Extended integrability régime for the supersymmetric U model

An extension of the supersymmetric U model for correlated electrons is given and integrability is established by demonstrating that the model can be constructed through the Quantum Inverse Scattering Method using an R-matrix without the difference property. Some general symmetry properties of the model are discussed and from the Bethe ansatz solution an expression for the energies is presented. The supersymmetric (SUSY) U model was first introduced in [1] as an example of a system of correlated electrons which is integrable in one dimension as a consequence of the Quantum Inverse Scattering Method (QISM) (e.g. see [2]). Such models, which can be solved exactly by the Bethe ansatz method, are important in that the exact solutions offer non-perturbative results concerning physical behaviour. For the SUSY U model, Bethe ansatz solutions have been studied [3, 4, 5, 6, 7, 8, 9] and several analyses into the physical characteristics that the model describes have been undertaken [4, 5, 7, 10]. The construction of the SUSY U model is based on anR-matrix satisfying the YangBaxter equation associated with the one-parameter family of minimal typical representations of the Lie superalgebra gl(2|1). In terms of the standard notation for electron creation, annihilation and occupation operators the local (two-site) Hamiltonian for the model reads hi,i+1 = − ∑ σ (c†iσci+1σ + h.c.)(1 + U) 1/2(ni,−σ+ni+1,−σ) +U(c†i↓c † i↑ci+1↑ci+1↓ + h.c.) +U(ni↑ni↓ + ni+1↑ni+1↓)− 2 +ni↑ + ni↓ + ni+1↑ + ni+1↓, where U is an arbitrary free parameter. The local Hamiltonian is also invariant with respect to the Lie superalgebra gl(2|1) (hence the name). Below, an extension of this model will be derived in such a way that integrability is maintained. The local Hamiltonian of the new model reads hi,i+1 = − ∑ σ ( (1− it)c†iσci+1σ + h.c. ) (1 + U)i,−σi+1,−σ +U ( (1− it)c†i↓c † i↑ci+1↑ci+1↓ + h.c. ) +U(ni↑ni↓ + ni+1↑ni+1↓)− 2 +ni↑ + ni↓ + ni+1↑ + ni+1↓ (1) where now t is an additional free variable which when chosen to be real (along with U real and U > −1) results in a Hermitian Hamiltonian. For the case t = 0 the usual SUSY U model is recovered. The extended model bears some similarity with the multiparametric SUSY U model constructed in [11] but is in fact inherently different. The construction of the above model is through the use of a solution of the YangBaxter equation without difference property in the spectral parameter. It is known that theHubbardmodelmay be derived via the QISMusing anR-matrix which is also without the difference property [12, 13]. However, for the Hubbard model the Lax operator 1 is given as a particular coupling of two auxilliary Lax operators of six-vertex type. The construction employed here appears more akin to the generalized chiral Potts models given in [14] based on representations of quantum algebras at roots of unity. For these models and the one discussed here the spectral parameters without difference property originate from the representation of the underlying algebraic structure. In order to demonstate integrability of this model, we begin with the rational limit of the Uq(gl(2|1)) invariant (i.e. gl(2|1) invariant) solution of the Yang-Baxter equation with additional spectral paramters constructed in [15, 16]. This solution may bewritten in the form R(u, β, α) = u− α− β u+ α + β P1 + P2 + u+ α + β + 2 u− α− β − 23 (2) and satisfies the Yang-Baxter equation R12(u− v, β, γ)R13(u, β, α)R23(v, γ, α) = R23(v, γ, α)R13(u, β, α)R12(u− v, β, γ). Note that this solution of the Yang-Baxter equation has multiple spectral parameters, one of which has the difference property while the other two do not. It is worth remarking at this point that the above solution is just one of a plethora of solutions of this type which arise naturally as a consequence of the representation theory of the type I quantum superalgebras [16, 17]. In the above expression for the R-matrix the operators Pi are gl(2|1) invariant projection operators. To explain their actions we begin by recalling that gl(2|1) has generators E j , i, j = 1, 2, 3 satisfying the super commutator relations [E j , E k l ] = δ k jE i l − (−1)δ lE j . Above, [1] = [2] = 0, [3] = 1. Choosing a four dimensional space with basis {|i〉 : i = 1, 2, 3, 4}, a representation πα of gl(2|1) acting on this space exists with the action of the generators given by