G von Gehlen

Zn-symmetric quantum chains with an infinite set of conserved charges and Zn zero modes

Abstract A set of special Z n -symmetric quantum chains is proposed which generalize several of the peculiar properties of the Z 2 -symmetric Ising quantum chain: (i) the models are self-dual and have an infinite number of commuting conserved charges, and (ii) they present zero modes. At the critical point all states are n -fold degenerate, away from the critical point the splitting is linear in the coupling constant.

Conformal invariance and finite one-dimensional quantum chains

Based on previous work of Cardy (1984), the authors show in a systematic way how using conformal invariance one can determine the anomalous dimensions of various operators from finite quantum chains with different boundary conditions. The method is illustrated in the case of the three- and four-state Potts models where the anomalous dimensions of the para-fermionic operators are found.

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.

The Superintegrable Chiral Potts Quantum Chain and Generalized Chebyshev Polynomials

Finite-dimensional representations of Onsager’s algebra are characterized by the zeros of truncation polynomials. The Z N-chiral Potts quantum chain hamiltonians (of which the Ising chain hamiltonian is the N = 2 case) are the main known interesting representations of Onsager’s algebra and the corresponding polynomials have been found by Baxter and Albertini, McCoy and Perk in 1987-89 considering the Yang-Baxter-integrable 2-dimensional chiral Potts model. We study the mathematical nature of these polynomials. We find that for N ≥ 3 and fixed charge Q these don’t form classical orthogonal sets because their pure recursion relations have at least N + 1-terms. However, several basic properties are very similar to those required for orthogonal polynomials. The N + 1-term recursions are of the simplest type: like for the Chebyshev polynomials the coefficients are independent of the degree. We find a remarkable partial orthogonality, for N = 3, 5 with respect to Jacobi-, and for N = 4, 6 with respect to Chebyshev weight functions. The separation properties of the zeros known from orthogonal polynomials are violated only by the extreme zero at one end of the interval

Finite-size effects in the three-state quantum asymmetric clock model

Abstract The one-dimensional quantum hamiltonian of the asymmetric three-state clock model is studied using finite-size scaling. Various boundary conditions are considered on chains containing up to eight sites. We calculate the boundary of the commensurate phase and the mass gap index. The model shows an interesting finite-size dependence in connexion with the presence of the incommensurate phase indicating that for the infinite system there is no Lifshitz point.

Off-criticality behaviour of the Blume-Capel quantum chain as a check of Zamolodchikov's conjecture

Abstract Using finite-size numerical calculations, we study the off-criticality behaviour of the Blume-Capel quantum chain in the neighbourhood of the c = 7 10 tricritical Ising point. Moving from the tricritical point in the ( 1 10, 1 10 )- and ( 3 5, 3 5 )- directions into the disordered region, we find masses and thresholds in agreement with the structure proposed by Zamolodchikov from conformal field theory. Moving in opposite directions, the spectrum is degenerate between the Z2-even and Z2-odd sectors, suggesting an underlying supersymmetry. The free-particle energy-momentum relation and the scaling properties off criticality are checked.

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.