11 research outputs found
An Analytic Formula for the A₂ Jack Polynomials
In this letter I shall review my joint results with Vadim Kuznetsov and Evgeny Sklyanin [Indag. Math. 14 (2003), 451-482] on separation of variables (SoV) for the An Jack polynomials. This approach originated from the work [RIMS Kokyuroku 919 (1995), 27-34] where the integral representations for the A2 Jack polynomials was derived. Using special polynomial bases I shall obtain a more explicit expression for the A2 Jack polynomials in terms of generalised hypergeometric functions
Particles and strings in a (2+1)-D integrable quantum model
We give a review of some recent work on generalization of the Bethe ansatz in
the case of -dimensional models of quantum field theory. As such a model,
we consider one associated with the tetrahedron equation, i.e. the
-dimensional generalization of the famous Yang--Baxter equation. We
construct some eigenstates of the transfer matrix of that model. There arise,
together with states composed of point-like particles analogous to those in the
usual -dimensional Bethe ansatz, new string-like states and
string-particle hybrids
Some exact results for the three-layer Zamolodchikov model
In this paper we continue the study of the three-layer Zamolodchikov model
started in our previous works. We analyse numerically the solutions to the
Bethe ansatz equations. We consider two regimes I and II which differ by the
signs of the spherical sides (a1,a2,a3)->(-a1,-a2,-a3). We accept the two-line
hypothesis for the regime I and the one-line hypothesis for the regime II. In
the thermodynamic limit we derive integral equations for distribution densities
and solve them exactly. We calculate the partition function for the three-layer
Zamolodchikov model and check a compatibility of this result with the
functional relations. We also do some numerical checkings of our results.Comment: LaTeX, 27 pages, 9 figure
On exact solution of a classical 3D integrable model
We investigate some classical evolution model in the discrete 2+1 space-time.
A map, giving an one-step time evolution, may be derived as the compatibility
condition for some systems of linear equations for a set of auxiliary linear
variables. Dynamical variables for the evolution model are the coefficients of
these systems of linear equations. Determinant of any system of linear
equations is a polynomial of two numerical quasimomenta of the auxiliary linear
variables. For one, this determinant is the generating functions of all
integrals of motion for the evolution, and on the other hand it defines a high
genus algebraic curve. The dependence of the dynamical variables on the
space-time point (exact solution) may be expressed in terms of theta functions
on the jacobian of this curve. This is the main result of our paper
Spin chains with dynamical lattice supersymmetry
Spin chains with exact supersymmetry on finite one-dimensional lattices are
considered. The supercharges are nilpotent operators on the lattice of
dynamical nature: they change the number of sites. A local criterion for the
nilpotency on periodic lattices is formulated. Any of its solutions leads to a
supersymmetric spin chain. It is shown that a class of special solutions at
arbitrary spin gives the lattice equivalents of the N=(2,2) superconformal
minimal models. The case of spin one is investigated in detail: in particular,
it is shown that the Fateev-Zamolodchikov chain and its off-critical extension
admits a lattice supersymmetry for all its coupling constants. Its
supersymmetry singlets are thoroughly analysed, and a relation between their
components and the weighted enumeration of alternating sign matrices is
conjectured.Comment: Revised version, 52 pages, 2 figure
The eight-vertex model and lattice supersymmetry
We show that the XYZ spin chain along the special line of couplings
J_xJ_y+J_xJ_z+J_yJ_z=0 possesses a hidden N=(2,2) supersymmetry. This lattice
supersymmetry is non-local and changes the number of sites. It extends to the
full transfer matrix of the corresponding eight-vertex model. In particular, it
is shown how to derive the supercharges from Baxter's Bethe ansatz. This
analysis leads to new conjectures concerning the ground state for chains of odd
length. We also discuss a correspondence between the spectrum of this XYZ chain
and that of a manifestly supersymmetric staggered fermion chain.Comment: 40 pages, 6 figures, Tik
The Ising Susceptibility Scaling Function
We have dramatically extended the zero field susceptibility series at both
high and low temperature of the Ising model on the triangular and honeycomb
lattices, and used these data and newly available further terms for the square
lattice to calculate a number of terms in the scaling function expansion around
both the ferromagnetic and, for the square and honeycomb lattices, the
antiferromagnetic critical point.Comment: PDFLaTeX, 50 pages, 5 figures, zip file with series coefficients and
background data in Maple format provided with the source files. Vs2: Added
dedication and made several minor additions and corrections. Vs3: Minor
corrections. Vs4: No change to eprint. Added essential square-lattice series
input data (used in the calculation) that were removed from University of
Melbourne's websit