M.J. Martins

The Bethe ansatz approach for factorizable centrally extended S-matrices

We consider the Bethe ansatz solution of integrable models interacting through factorized S-matrices based on the central extension of the su(2|2) symmetry. The respective su(2|2) R-matrix is explicitly related to that of the covering Hubbard model through a spectral parameter dependent transformation. This mapping allows us to diagonalize inhomogeneous transfer matrices whose statistical weights are given in terms of su(2|2) S-matrices by the algebraic Bethe ansatz. As a consequence of that we derive the quantization condition on the circle for the asymptotic momenta of particles scattering by the su(2|2)⊗su(2|2) S-matrix. The result for the quantization rule may be of relevance in the study of the energy spectrum of the AdS5×S5 string sigma model in the thermodynamic limit.

One-parameter family of an integrable spl(2|1) vertex model: Algebraic Bethe ansatz and ground state structure

Abstract We formulate in terms of the quantum inverse scattering method the exact solution of a spl (2|1) invariant vertex model recently introduced in the literature. The corresponding transfer matrix is diagonalized by using the algebraic (nested) Bethe ansatz approach. The ground state structure is investigated and we argue that a Pokrovsky-Talapov transition is favored for a certain value of the 4-dimensional spl (2|1) parameter.

Algebraic Bethe ansatz approach for the one-dimensional Hubbard model

We formulate in terms of the quantum inverse scattering method the algebraic Bethe ansatz solution of the one-dimensional Hubbard model. The method developed is based on a new set of commutation relations which encodes a hidden symmetry of six-vertex type.

The exact solution and the finite-size behaviour of the Osp(1|2)-invariant spin chain

Abstract We have solved exactly the Osp(1|2) spin chain by the Bethe ansatz approach. Our solution is based on an equivalence between the Osp(1|2) chain and a certain special limit of the Izergin-Korepin vertex model. The completeness of the Bethe ansatz equations is discussed for a system with four sites and the appearance of special string structures is noted. The Bethe ansatz presents an important phase factor which distinguishes the even and odd sectors of the theory. The finite-size properties are governed by a conformal field theory with central charge c = 1.

New R-matrices from representations of braid-monoid algebras based on superalgebras

Abstract In this paper we discuss representations of the Birman–Wenzl–Murakami algebra as well as of its dilute extension containing several free parameters. These representations are based on superalgebras and their baxterizations permit us to derive novel trigonometric solutions of the graded Yang–Baxter equation. In this way we obtain the multiparametric R-matrices associated to the U q [ sl ( r | 2 m ) ( 2 ) ] , U q [ osp ( r | 2 m ) ( 1 ) ] and U q [ osp ( r = 2 n | 2 m ) ( 2 ) ] quantum symmetries. Two other families of multiparametric R-matrices not predicted before within the context of quantum superalgebras are also presented. The latter systems are indeed non-trivial generalizations of the U q [ D n + 1 ( 2 ) ] vertex model when both distinct edge variables statistics and extra free-parameters are admissible.

Conformal invariance and the operator content of the XXZ model with arbitrary spin

The authors are concerned with the critical properties of the anisotropic Heisenberg chain, or XXZ model, with arbitrary integer or half-integer spin. The eigenspectra of these Hamiltonians, with periodic boundaries, are calculated for finite chains by solving numerically their associated Bethe ansatz equations. The resulting spectra are found to be in accord with the predictions of conformal invariance and the operator content is identified, for lattices with an even and odd number of sites. The results for spin 1 and spin 3/2 indicate that the conformal anomaly c for these models, in the gapless regime, has the value c=3S/(1+S), independent of the anisotropy, and the exponents vary continuously with the anisotropy as in the eight-vertex model. The operator content of these models indicate that the underlying field theory governing these critical spin-S models is described by composite fields formed by the product of Gaussian and Z(N) fields, with N=2S. Finally some of the irrelevant operators which produce the leading finite-size corrections of the eigenenergies are identified.

Conformal invariance and critical exponents of the Takhtajan-Babujian models

The authors are concerned with the critical properties of antiferromagnetic Takhtajan-Babujian models with spin S=1, 3/2 and 2. The leading eigenenergies of this Hamiltonian, in a finite chain, are calculated by investigating numerically and analytically the Bethe ansatz equations for the finite system. The critical exponents and the conformal anomaly are obtained from their relations with the eigenspectrum of the finite Hamiltonian. The appearance of logarithmic corrections produces poor estimates. However, a combination of analytical and numerical methods produces very good estimates. Their results strongly support the conjecture that the Wess-Zumino-Witten-Novikov non-linear sigma models with topological charge k=2S are the underlying field theories for these spin-S statistical mechanics models.

Finite size properties of staggered Uq[sl(2|1)] superspin chains

Abstract Based on the exact solution of the eigenvalue problem for the U q [ sl ( 2 | 1 ) ] vertex model built from alternating three-dimensional fundamental and dual representations by means of the algebraic Bethe ansatz we investigate the ground state and low energy excitations of the corresponding mixed superspin chain for deformation parameter q = exp ( − i γ / 2 ) . The model has a line of critical points with central charge c = 0 and continua of conformal dimensions grouped into sectors with γ-dependent lower edges for 0 ⩽ γ π / 2 . The finite size scaling behavior is consistent with a low energy effective theory consisting of one compact and one non-compact bosonic degree of freedom. In the ‘ferromagnetic’ regime π γ ⩽ 2 π the critical theory has c = − 1 with exponents varying continuously with the deformation parameter. Spin and charge degrees of freedom are separated in the finite size spectrum which coincides with that of the U q [ osp ( 2 | 2 ) ] spin chain. In the intermediate regime π / 2 γ π the finite size scaling of the ground state energy depends on the deformation parameter.

Phase diagram of an integrable alternating Uq[sl(2|1)] superspin chain

Abstract We construct a family of integrable vertex model based on the typical four-dimensional representations of the quantum group deformation of the Lie superalgebra sl ( 2 | 1 ) . Upon alternation of such a representation with its dual this model gives rise to a mixed superspin Hamiltonian with local interactions depending on the representation parameter ±b and the deformation parameter γ. As a subsector this model contains integrable vertex models with ordinary symmetries for twisted boundary conditions. The thermodynamic limit and low energy properties of the mixed superspin chain are studied using a combination of analytical and numerical methods. Based on these results we identify the phases realized in this system as a function of the parameters b and γ. The different phases are characterized by the operator content of the corresponding critical theory. Only part of the spectrum of this effective theory can be understood in terms of the U ( 1 ) symmetries related to the physical degrees of freedom corresponding to spin and charge. The other modes lead to logarithmic finite-size corrections in the spectrum of the theory.

Algebraic Bethe ansatz for U(1) invariant integrable models: The method and general results

Abstract In this work we have developed the essential tools for the algebraic Bethe ansatz solution of integrable vertex models invariant by a unique U ( 1 ) charge symmetry. The formulation is valid for arbitrary statistical weights and respective number N of edge states. We show that the fundamental commutation rules between the monodromy matrix elements are derived by solving linear systems of equations. This makes possible the construction of the transfer matrix eigenstates by means of a new recurrence relation depending on N − 1 distinct types of creation fields. The necessary identities to solve the eigenvalue problem are obtained exploring the unitarity property and the Yang–Baxter equation satisfied by the R-matrix. The on-shell and off-shell properties of the algebraic Bethe ansatz are explicitly presented in terms of the arbitrary R-matrix elements. This includes the transfer matrix eigenvalues, the Bethe ansatz equations and the structure of the vectors not parallel to the eigenstates.