Bethe Ansatz Equations for the Broken ZNZ_{N}-Symmetric Model

We obtain the Bethe Ansatz equations for the broken ${\bf Z}_{N}$-symmetric model by constructing a functional relation of the transfer matrix of $L$-operators. This model is an elliptic off-critical extension of the Fateev-Zamolodchikov model. We calculate the free energy of this model on the basis of the string hypothesis.Comment: 43 pages, latex, 11 figure

Thermodynamics of the 3-State Potts Spin Chain

We demonstrate the relation of the infrared anomaly of conformal field theory with entropy considerations of finite temperature thermodynamics for the 3-state Potts chain. We compute the free energy and compute the low temperature specific heat for both the ferromagnetic and anti-ferromagnetic spin chains, and find the central charges for both.Comment: 18 pages, LaTex. Preprint # ITP-SB-92-60. References added and first section expande

Asymmetric XXZ chain at the antiferromagnetic transition: Spectra and partition functions

The Bethe ansatz equation is solved to obtain analytically the leading finite-size correction of the spectra of the asymmetric XXZ chain and the accompanying isotropic 6-vertex model near the antiferromagnetic phase boundary at zero vertical field. The energy gaps scale with size $N$ as $N^{-1/2}$ and its amplitudes are obtained in terms of level-dependent scaling functions. Exactly on the phase boundary, the amplitudes are proportional to a sum of square-root of integers and an anomaly term. By summing over all low-lying levels, the partition functions are obtained explicitly. Similar analysis is performed also at the phase boundary of zero horizontal field in which case the energy gaps scale as $N^{-2}$. The partition functions for this case are found to be that of a nonrelativistic free fermion system. From symmetry of the lattice model under $\pi /2$ rotation, several identities between the partition functions are found. The $N^{-1/2}$ scaling at zero vertical field is interpreted as a feature arising from viewing the Pokrovsky-Talapov transition with the space and time coordinates interchanged.Comment: Minor corrections only. 18 pages in RevTex, 2 PS figure

sl(N) Onsager's Algebra and Integrability

We define an $sl(N)$ analog of Onsager's Algebra through a finite set of relations that generalize the Dolan Grady defining relations for the original Onsager's Algebra. This infinite-dimensional Lie Algebra is shown to be isomorphic to a fixed point subalgebra of $sl(N)$ Loop Algebra with respect to a certain involution. As the consequence of the generalized Dolan Grady relations a Hamiltonian linear in the generators of $sl(N)$ Onsager's Algebra is shown to posses an infinite number of mutually commuting integrals of motion

Free Energy of the Eight Vertex Model with an Odd Number of Lattice Sites

We calculate the bulk contribution for the doubly degenerated largest eigenvalue of the transfer matrix of the eight vertex model with an odd number of lattice sites N in the disordered regime using the generic equation for roots proposed by Fabricius and McCoy. We show as expected that in the thermodynamic limit the result coincides with the one in the N even case.Comment: 11 pages LaTeX New introduction, Method change

Impact of positivity and complete positivity on accessibility of Markovian dynamics

We consider a two-dimensional quantum control system evolving under an entropy-increasing irreversible dynamics in the semigroup form. Considering a phenomenological approach to the dynamics, we show that the accessibility property of the system depends on whether its evolution is assumed to be positive or completely positive. In particular, we characterize the family of maps having different accessibility and show the impact of that property on observable quantities by means of a simple physical model.Comment: 11 pages, to appear in J. Phys.

The Importance of being Odd

In this letter I consider mainly a finite XXZ spin chain with periodic boundary conditions and \bf{odd} \rm number of sites. This system is described by the Hamiltonian $H_{xxz}=-\sum_{j=1}^{N}\{\sigma_j^{x}\sigma_{j+1}^{x} +\sigma_j^{y}\sigma_{j+1}^{y} +\Delta \sigma_j^z\sigma_{j+1}^z\}$. As it turned out, its ground state energy is exactly proportional to the number of sites $E=-3N/2$ for a special value of the asymmetry parameter $\Delta=-1/2$. The trigonometric polynomial $q(u)$, zeroes of which being the parameters of the ground state Bethe eigenvector is explicitly constructed. This polynomial of degree $n=(N-1)/2$ satisfy the Baxter T-Q equation. Using the second independent solution of this equation corresponding to the same eigenvalue of the transfer matrix, it is possible to find a derivative of the ground state energy w.r.t. the asymmetry parameter. This derivative is closely connected with the correlation function $=-1/2+3/2N^2$. In its turn this correlation function is related to an average number of spin strings for the ground state of the system under consideration: $= {3/8}(N-1/N)$. I would like to stress once more that all these simple formulas are \bf wrong \rm in the case of even number of sites. Exactly this case is usually considered.Comment: 9 pages, based on the talk given at NATO Advanced Research Workshop "Dynamical Symmetries in Integrable Two-dimensional Quantum Field Theories and Lattice Models", 25-30 September 2000, Kyiv, Ukraine. New references are added plus some minor correction

The Blob Algebra and the Periodic Temperley-Lieb Algebra

We determine the structure of two variations on the Temperley-Lieb algebra, both used for dealing with special kinds of boundary conditions in statistical mechanics models. The first is a new algebra, the `blob' algebra (the reason for the name will become obvious shortly!). We determine both the generic and all the exceptional structures for this two parameter algebra. The second is the periodic Temperley-Lieb algebra. The generic structure and part of the exceptional structure of this algebra have already been studied. Here we complete the analysis, using results from the study of the blob algebra.Comment: 12 page

The structure of quotients of the Onsager algebra by closed ideals

We study the Onsager algebra from the ideal theoretic point of view. A complete classification of closed ideals and the structure of quotient algebras are obtained. We also discuss the solvable algebra aspect of the Onsager algebra through the use of formal Lie algebras.Comment: 33 pages, Latex, small topos corrected-Journal versio

Three-Dimensional Vertex Model in Statistical Mechanics, from Baxter-Bazhanov Model

We find that the Boltzmann weight of the three-dimensional Baxter-Bazhanov model is dependent on four spin variables which are the linear combinations of the spins on the corner sites of the cube and the Wu-Kadanoff duality between the cube and vertex type tetrahedron equations is obtained explicitly for the Baxter-Bazhanov model. Then a three-dimensional vertex model is obtained by considering the symmetry property of the weight function, which is corresponding to the three-dimensional Baxter-Bazhanov model. The vertex type weight function is parametrized as the dihedral angles between the rapidity planes connected with the cube. And we write down the symmetry relations of the weight functions under the actions of the symmetry group $G$ of the cube. The six angles with a constrained condition, appeared in the tetrahedron equation, can be regarded as the six spectrums connected with the six spaces in which the vertex type tetrahedron equation is defined.Comment: 29 pages, latex, 8 pasted figures (Page:22-29