Spin operator matrix elements in the superintegrable chiral Potts quantum chain

We derive spin operator matrix elements between general eigenstates of the superintegrable Z_N-symmetric chiral Potts quantum chain of finite length. Our starting point is the extended Onsager algebra recently proposed by R.Baxter. For each pair of spaces (Onsager sectors) of the irreducible representations of the Onsager algebra, we calculate the spin matrix elements between the eigenstates of the Hamiltonian of the quantum chain in factorized form, up to an overall scalar factor. This factor is known for the ground state Onsager sectors. For the matrix elements between the ground states of these sectors we perform the thermodynamic limit and obtain the formula for the order parameters. For the Ising quantum chain in a transverse field (N=2 case) the factorized form for the matrix elements coincides with the corresponding expressions obtained recently by the Separation of Variables Method.Comment: 24 pages, 1 figur

The order parameter of the chiral Potts model

An outstanding problem in statistical mechanics is the order parameter of the chiral Potts model. An elegant conjecture for this was made in 1983. It has since been successfully tested against series expansions, but as far as the author is aware there is as yet no proof of the conjecture. Here we show that if one makes a certain analyticity assumption similar to that used to derive the free energy, then one can indeed verify the conjecture. The method is based on the ``broken rapidity line'' approach pioneered by Jimbo, Miwa and Nakayashiki.Comment: 29 pages, 7 figures. Citations made more explicit and some typos correcte

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.Comment: 20 pages, 2 Postscript figure

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

Analyticity and Integrabiity in the Chiral Potts Model

We study the perturbation theory for the general non-integrable chiral Potts model depending on two chiral angles and a strength parameter and show how the analyticity of the ground state energy and correlation functions dramatically increases when the angles and the strength parameter satisfy the integrability condition. We further specialize to the superintegrable case and verify that a sum rule is obeyed.Comment: 31 pages in harvmac including 9 tables, several misprints eliminate

Construction of some missing eigenvectors of the XYZ spin chain at the discrete coupling constants and the exponentially large spectral degeneracy of the transfer matrix

We discuss an algebraic method for constructing eigenvectors of the transfer matrix of the eight vertex model at the discrete coupling parameters. We consider the algebraic Bethe ansatz of the elliptic quantum group $E_{\tau, \eta}(sl_2)$ for the case where the parameter $\eta$ satisfies $2 N \eta = m_1 + m_2 \tau$ for arbitrary integers $N$, $m_1$ and $m_2$. When $m_1$ or $m_2$ is odd, the eigenvectors thus obtained have not been discussed previously. Furthermore, we construct a family of degenerate eigenvectors of the XYZ spin chain, some of which are shown to be related to the $sl_2$ loop algebra symmetry of the XXZ spin chain. We show that the dimension of some degenerate eigenspace of the XYZ spin chain on $L$ sites is given by $N 2^{L/N}$, if $L/N$ is an even integer. The construction of eigenvectors of the transfer matrices of some related IRF models is also discussed.Comment: 19 pages, no figure (revisd version with three appendices

Eigenvectors of Baxter-Bazhanov-Stroganov \tau^{(2)}(t_q) model with fixed-spin boundary conditions

The aim of this contribution is to give the explicit formulas for the eigenvectors of the transfer-matrix of Baxter-Bazhanov-Stroganov (BBS) model (N-state spin model) with fixed-spin boundary conditions. These formulas are obtained by a limiting procedure from the formulas for the eigenvectors of periodic BBS model. The latter formulas were derived in the framework of the Sklyanin's method of separation of variables. In the case of fixed-spin boundaries the corresponding T-Q Baxter equations for the functions of separated variables are solved explicitly. As a particular case we obtain the eigenvectors of the Hamiltonian of Ising-like Z_N quantum chain model.Comment: 14 pages, paper submitted to Proceedings of the International Workshop "Classical and Quantum Integrable Systems" (Dubna, January, 2007

Nonequilibrium Forces Between Neutral Atoms Mediated by a Quantum Field

We study all known and as yet unknown forces between two neutral atoms, modeled as three dimensional harmonic oscillators, arising from mutual influences mediated by an electromagnetic field but not from their direct interactions. We allow as dynamical variables the center of mass motion of the atom, its internal degrees of freedom and the quantum field treated relativistically. We adopt the method of nonequilibrium quantum field theory which can provide a first principle, systematic and unified description including the intrinsic field fluctuations and induced dipole fluctuations. The inclusion of self-consistent back-actions makes possible a fully dynamical description of these forces valid for general atom motion. In thermal equilibrium we recover the known forces -- London, van der Waals and Casimir-Polder forces -- between neutral atoms in the long-time limit but also discover the existence of two new types of interatomic forces. The first, a `nonequilibrium force', arises when the field and atoms are not in thermal equilibrium, and the second, which we call an `entanglement force', originates from the correlations of the internal degrees of freedom of entangled atoms.Comment: 16 pages, 2 figure

Comment on `Series expansions from the corner transfer matrix renormalization group method: the hard-squares model'

Earlier this year Chan extended the low-density series for the hard-squares partition function $\kappa(z)$ to 92 terms. Here we analyse this extended series focusing on the behaviour at the dominant singularity $z_d$ which lies on on the negative fugacity axis. We find that the series has a confluent singularity of order 2 at $z_d$ with exponents $\theta=0.83333(2)$ and $\theta'= 1.6676(3)$. We thus confirm that the exponent $\theta$ has the exact value $\frac56$ as observed by Dhar.Comment: 5 pages, 1 figure, IoP macros. Expanded second and final versio

Scattering Rule in Soliton Cellular Automaton associated with Crystal Base of Uq(D4(3))U_q(D_4^{(3)})

In terms of the crystal base of a quantum affine algebra $U_q(\mathfrak{g})$, we study a soliton cellular automaton (SCA) associated with the exceptional affine Lie algebra $\mathfrak{g}=D_4^{(3)}$. The solitons therein are labeled by the crystals of quantum affine algebra $U_q(A_1^{(1)})$. The scatteing rule is identified with the combinatorial $R$ matrix for $U_q(A_1^{(1)})$-crystals. Remarkably, the phase shifts in our SCA are given by {\em 3-times} of those in the well-known box-ball system.Comment: 25 page