8,857 research outputs found
On the Degenerate Multiplicity of the 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 loop
algebra symmetry if the parameter is given by a root of unity,
, for an integer . 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 , which
leads to evaluation parameters . 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 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 at the discrete
coupling constants: , where and are
integers. Then we show that degenerate eigenvectors of the transfer matrix of
the six-vertex model at roots of unity in the sector (mod )
are derived from those of the 8V CSOS model, through the trigonometric limit.
They are associated with the complete strings. From the result we see that
the dimension of a given degenerate eigenspace in the sector
(mod ) of the six-vertex model at th roots of unity is given by
, where is the maximal value of the total spin
operator in the degenerate eigenspace.Comment: 7 pages, no figure, conference proceeding
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
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