N. Iorgov

Painlevé VI connection problem and monodromy of c = 1 conformal blocks

A bstractGeneric c = 1 four-point conformal blocks on the Riemann sphere can be seen as the coefficients of Fourier expansion of the tau function of Painlevé VI equation with respect to one of its integration constants. Based on this relation, we show that c = 1 fusion matrix essentially coincides with the connection coefficient relating tau function asymptotics at different critical points. Explicit formulas for both quantities are obtained by solving certain functional relations which follow from the tau function expansions. The final result does not involve integration and is given by a ratio of two products of Barnes G-functions with arguments expressed in terms of conformal dimensions/monodromy data. It turns out to be closely related to the volume of hyperbolic tetrahedron.

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.

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.

Spin operator matrix elements in the quantum Ising chain: fermion approach

Using some modification of the standard fermion technique we derive factorized formulas for spin operator matrix elements (form factors) between general eigenstates of the Hamiltonian of the quantum Ising chain in a transverse field of finite length. The derivation is based on the approach recently used to derive factorized formulas for ZN-spin operator matrix elements between ground eigenstates of the Hamiltonian of the ZN-symmetric superintegrable chiral Potts quantum chain. The obtained factorized formulas for the matrix elements of the Ising chain coincide with the corresponding expressions obtained by the separation of variables method.