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

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

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

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

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

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

Critical Behavior of an Ising System on the Sierpinski Carpet: A Short-Time Dynamics Study

The short-time dynamic evolution of an Ising model embedded in an infinitely ramified fractal structure with noninteger Hausdorff dimension was studied using Monte Carlo simulations. Completely ordered and disordered spin configurations were used as initial states for the dynamic simulations. In both cases, the evolution of the physical observables follows a power-law behavior. Based on this fact, the complete set of critical exponents characteristic of a second-order phase transition was evaluated. Also, the dynamic exponent $\theta$ of the critical initial increase in magnetization, as well as the critical temperature, were computed. The exponent $\theta$ exhibits a weak dependence on the initial (small) magnetization. On the other hand, the dynamic exponent $z$ shows a systematic decrease when the segmentation step is increased, i.e., when the system size becomes larger. Our results suggest that the effective noninteger dimension for the second-order phase transition is noticeably smaller than the Hausdorff dimension. Even when the behavior of the magnetization (in the case of the ordered initial state) and the autocorrelation (in the case of the disordered initial state) with time are very well fitted by power laws, the precision of our simulations allows us to detect the presence of a soft oscillation of the same type in both magnitudes that we attribute to the topological details of the generating cell at any scale.Comment: 10 figures, 4 tables and 14 page

Topological Effects caused by the Fractal Substrate on the Nonequilibrium Critical Behavior of the Ising Magnet

The nonequilibrium critical dynamics of the Ising magnet on a fractal substrate, namely the Sierpinski carpet with Hausdorff dimension $d_H$ =1.7925, has been studied within the short-time regime by means of Monte Carlo simulations. The evolution of the physical observables was followed at criticality, after both annealing ordered spin configurations (ground state) and quenching disordered initial configurations (high temperature state), for three segmentation steps of the fractal. The topological effects become evident from the emergence of a logarithmic periodic oscillation superimposed to a power law in the decay of the magnetization and its logarithmic derivative and also from the dependence of the critical exponents on the segmentation step. These oscillations are discussed in the framework of the discrete scale invariance of the substrate and carefully characterized in order to determine the critical temperature of the second-order phase transition and the critical exponents corresponding to the short-time regime. The exponent $\theta$ of the initial increase in the magnetization was also obtained and the results suggest that it would be almost independent of the fractal dimension of the susbstrate, provided that $d_H$ is close enough to d=2.Comment: 9 figures, 3 tables, 10 page

Quantifying pCO₂ in biological ocean acidification experiments: a comparison of four methods

Quantifying the amount of carbon dioxide (CO₂) in seawater is an essential component of ocean acidification research; however, equipment for measuring CO₂ directly can be costly and involve complex, bulky apparatus. Consequently, other parameters of the carbonate system, such as pH and total alkalinity (AT), are often measured and used to calculate the partial pressure of CO₂ (pCO₂) in seawater, especially in biological CO₂-manipulation studies, including large ecological experiments and those conducted at field sites. Here we compare four methods of pCO₂ determination that have been used in biological ocean acidification experiments: 1) Versatile INstrument for the Determination of Total inorganic carbon and titration Alkalinity (VINDTA) measurement of dissolved inorganic carbon (CT) and AT, 2) spectrophotometric measurement of pHT and AT, 3) electrode measurement of pHNBS and AT, and 4) the direct measurement of CO₂ using a portable CO₂ equilibrator with a non-dispersive infrared (NDIR) gas analyser. In this study, we found these four methods can produce very similar pCO₂ estimates, and the three methods often suited to field-based application (spectrophotometric pHT, electrode pHNBS and CO₂ equilibrator) produced estimated measurement uncertainties of 3.5–4.6% for pCO₂. Importantly, we are not advocating the replacement of established methods to measure seawater carbonate chemistry, particularly for high-accuracy quantification of carbonate parameters in seawater such as open ocean chemistry, for real-time measures of ocean change, nor for the measurement of small changes in seawater pCO₂. However, for biological CO₂-manipulation experiments measuring differences of over 100 μatm pCO₂ among treatments, we find the four methods described here can produce similar results with careful use

Identification of hydrocarbons in chalk reservoirs from surface seismic data: South Arne field, North Sea

Seismic data are mainly used to map out structures in the subsurface, but are also increasingly used to detect differences in porosity and in the fluids that occupy the pore space in sedimentary rocks. Hydrocarbons are generally lighter than brine, and the bulk density and sonic velocity (speed of pressure waves or P-wave velocity) of hydrocarbon-bearing sedimentary rocks are therefore reduced compared to non-reservoir rocks. However, sound is transmitted in different wave forms through the rock, and the shear velocity (speed of shear waves or S-wave velocity) is hardly affected by the density of the pore fluid. In order to detect the presence of hydrocarbons from seismic data, it is thus necessary to investigate how porosity and pore fluids affect the acoustic properties of a sedimentary rock. Much previous research has focused on describing such effects in sandstone (see Mavko et al. 1998), and only in recent years have corresponding studies on the rock physics of chalk appeared (e.g. Walls et al. 1998; Røgen 2002; Fabricius 2003; Gommesen 2003; Japsen et al. 2004). In the North Sea, chalk of the Danian Ekofisk Formation and the Maastrichtian Tor Formation are important reservoir rocks. More information could no doubt be extracted from seismic data if the fundamental physical properties of chalk were better understood. The presence of gas in chalk is known to cause a phase reversal in the seismic signal (Megson 1992), but the presence of oil in chalk has only recently been demonstrated to have an effect on surface seismic data (Japsen et al. 2004). The need for a better link between chalk reservoir parameters and geophysical observations has, however, strongly increased since the discovery of the Halfdan field proved major reserves outside four-way dip closures (Jacobsen et al. 1999; Vejbæk & Kristensen 2000)