Searcharxiv⌕ Search

arXiv subjects

Barry M. McCoy

Publications and source records attributed to Barry M. McCoy.

At least 19 recordsLinked to original sources
arXiv2011

The Romance of the Ising Model

The essence of romance is mystery. In this talk, given in honor of the 60th birthday of Michio Jimbo, I will explore the meaning of this for the Ising model beginning in 1946 with Bruria Kaufman and Willis Lamb, continuing with the wedding by Jimbo and Miwa in 1980 of the Ising model with the Painlev{é} VI equation which had been first discovered by Picard in 1889. I will conclude with the current fascination of the magnetic susceptibility and explore some of the mysteries still outstanding.

math-ph↗
arXiv2008

New Q matrices and their functional equations for the eight vertex model at elliptic roots of unity

The Q matrix invented by Baxter in 1972 to solve the eight vertex model at roots of unity exists for all values of N, the number of sites in the chain, but only for a subset of roots of unity. We show in this paper that a new Q matrix, which has recently been introduced and is non zero only for N even, exists for all roots of unity. In addition we consider the relations between all of the known Q matrices of the eight vertex model and conjecture functional equations for them.

cond-mat.stat-mech↗
arXiv2007

The Eight Vertex Model.New results

Whereas the tools to determine the eigenvalues of the eight-vertex transfer matrix T are well known there has been until recently incomplete knowledge about the eigenvectors of T. We describe the construction of eigenvectors of T corresponding to degenerate eigenvalues and discuss the related hidden elliptic symmetry.

cond-mat.stat-mech↗
arXiv2007

The TQ equation of the 8 vertex model for complex elliptic roots of unity

We extend our studies of the TQ equation introduced by Baxter in his 1972 solution of the 8 vertex model with parameter $η$ given by $2Lη=2m_1K+im_2K'$ from $m_2=0$ to the more general case of complex $η.$ We find that there are several different cases depending on the parity of $m_1$ and $m_2$.

cond-mat.stat-mech↗
arXiv2007

New Developments in the Eight Vertex Model II. Chains of odd length

We study the transfer matrix of the 8 vertex model with an odd number of lattice sites $N.$ For systems at the root of unity points $η=mK/L$ with $m$ odd the transfer matrix is known to satisfy the famous ``$TQ$'' equation where ${\bf Q}(v)$ is a specifically known matrix. We demonstrate that the location of the zeroes of this ${\bf Q}(v)$ matrix is qualitatively different from the case of even $N$ and in particular they satisfy a previously unknown equation which is more general than what is often called ``Bethe's equation.'' For the case of even $m$ where no ${\bf Q}(v)$ matrix is known we demonstrate that there are many states which are not obtained from the formalism of the SOS model but which do satisfy the $TQ$ equation. The ground state for the particular case of $η=2K/3$ and $N$ odd is investigated in detail.

cond-mat.stat-mech↗
arXiv2006

An elliptic current operator for the 8 vertex model

We compute the operator which creates the missing degenerate states in the algebraic Bethe ansatz of the 8 vertex model at roots of unity and relate it to the concept of an elliptic current operator. We find that in sharp contrast with the corresponding formalism in the six-vertex model at roots of unity the current operator is not nilpotent with the consequence that in the construction of degenerate eigenstates of the transfer matrix an arbitrary number of exact strings can be added to the set of regular Bethe roots. Thus the original set of free parameters {s,t} of an eigenvector of T is enlarged to become {s,t,λ_{c,1}, ..., λ_{c,n}\} with arbitrary string centers λ_{c,j} and arbitrary n.

cond-mat.stat-mech↗
arXiv2004

Root of unity symmetries in the 8 and 6 vertex models

We review the recently discovered symmetries of the 8 and 6 vertex models which exist at roots of unity and present their relation with representation theory of affine Lie algebras, Drinfeld polynomials and Bethe vectors.

