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"McCoy, Barry M"
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Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions
We evaluate the virial coefficients Bk 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.
Journal Article
Bethe's Equation Is Incomplete for the XXZ Model at Roots of Unity
We demonstrate for the six vertex and XXZ model parameterized by Δ= −(q+q-1)/2≠±1 that when q2N=1 for integer N≥2 the Bethe's ansatz equations determine only the eigenvectors which are the highest weights of the infinite dimensional sl2 loop algebra symmetry group of the model. Therefore in this case the Bethe's ansatz equations are incomplete and further conditions need to be imposed in order to completely specify the wave function. We discuss how the evaluation parameters of the finite dimensional representations of the sl2 loop algebra can be used to complete this specification.
Journal Article
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 sl2 loop algebra symmetry is sufficiently powerful to determine that the highest weight of each irreducible representation is given by Bethe's ansatz.
Journal Article
The sl2 Loop Algebra Symmetry of the Six-Vertex Model at Roots of Unity
We demonstrate that the six vertex model (XXZ spin chain) with Δ=(q+q-1)/2 and q2N=1 has an invariance under the loop algebra of sl2 which produces a special set of degenerate eigenvalues. For Δ=0 we compute the multiplicity of the degeneracies using Jordan–Wigner techniques.
Journal Article
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.
Journal Article
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 pointsη=mK/L with m odd the transfer matrix is known to satisfy the famous ‘‘TQ’’ equation where Q(υ) is a specifically known matrix. We demonstrate that the location of the zeroes of this Q(υ) matrix is qualitatively different from the case of evenN 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 Q(υ) 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.
Journal Article
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.
Journal Article
Finite Size Corrections for the Ising Model on Higher Genus Triangular Lattices
We study the topology dependence of the 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.
Journal Article
Rogers–Schur–Ramanujan Type Identities for the M(p,p′) Minimal Models of Conformal Field Theory
We present and prove Rogers–Schur–Ramanujan (Bose/Fermi) type identities for the Virasoro characters of the minimal model M(p,p′). The proof uses the continued fraction decomposition of p′/p introduced by Takahashi and Suzuki for the study of the Bethe's Ansatz equations of the XXZ model and gives a general method to construct polynomial generalizations of the fermionic form of the characters which satisfy the same recursion relations as the bosonic polynomials of Forrester and Baxter. We use this method to get fermionic representations of the characters for many classes of r and s.
Journal Article