We examine in this paper the two parameter deformed superalgebra 𝒰qs(sl(1‖2)) and use the results in the construction of quantum chain Hamiltonians. This study is done both in the framework of the Serre presentation and in the R-matrix scheme of Faddeev, Reshetikhin, and Takhtajan (FRT). We show that there exists an infinite number of Casimir operators, indexed by integers p≥2 in the undeformed case and by p∈Z in the deformed case, which obey quadratic relations. The construction of the dual superalgebra of functions of SLqs(1‖2) is also given and higher tensor product representations are discussed. Finally, we construct quantum chain Hamiltonians based on the Casimir operators. In the deformed case we find two Hamiltonians which describe deformed t−J models.
The centrally extended superalgebra was shown to play an important role for the integrable structures of the one-dimensional Hubbard model and of the planar AdS/CFT correspondence. Here we consider its quantum deformation and derive the fundamental R -matrix. From the latter we deduce an integrable spin-chain Hamiltonian with three independent parameters and the corresponding Bethe equations to describe the spectrum on periodic chains. We relate our Hamiltonian to a two-parametric Hamiltonian proposed by Alcaraz and Bariev which can be considered a quantum deformation of the one-dimensional Hubbard model.
We study an integrable non-compact superspin chain model that emerged in recent studies of the dilatation operator in the super-Yang–Mills theory. It was found that the latter can be mapped into a homogeneous Heisenberg magnet with the quantum space in all sites corresponding to infinite dimensional representations of the SL (2|1) group. We extend the method of the Baxter Q -operator to spin chains with supergroup symmetry and apply it to determine the eigenspectrum of the model. Our analysis relies on a factorization property of the -operators acting on the tensor product of two generic infinite dimensional SL (2|1) representations. It allows us to factorize an arbitrary transfer matrix into a product of three ‘elementary’ transfer matrices which we identify as Baxter Q -operators. We establish functional relations between transfer matrices and use them to derive the T – Q relations for the Q -operators. The proposed construction can be generalized to integrable models based on supergroups of higher rank and, as distinct from the Bethe ansatz, it is not sensitive to the existence of the pseudovacuum state in the quantum space of the model.
We develop an approach for constructing the Baxter -operators for generic sl ( N ) spin chains. The key element of our approach is the possibility of representing a solution of the Yang–Baxter equation in the factorized form. We prove that such a representation holds for a generic sl ( N ) invariant -operator and find the explicit expression for the factorizing operators. Taking trace of monodromy matrices constructed of the factorizing operators one defines a family of commuting (Baxter) operators on the quantum space of the model. We show that a generic transfer matrix factorizes into the product of N Baxter -operators and discuss an application of this representation for a derivation of functional relations for transfer matrices.
All classical Lie algebras can be realized a la Schwinger (1965) in terms of fermionic oscillators. We show that the same can be done for their q-deformed counterparts by simply replacing the fermionic oscillators with anyonic ones defined on a two-dimensional lattice. The deformation parameter q is a phase related to the anyonic statistical parameter. A crucial role in this construction is played by a sort of bosonization formula which gives the generators of the quantum algebras in terms of the undeformed ones. The entire procedure works even on one-dimensional chains; in such a case q can also be real.
A bstract We introduce new
Uqsl2 U q sl 2 -invariant boundary conditions for the open XXZ spin chain. For generic values of
q q we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for
q=epiπ q = e iπ p a 2 p -th root of unity — to its p -dimensional analogue. Both cases are parametrised by a continuous “spin” α ∈ ℂ. To motivate our construction, we first specialise to
q q = i , where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly using free fermions. In the continuum, this model is identified with the ( η, ξ ) ghost CFT on the upper-half plane with a continuum of conformally invariant boundary conditions on the real axis. The different sectors of the Hamiltonian are identified with irreducible Virasoro representations. Going back to generic q we investigate the algebraic properties of the underlying lattice algebras. We show that if
q q α ∉ ±
q q ℤ , the new boundary coupling provides a faithful representation of the blob algebra which is Schur-Weyl dual to
Uqsl2 U q sl 2 . Then, modifying the boundary conditions on both the left and the right, we obtain a representation of the universal two-boundary Temperley-Lieb algebra. The generators and parameters of these representations are computed explicitly in terms of
q q and α . Finally, we conjecture the general form of the Schur-Weyl duality in this case. This paper is the first in a series where we will study, at all values of the parameters, the spectrum and its continuum limit, the representation content of the relevant lattice algebras and the fusion properties of these new spin chains.