cond-mat.stat-mech↗
arXiv2003

Functional Equations and Fusion Matrices for the Eight Vertex Model

We derive sets of functional equations for the eight vertex model by exploiting an analogy with the functional equations of the chiral Potts model. From these equations we show that the fusion matrices have special reductions at certain roots of unity. We explicitly exhibit these reductions for the 3,4 and 5 order fusion matrices and compare our formulation with the algebra of Sklyanin.

cond-mat.stat-mech↗
arXiv2003

Finite size corrections for the Ising model on higher genus triangular lattices

We study the topology dependence of finite size corrections to the Ising model partition function by considering the model on a triangular lattice embedded on a genus two surface. At criticality we observe a universal shape dependent correction, expressible in terms of Riemann theta functions, that reproduces the modular invariant partition function of the corresponding conformal field theory. The period matrix characterizing the moduli parameters of the limiting Riemann surface is obtained by a numerical study of the lattice continuum limit. The same results are reproduced using a discrete holomorphic structure.

cond-mat.stat-mech↗
arXiv2002

New Developments in the Eight Vertex Model

We demonstrate that the Q matrix introduced in Baxter's 1972 solution of the eight vertex model has some eigenvectors which are not eigenvectors of the spin reflection operator and conjecture a new functional equation for Q(v) which both contains the Bethe equation that gives the eigenvalues of the transfer matrix and computes the degeneracies of these eigenvalues.

cond-mat.stat-mech↗
arXiv2001

Dimers and the Critical Ising Model on Lattices of genus>1

We study the partition function of both Close-Packed Dimers and the Critical Ising Model on a square lattice embedded on a genus two surface. Using numerical and analytical methods we show that the determinants of the Kasteleyn adjacency matrices have a dependence on the boundary conditions that, for large lattice size, can be expressed in terms of genus two theta functions. The period matrix characterizing the continuum limit of the lattice is computed using a discrete holomorphic structure. These results relate in a direct way the lattice combinatorics with conformal field theory, providing new insight to the lattice regularization of conformal field theories on higher genus Riemann Surfaces.

hep-th↗
arXiv2001

Evaluation parameters and Bethe roots for the six vertex model at roots of unity

We propose an expression for the current form of the lowering operator of the $ {sl}_2$ loop algebra symmetry of the six vertex model (XXZ spin chain) at roots of unity. This operator has poles which correspond to the evaluation parameters of representation theory which are given as the roots of the Drinfeld polynomial. We explicitly compute these polynomials in terms of the Bethe roots which characterize the highest weight states for all values of $S^z$. From these polynomials we find that the Bethe roots satisfy sum rules for each value of S^z.

cond-mat.stat-mech↗
arXiv2001

Loop symmetry of integrable vertex models at roots of unity

It has been recently discovered in the context of the six vertex or XXZ model in the fundamental representation that new symmetries arise when the anisotropy parameter $(q+q^{-1})/2$ is evaluated at roots of unity $q^{N}=1$. These new symmetries have been linked to an $U(A^{(1)}_1)$ invariance of the transfer matrix and the corresponding spin-chain Hamiltonian.In this paper these results are generalized for odd primitive roots of unity to all vertex models associated with trigonometric solutions of the Yang-Baxter equation by invoking representation independent methods which only take the algebraic structure of the underlying quantum groups $U_q(\hat g)$ into account. Here $\hat g$ is an arbitrary Kac-Moody algebra. Employing the notion of the boost operator it is then found that the Hamiltonian and the transfer matrix of the integrable model are invariant under the action of $U(\hat{g})$. For the simplest case $\hat g=A_1^{(1)}$ the discussion is also extended to even primitive roots of unity.

hep-th↗
arXiv2001

Completing Bethe's equations at roots of unity

In a previous paper we demonstrated that Bethe's equations are not sufficient to specify the eigenvectors of the XXZ model at roots of unity for states where the Hamiltonian has degenerate eigenvalues. We here find the equations which will complete the specification of the eigenvectors in these degenerate cases and present evidence that the $sl_2$ loop algebra symmetry is sufficiently powerful to determine that the highest weight of each irreducible representation is given by Bethe's ansatz.

cond-mat.stat-mech↗