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.Comment: 12 pages. Final version which will be published in J. Stat. Phy

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

The Q-operator and Functional Relations of the Eight-vertex Model at Root-of-unity η=2mKN\eta = \frac{2m K}{N} for odd N

Following Baxter's method of producing Q_{72}-operator, we construct the Q-operator of the root-of-unity eight-vertex model for the crossing parameter $\eta = \frac{2m K}{N}$ with odd $N$ where Q_{72} does not exist. We use this new Q-operator to study the functional relations in the Fabricius-McCoy comparison between the root-of-unity eight-vertex model and the superintegrable N-state chiral Potts model. By the compatibility of the constructed Q-operator with the structure of Baxter's eight-vertex (solid-on-solid) SOS model, we verify the set of functional relations of the root-of-unity eight-vertex model using the explicit form of the Q-operator and fusion weights of SOS model.Comment: Latex 28 page; Typos corrected, minor changes in presentation, References added and updated-Journal versio

The stellar populations of the central region of M31

We continue the analysis of the dataset of our spectroscopic observation campaign of M31, by deriving simple stellar population properties (age metallicity and alpha-elements overabundance) from the measurement of Lick/IDS absorption line indices. We describe their two-dimensional maps taking into account the dust distribution in M31. 80\% of the values of our age measurements are larger than 10 Gyr. The central 100 arcsec of M31 are dominated by the stars of the classical bulge of M31. They are old (11-13 Gyr), metal-rich (as high as [Z/H]~0.35 dex) at the center with a negative gradient outwards and enhanced in alpha-elements ([alpha/Fe]~ 0.28+- 0.01 dex). The bar stands out in the metallicity map, where an almost solar value of [Z/H] (~0.02+-0.01 dex) with no gradient is observed along the bar position angle (55.7 deg) out to 600 arcsec from the center. In contrast, no signature of the bar is seen in the age and [alpha/Fe] maps, that are approximately axisymmetric, delivering a mean age and overabundance for the bar and the boxy-peanut bulge of 10-13 Gyr and 0.25-0.27 dex, respectively. The boxy/peanut-bulge has almost solar metallicity (-0.04+- 0.01 dex). The mass-to-light ratio of the three components is approximately constant at M/LV ~ 4.4-4.7 Msol/Lsol. The disk component at larger distances is made of a mixture of stars, as young as 3-4 Gyr, with solar metallicity and smaller M/LV (~3+-0.1 Msol/Lsol). We propose a two-phase formation scenario for the inner region of M31, where most of the stars of the classical bulge come into place together with a proto-disk, where a bar develops and quickly transforms it into a boxy-peanut bulge. Star formation continues in the bulge region, producing stars younger than 10 Gyr, in particular along the bar, enhancing its metallicity. The disk component appears to build up on longer time-scales.Comment: Language-edited version, Accepted for publication in A&

XXZ Bethe states as highest weight vectors of the sl2sl_2 loop algebra at roots of unity

We show that every regular Bethe ansatz eigenvector of the XXZ spin chain at roots of unity is a highest weight vector of the $sl_2$ loop algebra, for some restricted sectors with respect to eigenvalues of the total spin operator $S^Z$, and evaluate explicitly the highest weight in terms of the Bethe roots. We also discuss whether a given regular Bethe state in the sectors generates an irreducible representation or not. In fact, we present such a regular Bethe state in the inhomogeneous case that generates a reducible Weyl module. Here, we call a solution of the Bethe ansatz equations which is given by a set of distinct and finite rapidities {\it regular Bethe roots}. We call a nonzero Bethe ansatz eigenvector with regular Bethe roots a {\it regular Bethe state}.Comment: 40pages; revised versio