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

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 $sl_2$ loop algebra symmetry if the $q$ parameter is given by a root of unity, $q_0^{2N}=1$, for an integer $N$. 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 ${\bar d}_k^{\pm}$, which leads to evaluation parameters $a_j$. 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

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

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

High-speed shear driven dynamos. Part 2. Numerical analysis

This paper aims to numerically verify the large Reynolds number asymptotic theory of magneto-hydrodynamic (MHD) flows proposed in the companion paper Deguchi (2019). To avoid any complexity associated with the chaotic nature of turbulence and flow geometry, nonlinear steady solutions of the viscous-resistive magneto-hydrodynamic equations in plane Couette flow have been utilised. Two classes of nonlinear MHD states, which convert kinematic energy to magnetic energy effectively, have been determined. The first class of nonlinear states can be obtained when a small spanwise uniform magnetic field is applied to the known hydrodynamic solution branch of the plane Couette flow. The nonlinear states are characterised by the hydrodynamic/magnetic roll-streak and the resonant layer at which strong vorticity and current sheets are observed. These flow features, and the induced strong streamwise magnetic field, are fully consistent with the vortex/Alfv\'en wave interaction theory proposed in Deguchi (2019). When the spanwise uniform magnetic field is switched off, the solutions become purely hydrodynamic. However, the second class of `self-sustained shear driven dynamos' at the zero-external magnetic field limit can be found by homotopy via the forced states subject to a spanwise uniform current field. The discovery of the dynamo states has motivated the corresponding large Reynolds number matched asymptotic analysis in Deguchi (2019). Here, the reduced equations derived by the asymptotic theory have been solved numerically. The asymptotic solution provides remarkably good predictions for the finite Reynolds number dynamo solutions

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

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 $E_{\tau, \eta}(sl_2)$ at the discrete coupling constants: $2N \eta = m_1 + i m_2 \tau$, where $N, m_1$ and $m_2$ are integers. Then we show that degenerate eigenvectors of the transfer matrix of the six-vertex model at roots of unity in the sector $S^Z \equiv 0$ (mod $N$) are derived from those of the 8V CSOS model, through the trigonometric limit. They are associated with the complete $N$ strings. From the result we see that the dimension of a given degenerate eigenspace in the sector $S^Z \equiv 0$ (mod $N$) of the six-vertex model at $N$th roots of unity is given by $2^{2S_{max}^Z/N}$, where $S_{max}^Z$ is the maximal value of the total spin operator $S^Z$ in the degenerate eigenspace.Comment: 7 pages, no figure, conference proceeding

Atiyah-Singer Index Theorem in an SO(3) Yang-Mills-Higgs system and derivation of a charge quantization condition

The Atiyah-Singer index theorem is generalized to a two-dimensional SO(3) Yang-Mills-Higgs (YMH) system. The generalized theorem is proven by using the heat kernel method and a nonlinear realization of SU(2) gauge symmetry. This theorem is applied to the problem of deriving a charge quantization condition in the four-dimensional SO(3) YMH system with non-Abelian monopoles. The resulting quantization condition, eg=n (n: integer), for an electric charge e and a magnetic charge g is consistent with that found by Arafune, Freund and Goebel. It is shown that the integer n is half of the index of a Dirac operator.Comment: 18pages, no figures, minor corrections, published versio