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

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

Generalized Supersymmetric Perturbation Theory

Using the basic ingredient of supersymmetry, we develop a simple alternative approach to perturbation theory in one-dimensional non-relativistic quantum mechanics. The formulae for the energy shifts and wave functions do not involve tedious calculations which appear in the available perturbation theories. The model applicable in the same form to both the ground state and excited bound states, unlike the recently introduced supersymmetric perturbation technique which, together with other approaches based on logarithmic perturbation theory, are involved within the more general framework of the present formalism.Comment: 13 pages article in LaTEX (uses standard article.sty). No Figures. Sent to Ann. Physics (2004

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 the duality relation for correlation functions of the Potts model

We prove a recent conjecture on the duality relation for correlation functions of the Potts model for boundary spins of a planar lattice. Specifically, we deduce the explicit expression for the duality of the n-site correlation functions, and establish sum rule identities in the form of the M\"obius inversion of a partially ordered set. The strategy of the proof is by first formulating the problem for the more general chiral Potts model. The extension of our consideration to the many-component Potts models is also given.Comment: 17 pages in RevTex, 5 figures, submitted to J. Phys.

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

Symmetries of Large N Matrix Models for Closed Strings

We obtain the symmetry algebra of multi-matrix models in the planar large N limit. We use this algebra to associate these matrix models with quantum spin chains. In particular, certain multi-matrix models are exactly solved by using known results of solvable spin chain systems.Comment: 12 pages, 1 eps figure, RevTex, some minor typos in the publised version are correcte

Eigenvectors in the Superintegrable Model II: Ground State Sector

In 1993, Baxter gave $2^{m_Q}$ eigenvalues of the transfer matrix of the $N$-state superintegrable chiral Potts model with spin-translation quantum number $Q$, where $m_Q=\lfloor(NL-L-Q)/N\rfloor$. In our previous paper we studied the Q=0 ground state sector, when the size $L$ of the transfer matrix is chosen to be a multiple of $N$. It was shown that the corresponding $\tau_2$ matrix has a degenerate eigenspace generated by the generators of $r=m_0$ simple $sl_2$ algebras. These results enable us to express the transfer matrix in the subspace in terms of these generators $E_m^{\pm}$ and $H_m$ for $m=1,...,r$. Moreover, the corresponding $2^r$ eigenvectors of the transfer matrix are expressed in terms of rotated eigenvectors of $H_m$.Comment: LaTeX 2E document, using iopart.cls with iopams packages. 17 pages, uses eufb10 and eurm10 fonts. Typeset twice! vs2: Many changes and additions, adding 7 pages. vs3: minor corrections. vs4 minor improvement

Logarithmic perturbation theory for quasinormal modes

Logarithmic perturbation theory (LPT) is developed and applied to quasinormal modes (QNMs) in open systems. QNMs often do not form a complete set, so LPT is especially convenient because summation over a complete set of unperturbed states is not required. Attention is paid to potentials with exponential tails, and the example of a Poschl-Teller potential is briefly discussed. A numerical method is developed that handles the exponentially large wavefunctions which appear in dealing with QNMs.Comment: 24 pages, 4 Postscript figures, uses ioplppt.sty and epsfig.st