New R-matrices with non-additive spectral parameters and integrable models of strongly correlated fermions

We present a general formula for constructing R-matrices with non-additive spectral parameters associated with a type-I quantum superalgebra. The spectral parameters originate from two oneparameter families of inequivalent finite-dimensional irreducible representations of the quantum superalgebra upon which the R-matrix acts. Applying to the quantum superalgebra Uq(gl(2|1)), we obtain the explicit expression for the Uq(gl(2|1))-invariant R-matrix which is of non-difference form in spectral parameters. Using this R-matrix we derive a new two-parameter integrable model of strongly correlated electrons with pure imaginary pair hopping terms. PACS numbers: 02.30.Ik, 71.10.Fd, 75.10.Pq, 03.65.Fd


Introduction
Type-I quantum superalgebras are known to admit non-trivial one-parameter families of inequivalent finite dimensional irreducible representations (irreps), even for generic q [1]. Using such irreps, R-matrices which depend not only on a spectral parameter but in addition on further continuous parameters were constructed in [2,3]. These extra parameters enter the quantum Yang-Baxter equation (QYBE) in non-additive form. The freedom of having such further parameters opens up new and exciting possibilities. For example, by using the rational R-matrix associated with the 4-dimensional irreps of superalgebra gl(2|1), the supersymmetric U model for correlated electrons was introduced in [4]. The Bethe ansatz solutions of this model were later derived in [5,6]. A two-parameter version was proposed in [7] and was shown in [8] to come from the trigonometric R-matrix associated with quantum superalgebra U q (gl(2|1)).
There is a different way of treating the extra parameters in the R-matrices obtained in [2,3].
Instead of regarding them as free parameters, one can treat them as spectral parameters. This approach was applied in [9] to the gl(2|1)-invariant rational R-matrix to derive another extension of the supersymmetric U model. Similar ideas recently also appeared in the construction of Rmatrices associated with the central extension of su(2|2) [10,11]. There the values of the central elements, which parametrize the representations of the centrally extended superalgebra, give rise to spectral parameters. R-matrices obtained from such a construction are also of non-difference form and have found applications in various areas (see e.g. [12,13,14] and references therein).
In this paper we implement a similar strategy for the one-parameter family of irreps of a general type-I quantum superalgebra. From the results obtained in [2,3], we deduce a formula for quantum superalgebra invariant R-matrices with non-additive spectral parameters (without the need of acquiring a spectral parameter via affinization of the quantum superalgebra). Applying our results to the two 4-dimensional irreps of U q (gl(2|1)), we obtain the explicit expression of a new 36-vertex R-matrix without difference property of the spectral parameters.
The obtained R-matrices can be used to construct new integrable models of correlated fermions. As the R-matrices are of non-difference form, the actual values of the spectral parameters are important in the corresponding model construction. We will restrict our attention to the homogeneous case in which the parameter of each of the representations is the same at each site. Then the two-site local Hamiltonian of the integrable model corresponding to bivariate where P is the graded permutation operator. Using the U q (gl(2|1))-invariant R-matrix derived in this paper, we obtain the local Hamiltonian in terms of the standard electron creation and destruction operators, 2 R-matrices with non-additive spectral parameters We apply the technique developed in [2,3] to find new solutions to the QYBE associated with the one-parameter family of irreps of type-I quantum superalgebras, thus obtaining R-matrices which depend on non-additive spectral parameters.
Let G denote a simple type-I Lie superalgebra of rank r with generators {e i , f i , h i } and let α i be its simple roots. Then the quantum superalgebra U q (G) can be defined with the structure of a Z 2 -graded quasi-triangular Hopf algebra. We will not give the full defining relations for U q (G) here but mention that U q (G) has a coproduct structure given by The multiplication rule for the tensor product of elements a, b, c, d ∈ U q (G) is defined by where [a] ∈ Z 2 denotes the degree of the element a.
Let π α be a one-parameter family of irreps of U q (G) afforded by the irreducible module V (Λ α ) in such a way that the highest weight of the irrep depends on the parameter α. Consider , where α, β are two parameters from the two irreps π α , π β , which serve as the spectral parameters for the R-matrix. Then from Jimbo's results [15] a solution to the linear equations satisfies the QYBE with non-difference property for the spectral parameters α, β, γ, in the tensor of three irreps from the one-parameter family: In the above,∆ = T ·∆, with T the twist map defined by In all our equations we implicitly use the "graded" multiplication rule of eq. (3). Thus the R-matrix of a quantum superalgebra satisfies a "graded" QYBE which, when written as an ordinary matrix equation, contains extra signs: Introduce the graded permutation operator P on the tensor product module and setŘ (α, β) = P R(α, β).
Then (4) can be rewritten aš and in terms ofŘ(α, β) the non-additive-parameter-dependent QYBE becomes We will normalize the R-matrixŘ(α, β) in such a way thať They are usually called the regularity and unitarity conditions, respectively, in the literature.
In the case of a multiplicity-free tensor product decomposition where µ denotes a highest weight depending on the parameters α and β, the R-matrixŘ(α, β) can be obtained by applying the techniques in [2]. Let P αβ Obviously, when α = β, P αα µ are the usual projection operators having the property that µ P αα µ = I. Then the R-matrixŘ(α, β) has the particularly simple form, This is shown as follows. Firstly the property (13) of the elementary intertwiners enables us to Then using the unitarity conditionŘ(α, β)Ř(β, α) = I we at once see that ρ µ satisfies (ρ µ ) 2 = 1, so that ρ µ = ±1. Finally the regularity conditionŘ(α, α) = I and the property µ P αα µ = I mean that ρ µ appearing in (15) must equal to 1 identically, thus completing the proof.
Explicit expressions for the elementary intertwiners P αβ µ can be constructed as follows [2].
i β⊗α } be the corresponding orthonormal basis. Using these bases the operators P αβ µ can be expressed as The simple formula (14) for the U q (G)-invariant bivariate R-matricesŘ(α, β) is one of the main results in this paper. It gives solutions to the QYBE (10) without difference property of the spectral parameters.
3 Explicit expression for U q (gl(2|1))-invariant new R-matrix We now apply the above formalism to the one-parameter family of 4-dimensional irreps of U q (gl(2|1)) with hight weight Λ α = (0, 0|α). The quantum superalgebra U q (gl(2|1)) has genera- v=1 denote an orthonormal basis for a 4-dimensional U q (gl(2|1)) module V . Consistent with the Z 2 -grading of the generators we may assign a grading on the basis states by Then the generators {E i j } 3 i,j=1 act on this module according to where . Note that the representation depends upon a parameter α ∈ C. For α > 0 we have E i j † = E j i and we call the representation unitary of type I. For E j i and we say the representation is unitary of type II. Hereafter we assume that α is restricted to either of the above ranges.
Let Ψ 1 k α⊗β , Ψ 3 k α⊗β , k = 1, 2, 3, 4 and Ψ 2 l α⊗β , l = 1, 2, · · · , 8 form symmetry adapted bases for the spaces V 1 , V 3 and V 2 respectively. By means of the representation (18) and the co-product action (19) we obtain The dual vectors of these basis vectors are defined using the following rules, It can be checked that (21) forms an orthonormal basis 2 . It then follows from (14) and (16) that the U q (gl(2|1))-invariant R-matrix depending on the spectral parameters α, β is given by By means of the orthonormal basis vectors (21) we obtain the following explicit expression for the U q (gl(2|1))-invariant R-matrix which satisfies the QYBE (10), where we have used the notation e ij ≡ |i j| and It can be easily checked that the above R-matrix satisfies the regularity and unitarity properties (11), as required.
2 A set of non mutually orthogonal basis vectors for Vi in the decomposition (20) was obtained in [16].

