Quantum Loop Subalgebra and Eigenvectors of the Superintegrable Chiral Potts Transfer Matrices

It has been shown in earlier works that for Q=0 and L a multiple of N, the ground state sector eigenspace of the superintegrable tau_2(t_q) model is highly degenerate and is generated by a quantum loop algebra L(sl_2). Furthermore, this loop algebra can be decomposed into r=(N-1)L/N simple sl_2 algebras. For Q not equal 0, we shall show here that the corresponding eigenspace of tau_2(t_q) is still highly degenerate, but splits into two spaces, each containing 2^{r-1} independent eigenvectors. The generators for the sl_2 subalgebras, and also for the quantum loop subalgebra, are given generalizing those in the Q=0 case. However, the Serre relations for the generators of the loop subalgebra are only proven for some states, tested on small systems and conjectured otherwise. Assuming their validity we construct the eigenvectors of the Q not equal 0 ground state sectors for the transfer matrix of the superintegrable chiral Potts model.Comment: LaTeX 2E document, using iopart.cls with iopams packages. 28 pages, uses eufb10 and eurm10 fonts. Typeset twice! Version 2: Details added, improvements and minor corrections made, erratum to paper 2 included. Version 3: Small paragraph added in introductio

Duality and Symmetry in Chiral Potts Model

We discover an Ising-type duality in the general $N$-state chiral Potts model, which is the Kramers-Wannier duality of planar Ising model when N=2. This duality relates the spectrum and eigenvectors of one chiral Potts model at a low temperature (of small $k'$) to those of another chiral Potts model at a high temperature (of $k'^{-1}$). The $\tau^{(2)}$-model and chiral Potts model on the dual lattice are established alongside the dual chiral Potts models. With the aid of this duality relation, we exact a precise relationship between the Onsager-algebra symmetry of a homogeneous superintegrable chiral Potts model and the $sl_2$-loop-algebra symmetry of its associated spin-$\frac{N-1}{2}$ XXZ chain through the identification of their eigenstates.Comment: Latex 34 pages, 2 figures; Typos and misprints in Journal version are corrected with minor changes in expression of some formula

Eigenvectors in the Superintegrable Model I: sl_2 Generators

In order to calculate correlation functions of the chiral Potts model, one only needs to study the eigenvectors of the superintegrable model. Here we start this study by looking for eigenvectors of the transfer matrix of the periodic tau_2(t)model which commutes with the chiral Potts transfer matrix. We show that the degeneracy of the eigenspace of tau_2(t) in the Q=0 sector is 2^r, with r=(N-1)L/N when the size of the transfer matrix L is a multiple of N. We introduce chiral Potts model operators, different from the more commonly used generators of quantum group U-tilde_q(sl-hat(2)). From these we can form the generators of a loop algebra L(sl(2)). For this algebra, we then use the roots of the Drinfeld polynomial to give new explicit expressions for the generators representing the loop algebra as the direct sum of r copies of the simple algebra sl(2).Comment: LaTeX 2E document, 11 pages, 1 eps figure, using iopart.cls with graphicx and iopams packages. v2: Appended text to title, added acknowledgments and made several minor corrections v3: Added reference, eliminated ambiguity, corrected a few misprint

New Results for the Correlation Functions of the Ising Model and the Transverse Ising Chain

In this paper we show how an infinite system of coupled Toda-type nonlinear differential equations derived by one of us can be used efficiently to calculate the time-dependent pair-correlations in the Ising chain in a transverse field. The results are seen to match extremely well long large-time asymptotic expansions newly derived here. For our initial conditions we use new long asymptotic expansions for the equal-time pair correlation functions of the transverse Ising chain, extending an old result of T.T. Wu for the 2d Ising model. Using this one can also study the equal-time wavevector-dependent correlation function of the quantum chain, a.k.a. the q-dependent diagonal susceptibility in the 2d Ising model, in great detail with very little computational effort.Comment: LaTeX 2e, 31 pages, 8 figures (16 eps files). vs2: Two references added and minor changes of style. vs3: Corrections made and reference adde

The Onsager Algebra Symmetry of τ(j)\tau^{(j)}-matrices in the Superintegrable Chiral Potts Model

