Spin-Anisotropy Commensurable Chains: Quantum Group Symmetries and N = 2 SUSY

In this paper we consider a class of the 2D integrable models. These models are higher spin XXZ chains with an extra condition of the commensurability between spin and anisotropy. The mathematics underlying this commensurability is provided by the quantum groups with deformation parameter being an Nth root of unity. Our discussion covers a range of topics including new integrable deformations, thermodynamics, conformal behaviour, S-matrices and magnetization. The emerging picture strongly depends on the N -parity. For the N even case at the commensurable point, S-matrices factorize into N = 2 supersymmetric Sine-Gordon matrix and an RSOS piece. The physics of an N odd case is rather different. Here, the supersymmetry does not manifest itself and the bootstrap hypothesis fails. Away from the commensurable point, we find an unusual magnetic behaviour. The magnetization of our chains depends on the sign of the external magnetic field. Introduction and Discussion Symmetry is the driving concept in particle physics. In Quantum Field Theory the particles are defined as finite dimensional irreps of the space-time and internal symmetry groups. In Statistical Mechanics the notion of symmetry has also played a very important role in the past, as a way of characterizing degrees of freedom and types of interaction. The recently introduced quantum groups are another stride in this direction, which deepens our understanding of symmetry in systems with an infinite number of degrees of freedom. Quantum groups shed new light on the number of difficult problems which are, essentially, non perturbative in nature; determination of the particle spectrum, scattering matrices, correlation functions, to name just a few. An important family of integrable models is based on the quantum affine algebras Uq(Ĝ) [1]. These models are characterized by the quantum deformation parameter q and a finite dimensional irrep of Ĝ. The famous six-vertex model corresponds to Ĝ = ̂ Sl(2) and the fundamental spin 1/2 irrep. [2]. The higher spin versions of Uq( ̂ Sl(2)) [3], [4] can be constructed by fusion procedure [5]. Generally, the spectrum of these models is expected to satisfy the bootstrap axioms [6]: there is a fundamental particle such that all others can be interpreted as bound states. For the isotropic models of spin j (q = 1) the particle spectrum consists of a fundamental spin 1/2 particle which has (for j > 1/2) extra hidden RSOS spin [7], [8]. The corresponding physical S-matrix factorizes into the product of the XXX S-matrix and an RSOS piece. This picture remains essentially unchanged in the semiclassical weak anisotropic regime [8], [9]. The central extension for these nearly isotropic models is c = 3j j+1 , i.e. that of a SU(2)k WZW model with level k = 2j. Moreover, the excitations above the ground state can be understood in terms of the free bosonic and Z2j-parafermionic [10] degrees of freedom [11]. There is a connection between N = 2 integrable field theories in two dimensions and quantum affine algebras at roots of unity. This became clear in the study of the Sine-Gordon model in ref. [12] and more recently in [13]. The N = 2 structure of the Sine-Gordon models appears at the special value β = 8π3/2 which yields q = 1 ( q = e , p = β/(8π − β)) with the SUSY realized in a non local way [14], namely, the one defined by the non local action of the quantum group generators [15]. The solitonic S-matrix of the N = 2 theories factorizes into an N = 0 RSOS S-matrix and the N = 2 S-matrix of the Sine-Gordon model at β = 8π3/2 [16].