S Pakuliak

The Baxter–Bazhanov–Stroganov model: separation of variables and the Baxter equation

The Baxter–Bazhanov–Stroganov model (also known as the τ(2) model) has attracted much interest because it provides a tool for solving the integrable chiral -Potts model. It can be formulated as a face spin model or via cyclic L-operators. Using the latter formulation and the Sklyanin–Kharchev–Lebedev approach, we give the explicit derivation of the eigenvectors of the component Bn(λ) of the monodromy matrix for the fully inhomogeneous chain of finite length. For the periodic chain, we obtain the Baxter T-Q-equations via separation of variables. The functional relations for the transfer matrices of the τ(2) model guarantee nontrivial solutions to the Baxter equations. For the N = 2 case, which is the free fermion point of a generalized Ising model, the Baxter equations are solved explicitly.

Integrable Structures of Exactly Solvable Two-Dimensional Models of Quantum Field Theory

A new basis for Bethe vectors of the Heisenberg model T.-D. Albert, K. Ruhlig. The form factors and quantum equation of motion in the sine-Gordon model H. Babudjian, M. Karowski. Instantons, Hilbert schemes and integrability H.W. Braden, N.A. Nekrasov. Low-temperature behaviour of 2D lattice SU(2) spin model O. Borisenko, V. Kushnir. Form factor representation of the correlation functions of the two dimensional Ising model on a cylinder A.I. Bugrij. Aspects of integrable quantum field theories with boundaries E. Corrigan. Functional realization of some elliptic Hamiltonian structures and bosonization of the corresponding quantum algebras B.L. Feigin, A.V. Odesskii. Quantized moduli spaces of the bundles on the elliptic curve and their applications B.L. Feigin, A.V. Odesskii. Thermodynamic Bethe ansatz and form factors for the homogeneous sine-Gordon models A. Fring. The superintegrable chiral Potts quantum chain and generalized Chebyshev polynomials G. von Gehlen, S.-S. Roan. Dualities in integrable systems: geometrical aspects A. Gorsky, V. Rubtsov. Integrable evolutionary equations via Lie algebras on hyperelliptic curves P. Holod, T. Skrypnyk. The quantum dilogarithm and Dehn twists in quantum Teichmuller theory R. Kashaev. Unitary representations of the modular and two-particle q-deformed Toda chains S. Kharchev, et al. The Algebraic Bethe Ansatz and the correlation functions of the Heisenberg magnet N.A. Kitanine, N.A. Slavnov. Dual algebras with non-linear Posson brackets A. Korovnichenko, A. Zhedanov. Sine-Gordon solitons vs. relativistis Calogero-Moser particles S.N.M. Ruijsenaars. Integrable three dimensional models in wholly discrete space-time S. Sergeev. Elliptic beta integrals andspecial functions of hypergeometric type V.P. Spiridonov. The 9-vertex model with a special value of the crossing parameter and the related XYZ spin chain Y. Stroganov. Correspondence between the XXZ model in roots of unity and the one-dimensional quantum Ising chain with different boundary conditions R.A. Usmanov. Index. List of the Workshop Participants.

Form-factors in the Baxter–Bazhanov–Stroganov model II: Ising model on the finite lattice

We continue our investigation of the Baxter–Bazhanov–Stroganov or τ(2)-model using the method of separation of variables (von Gehlen et al 2006 J. Phys. A: Math. Gen. 39 7257, 2007 J. Phys. A: Math. Theor. 40 14117). In this paper we derive for the first time the factorized formula for form-factors of the Ising model on a finite lattice conjectured previously by Bugrij and Lisovyy (2003 Phys. Lett. A 319 390, 2003 J. Theor. Math. Phys. 140 987). We also find the matrix elements of the spin operator for the finite quantum Ising chain in a transverse field.Dedicated to Professor Anatoly Bugrij on the occasion of his 60-th birthday

Form-factors in the Baxter-Bazhanov-Stroganov model I: Norms and matrix elements

We continue our investigation of the -Baxter–Bazhanov–Stroganov model using the method of separation of variables [1]. In this paper, we calculate the norms and matrix elements of a local -spin operator between eigenvectors of the auxiliary problem. For the norm the multiple sums over the intermediate states are performed explicitly. In the case N = 2, we solve the Baxter equation and obtain form-factors of the spin operator of the periodic Ising model on a finite lattice.

Spin operator matrix elements in the superintegrable chiral Potts quantum chain

We derive spin operator matrix elements between general eigenstates of the superintegrable ℤN-symmetric chiral Potts quantum chain of finite length. Our starting point is the extended Onsager algebra recently proposed by Baxter. For each pair of spaces (Onsager sectors) of the irreducible representations of the Onsager algebra, we calculate the spin matrix elements between the eigenstates of the Hamiltonian of the quantum chain in factorized form, up to an overall scalar factor. This factor is known for the ground state Onsager sectors. For the matrix elements between the ground states of these sectors we perform the thermodynamic limit and obtain the formula for the order parameters. For the Ising quantum chain in a transverse field (N=2 case) the factorized form for the matrix elements coincides with the corresponding expressions obtained recently by the Separation of Variables method.

Factorized finite-size Ising model spin matrix elements from separation of variables

Using the Sklyanin?Kharchev?Lebedev method of separation of variables adapted to the cyclic Baxter?Bazhanov?Stroganov or the ?(2)-model, we derive factorized formulae for general finite-size Ising model spin matrix elements, proving a recent conjecture by Bugrij and Lisovyy.

Explicit free parametrization of the modified tetrahedron equation

The modified tetrahedron equation (MTE) with affine Weyl quantum variables at the Nth root of unity is solved by a rational mapping operator which is obtained from the solution of a linear problem. We show that the solutions can be parametrized in terms of eight free parameters and 16 discrete phase choices, thus providing a broad starting point for the construction of three-dimensional integrable lattice models. The Fermat-curve points parametrizing the representation of the mapping operator in terms of cyclic functions are expressed in terms of the independent parameters. An explicit formula for the density factor of the MTE is derived. For the example N = 2 we write the MTE in full detail.

THE MODIFIED TETRAHEDRON EQUATION AND ITS SOLUTIONS

We present a pedagogical review of particle physics models that are based on the noncommutativity of space-time,

Theta-function parametrization and fusion for 3D integrable Boltzmann weights

[\hat{x}_\mu,\hat{x}_\nu]=i \theta_{\mu \nu}

3-dimensional Integrable Lattice Models and the Bazhanov-Stroganov Model

, with specific attention to the phenomenology these models predict in particle experiments either in existence or under development. We summarize results obtained for high energy scattering such as would occur for example in a future