoint this corresponds to the emergence of direct sums of differential operators homomorphic to symmetric powers of a second order operator associated with elliptic curve.

math-ph↗
arXiv2010

The energy density of an Ising half plane lattice

We compute the energy density at arbitrary temperature of the half plane Ising lattice with a boundary magnetic field $H_b$ at a distance $M$ rows from the boundary and compare limiting cases of the exact expression with recent calculations at $T=T_c$ done by means of discrete complex analysis methods.

math-ph↗
arXiv2010

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 $q$ and the modulus $k$ are compared and contrasted. The $λ$ generalized correlations $C(M,N;λ)$ are defined and explicitly computed in terms of theta functions for $M=N=0,1$.

math-ph↗
arXiv2007

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 $χ_{d}^{(1)}$ and $χ_{d}^{(2)}$ of the corresponding susceptibility, and obtain linear differential equations for the three and four particle contributions, as well as the five particle contribution $χ^{(5)}_d(t)$, 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 $ n$-particle contributions $χ_{d}^{(n)}$ are quite directly inherited from the direct sum structure on the form factors $ f^{(n)}$. We show that the $ n^{th}$ particle contributions $χ_{d}^{(n)}$ have their singularities at roots of unity. These singularities become dense on the unit circle $|\sinh2E_v/kT \sinh 2E_h/kT|=1$ as $ n\to \infty$.

math-ph↗
arXiv2007

Fuchs versus Painlevé

We briefly recall the Fuchs-Painlevé elliptic representation of Painlevé VI. We then show that the polynomiality of the expressions of the correlation functions (and form factors) in terms of the complete elliptic integral of the first and second kind, $ K$ and $ E$, is a straight consequence of the fact that the differential operators corresponding to the entries of Toeplitz-like determinants, are equivalent to the second order operator $ L_E$ which has $ E$ as solution (or, for off-diagonal correlations to the direct sum of $ L_E$ and $ d/dt$). We show that this can be generalized, mutatis mutandis, to the anisotropic Ising model. The singled-out second order linear differential operator $ L_E$ being replaced by an isomonodromic system of two third-order linear partial differential operators associated with $ Π_1$, the Jacobi's form of the complete elliptic integral of the third kind (or equivalently two second order linear partial differential operators associated with Appell functions, where one of these operators can be seen as a deformation of $ L_E$). We finally explore the generalizations, to the anisotropic Ising models, of the links we made, in two previous papers, between Painlevé non-linear ODE's, Fuchsian linear ODE's and elliptic curves. In particular the elliptic representation of Painlevé VI has to be generalized to an ``Appellian'' representation of Garnier systems.

math-ph↗
arXiv2006

Holonomy of the Ising model form factors

We study the Ising model two-point diagonal correlation function $ C(N,N)$ 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 $λ$, the $j$-particle contributions, $ f^{(j)}_{N,N}$. The corresponding $ λ$ extension of the two-point diagonal correlation function, $ C(N,N; λ)$, is shown, for arbitrary $λ$, to be a solution of the sigma form of the Painlev{é} VI equation introduced by Jimbo and Miwa. Linear differential equations for the form factors $ f^{(j)}_{N,N}$ 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 $ E$. Each $ f^{(j)}_{N,N}$ is expressed polynomially in terms of the elliptic integrals $ E$ and $ K$. The scaling limit of these differential operators breaks the direct sum structure but not the ``Russian doll'' structure. The previous $ λ$-extensions, $ C(N,N; λ)$ are, for singled-out values $ λ= \cos(πm/n)$ ($m, n$ integers), also solutions of linear differential equations. These solutions of Painlevé VI are actually algebraic functions, being associated with modular curves.

math-ph↗
arXiv2006

Painleve versus Fuchs

The sigma form of the Painlev{é} VI equation contains four arbitrary parameters and generically the solutions can be said to be genuinely ``nonlinear'' because they do not satisfy linear differential equations of finite order. However, when there are certain restrictions on the four parameters there exist one parameter families of solutions which do satisfy (Fuchsian) differential equations of finite order. We here study this phenomena of Fuchsian solutions to the Painlev{é} equation with a focus on the particular PVI equation which is satisfied by the diagonal correlation function C(N,N) of the Ising model. We obtain Fuchsian equations of order $N+1$ for C(N,N) and show that the equation for C(N,N) is equivalent to the $N^{th}$ symmetric power of the equation for the elliptic integral $E$. We show that these Fuchsian equations correspond to rational algebraic curves with an additional Riccati structure and we show that the Malmquist Hamiltonian $p,q$ variables are rational functions in complete elliptic integrals. Fuchsian equations for off diagonal correlations $C(N,M)$ are given which extend our considerations to discrete generalizations of Painlev{é}.

math-ph↗
arXiv2005

Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions

We evaluate the virial coefficients B_k for k<=10 for hard spheres in dimensions D=2,...,8. Virial coefficients with k even are found to be negative when D>=5. This provides strong evidence that the leading singularity for the virial series lies away from the positive real axis when D>=5. Further analysis provides evidence that negative virial coefficients will be seen for some k>10 for D=4, and there is a distinct possibility that negative virial coefficients will also eventually occur for D=3.

cond-mat.stat-mech↗