New integrable model of strongly correlated electrons
The new R-matrix (25) can be used to define an integrable model of correlated electrons. On the N -fold tensor product space we denotě and define a local Hamiltonian by From the explicit expression of the R-matrix (25) and via a direct computation, we find the two-site local Hamiltonian In view of the grading (17) we now make the assignment which allows us to express e ij ≡ |i j| in terms of the canonical fermion operators. We can show that we have Here and throughout, we have used the standard notation for electron spins: σ = + (or ↑), − (or ↓). Note that the local Hamiltonian is hermitian only for α > 0 or α < −1; i.e. when the underlying representation is unitary.
Under the unitary transformation On a periodic lattice, the corresponding global Hamiltonian of our electron model (which is integrable with periodic boundary conditions) reads Observing that ξ, η and U are related to two independent parameters α and q through (33) above, we can derive two distinct and non-trivial special cases corresponding to the following limits of q and α values, respectively. In the limit q → 1 so that η = 0, our model (34) becomes where Obviously this special case gives an non-trivial integrable electron model with pure imaginary pair hopping terms. It can be checked that this Hamiltonian is also obtainable directly from the R-matrix (26) which is the q → 1 limit of (25).
On the other hand, in the limit α → ∞ so that ξ = η, our model (34) reduces to the Bariev model [17] whose Hamiltonian on the periodic lattice can be written in our notation as In this sense, our electron model (34)

Conclusions and discussions
We have demonstrated in this paper that new solutions to the QYBE without difference property may be naturally constructed through R-matrices associated with one-parameter family of irreps of type I quantum superalgebras. As an example we have considered one of the simplest cases and derived the explicit expression of a new 36-vertex bivariate R-matrix associated with the 4-dimensional irreps of the quantum superalgebra U q (gl(2|1)). Using this R-matrix, we have obtained the Hamiltonian of the new two-parameter integrable model of strongly correlated electrons with pure imaginary pair hopping terms. Our model contains two non-trivial and distinct one-parameter electron models as its special cases.
Open problems for future work include solving our new model by means of the (algebraic) Bethe ansatz method, and constructing and solving the corresponding open chain models with boundary conditions defined by boundary K-matrices using an appropriate modification of Sklyanin's method [20]. Of particular interest is to consider twisted or anti-periodic boundary condition, i.e. a ring of electrons with Möbius like topological boundary condition, by means of the Cao-Yang-Shi-Wang method for a topological spin ring [21]. It is also important to derive explicit bivariate R-matrices associate with higher rank type-I quantum superalgebras and construct the corresponding integrable models of correlated fermions. Finally it seems that in [10,11], the bivariate R-matrices associated with the centrally extended su(2|2) and U q (su(2|2)) superalgebras were derived somewhat by brute force. It would be interesting to investigate whether or not the superalgebra representation theory approach presented in this paper can also be applied to derive the bivariate R-matrices associated with central extensions of certain superalgebras. We hope to examine some of the above problems and present the obtained results elsewhere.