We demonstrate that the $\tau^{(j)}$-matrices in the superintegrable chiral Potts model possess the Onsager algebra symmetry for their degenerate eigenvalues. The Fabricius-McCoy comparison of functional relations of the eight-vertex model for roots of unity and the superintegrable chiral Potts model has been carefully analyzed by identifying equivalent terms in the corresponding equations, by which we extract the conjectured relation of $Q$-operators and all fusion matrices in the eight-vertex model corresponding to the $T\hat{T}$-relation in the chiral Potts model.Comment: Latex 21 pages; Typos added, References update

On τ(2)\tau^{(2)}-model in Chiral Potts Model and Cyclic Representation of Quantum Group Uq(sl2)U_q(sl_2)

We identify the precise relationship between the five-parameter $\tau^{(2)}$-family in the $N$-state chiral Potts model and XXZ chains with $U_q (sl_2)$-cyclic representation. By studying the Yang-Baxter relation of the six-vertex model, we discover an one-parameter family of $L$-operators in terms of the quantum group $U_q (sl_2)$. When $N$ is odd, the $N$-state $\tau^{(2)}$-model can be regarded as the XXZ chain of $U_{\sf q} (sl_2)$ cyclic representations with ${\sf q}^N=1$. The symmetry algebra of the $\tau^{(2)}$-model is described by the quantum affine algebra $U_{\sf q} (\hat{sl}_2)$ via the canonical representation. In general for an arbitrary $N$, we show that the XXZ chain with a $U_q (sl_2)$-cyclic representation for $q^{2N}=1$ is equivalent to two copies of the same $N$-state $\tau^{(2)}$-model.Comment: Latex 11 pages; Typos corrected, Minor changes for clearer presentation, References added and updated-Journal versio

Factorized finite-size Ising model spin matrix elements from Separation of Variables

Using the Sklyanin-Kharchev-Lebedev method of Separation of Variables adapted to the cyclic Baxter--Bazhanov--Stroganov or $\tau^{(2)}$-model, we derive factorized formulae for general finite-size Ising model spin matrix elements, proving a recent conjecture by Bugrij and Lisovyy

Bethe Equation of τ(2)\tau^{(2)}-model and Eigenvalues of Finite-size Transfer Matrix of Chiral Potts Model with Alternating Rapidities

We establish the Bethe equation of the $\tau^{(2)}$-model in the $N$-state chiral Potts model (including the degenerate selfdual cases) with alternating vertical rapidities. The eigenvalues of a finite-size transfer matrix of the chiral Potts model are computed by use of functional relations. The significance of the "alternating superintegrable" case of the chiral Potts model is discussed, and the degeneracy of $\tau^{(2)}$-model found as in the homogeneous superintegrable chiral Potts model.Comment: Latex 25 pages; Typos corrected, Minor changes for clearer presentation, References added-Journal versio

Determination of the parameters of a Skyrme type effective interaction using the simulated annealing approach

We implement for the first time the simulated annealing method (SAM) to the problem of searching for the global minimum in the hyper-surface of the chi-square function which depends on the values of the parameters of a Skyrme type effective nucleon-nucleon interaction. We undertake a realistic case of fitting the values of the Skyrme parameters to an extensive set of experimental data on the ground state properties of many nuclei ranging from normal to exotic ones. The set of experimental data used in our fitting procedure includes the radii for the valence $1d_{5/2}$ and $1f_{7/2}$ neutron orbits in the $^{17}$O and $^{41}$Ca nuclei, respectively, and the breathing mode energies for several nuclei, in addition to the typically used data on binding energy, charge radii and spin-orbit splitting. We also include in the fit the critical density $\rho_{cr}$ and further constrain the values of the Skyrme parameters by requiring that (i) the quantity $P = 3\rho \frac{dS}{d\rho}$, directly related to the slope of the symmetry energy $S$, must be positive for densities up to $3\rho_0$ (ii) the enhancement factor $\kappa$, associated with the isovector giant dipole resonance, should lie in the range of $0.1 - 0.5$ and (iii) the Landau parameter $G_0^\prime$ is positive at $\rho = \rho_0$. We provide simple but consistent schemes to account for the center of mass corrections to the binding energy and charge radii.Comment: 33 pages, 4 figures, Phys. Rev. C (in press