8,857 research outputs found

    On the Degenerate Multiplicity of the sl2sl_2 Loop Algebra for the 6V Transfer Matrix at Roots of Unity

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    We review the main result of cond-mat/0503564. The Hamiltonian of the XXZ spin chain and the transfer matrix of the six-vertex model has the sl2sl_2 loop algebra symmetry if the qq parameter is given by a root of unity, q02N=1q_0^{2N}=1, for an integer NN. We discuss the dimensions of the degenerate eigenspace generated by a regular Bethe state in some sectors, rigorously as follows: We show that every regular Bethe ansatz eigenvector in the sectors is a highest weight vector and derive the highest weight dˉk±{\bar d}_k^{\pm}, which leads to evaluation parameters aja_j. If the evaluation parameters are distinct, we obtain the dimensions of the highest weight representation generated by the regular Bethe state.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

    The 8V CSOS model and the sl2sl_2 loop algebra symmetry of the six-vertex model at roots of unity

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    We review an algebraic method for constructing degenerate eigenvectors of the transfer matrix of the eight-vertex Cyclic Solid-on-Solid lattice model (8V CSOS model), where the degeneracy increases exponentially with respect to the system size. We consider the elliptic quantum group Eτ,η(sl2)E_{\tau, \eta}(sl_2) at the discrete coupling constants: 2Nη=m1+im2τ2N \eta = m_1 + i m_2 \tau, where N,m1N, m_1 and m2m_2 are integers. Then we show that degenerate eigenvectors of the transfer matrix of the six-vertex model at roots of unity in the sector SZ0S^Z \equiv 0 (mod NN) are derived from those of the 8V CSOS model, through the trigonometric limit. They are associated with the complete NN strings. From the result we see that the dimension of a given degenerate eigenspace in the sector SZ0S^Z \equiv 0 (mod NN) of the six-vertex model at NNth roots of unity is given by 22SmaxZ/N2^{2S_{max}^Z/N}, where SmaxZS_{max}^Z is the maximal value of the total spin operator SZS^Z in the degenerate eigenspace.Comment: 7 pages, no figure, conference proceeding

    High-speed shear driven dynamos. Part 1. Asymptotic analysis

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    Rational large Reynolds number matched asymptotic expansions of three-dimensional nonlinear magneto-hydrodynamic (MHD) states are concerned. The nonlinear MHD states, assumed to be predominantly driven by a unidirectional shear, can be sustained without any linear instability of the base flow and hence are responsible for subcritical transition to turbulence. Two classes of nonlinear MHD states are found. The first class of nonlinear states emerged out of a nice combination of the purely hydrodynamic vortex/wave interaction theory by Hall \& Smith (1991) and the resonant absorption theories on Alfv\'en waves, developed in the solar physics community (e.g. Sakurai et al. 1991; Goossens et al. 1995). Similar to the hydrodynamic theory, the mechanism of the MHD states can be explained by the successive interaction of the roll, streak, and wave fields, which are now defined both for the hydrodynamic and magnetic fields. The derivation of this `vortex/Alfv\'en wave interaction' state is rather straightforward as the scalings for both of the hydrodynamic and magnetic fields are identical. It turns out that the leading order magnetic field of the asymptotic states appears only when a small external magnetic field is present. However, it does not mean that purely shear-driven dynamos are not possible. In fact, the second class of `self-sustained shear driven dynamo theory' shows the magnetic generation that is slightly smaller size in the absence of any external field. Despite small size, the magnetic field causes the novel feedback mechanism in the velocity field through resonant absorption, wherein the magnetic wave becomes more strongly amplified than the hydrodynamic counterpart
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