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

Temperature dependent spatial oscillations in the correlations of the XXZ spin chain

We study the correlation $$ for the XXZ chain in the massless attractive (ferromagnetic) region at positive temperatures by means of a numerical study of the quantum transfer matrix. We find that there is a range of temperature where the behavior of the correlation for large separations is oscillatory with an incommensurate period which depends on temperature.Comment: 4 pages, REVTEX, 6 table

The TQ equation of the 8 vertex model for complex elliptic roots of unity

We extend our studies of the TQ equation introduced by Baxter in his 1972 solution of the 8 vertex model with parameter $\eta$ given by $2L\eta=2m_1K+im_2K'$ from $m_2=0$ to the more general case of complex $\eta.$ We find that there are several different cases depending on the parity of $m_1$ and $m_2$.Comment: 30 pages, LATE

The Q-operator for Root-of-Unity Symmetry in Six Vertex Model

We construct the explicit $Q$-operator incorporated with the $sl_2$-loop-algebra symmetry of the six-vertex model at roots of unity. The functional relations involving the $Q$-operator, the six-vertex transfer matrix and fusion matrices are derived from the Bethe equation, parallel to the Onsager-algebra-symmetry discussion in the superintegrable $N$-state chiral Potts model. We show that the whole set of functional equations is valid for the $Q$-operator. Direct calculations in certain cases are also given here for clearer illustration about the nature of the $Q$-operator in the symmetry study of root-of-unity six-vertex model from the functional-relation aspect.Comment: Latex 26 Pages; Typos and small errors corrected, Some explanations added for clearer presentation, References updated-Journal version with modified labelling of sections and formula

An elliptic current operator for the 8 vertex model

We compute the operator which creates the missing degenerate states in the algebraic Bethe ansatz of the 8 vertex model at roots of unity and relate it to the concept of an elliptic current operator. We find that in sharp contrast with the corresponding formalism in the six-vertex model at roots of unity the current operator is not nilpotent with the consequence that in the construction of degenerate eigenstates of the transfer matrix an arbitrary number of exact strings can be added to the set of regular Bethe roots. Thus the original set of free parameters {s,t} of an eigenvector of T is enlarged to become {s,t,\lambda_{c,1}, ..., \lambda_{c,n}\} with arbitrary string centers \lambda_{c,j} and arbitrary n.Comment: 16 pages, Latex typographic errors corrected, text added, reference added, accepted by Journal of Physics A,Mathematical and Genera

Injunction Against Prosecution of Divorce Actions in Other States

Aims: The formation scenario of extended counter-rotating stellar disks in galaxies is still debated. In this paper, we study the S0 galaxy IC 719 known to host two large-scale counter-rotating stellar disks in order to investigate their formation mechanism. Methods: We exploit the large field of view and wavelength coverage of the Multi Unit Spectroscopic Explorer (MUSE) spectrograph to derive two-dimensional (2D) maps of the various properties of the counter-rotating stellar disks, such as age, metallicity, kinematics, spatial distribution, the kinematical and chemical properties of the ionized gas, and the dust map. Results: Due to the large wavelength range, and in particular to the presence of the Calcium Triplet \u3bb\u3bb8498, 8542, 8662 \uc5 (CaT hereafter), the spectroscopic analysis allows us to separate the two stellar components in great detail. This permits precise measurement of both the velocity and velocity dispersion of the two components as well as their spatial distribution. We derived a 2D map of the age and metallicity of the two stellar components, as well as the star formation rate and gas-phase metallicity from the ionized gas emission maps. Conclusions: The main stellar disk of the galaxy is kinematically hotter, older, thicker and with larger scale-length than the secondary disk. There is no doubt that the latter is strongly linked to the ionized gas component: they have the same kinematics and similar vertical and radial spatial distribution. This result is in favor of a gas accretion scenario over a binary merger scenario to explain the origin of counter-rotation in IC 719. One source of gas that may have contributed to the accretion process is the cloud that surrounds IC 719

Thermodynamical Properties of a Spin 1/2 Heisenberg Chain Coupled to Phonons

We performed a finite-temperature quantum Monte Carlo simulation of the one-dimensional spin-1/2 Heisenberg model with nearest-neighbor interaction coupled to Einstein phonons. Our method allows to treat easily up to 100 phonons per site and the results presented are practically free from truncation errors. We studied in detail the magnetic susceptibility, the specific heat, the phonon occupation, the dimerization, and the spin-correlation function for various spin-phonon couplings and phonon frequencies. In particular we give evidence for the transition from a gapless to a massive phase by studying the finite-size behavior of the susceptibility. We also show that the dimerization is proportional to $g^2/\Omega$ for $T<2J$.Comment: 10 pages, 17 Postscript Figure