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 one-parameter 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.


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 R-matrices 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. 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.

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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 R-matrix R(α, β) is obtained via H 12 ∼ √ −1 P ∂ ∂α R(α, β) β=α , 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, where ξ, η and U are real parameters and σ = + (or ↑), − (or ↓). To our knowledge, this is a new integrable model of correlated electrons with pure imaginary pair hopping terms.

R-matrices with non-additive spectral parameters
In this section we deduce from the results in [2,3] an expression for solutions of the QYBE with non-additive spectral parameters. Let G be a simple type-I Lie superalgebra of rank r with generators {e i , f i , h i , i = 1, · · · , r} and U q (G) the corresponding quantum superalgebra. We will not give the defining relations for U q (G) here but mention that it is a Z 2 -graded quasi-triangular Hopf algebra with coproduct Note that U q (G) also has an opposite coproduct structure defined by∆ = T · ∆, with T being the twist map: ∈ Z 2 denotes the grading of the element a. The multiplication rule for the tensor product is defined by Let π α be a one-parameter family of irreducible representations (irreps) provided by the irreducible U q (G) module V (Λ α ), with the highest weight Λ α depending on the parameter α. Assume that for any α the irrep π α is affinizable, i.e. it can be extended to become a loop representation of the corresponding untwisted quantum affine superalgebra U q (Ĝ). Let x ∈ C be the associated loop parameter (the spectral parameter), and let R(x|α, β) ∈ End(V (Λ α ) ⊗ V (Λ β )) be an operator (R-matrix) associated with two affinizable irreps π α , π β from the one-parameter family. Leť where P is the graded permutation operator in V (Λ α ) ⊗ V (Λ β ) such that
In [3], α, β, γ were treated as extra free parameters although they enter in the QYBE in non-additive form.
A systematic method was developed in [3] for constructing solutions to (2.4) which satisfy the QYBE (2.5). In this work, we focus on the x = 1 case, i.e. we fix the parameter x to 1, but instead treat the parameters α, β as the spectral parameters. Seť It then follows from (2.4) and (2.5) that the linear relationš admit a unique solutionŘ(α, β) (under suitable normalization) for any given one-parameter family of irreps of U q (G) which satisfies the QYBE with non-additive spectral parameters α, β, γ, provided such irreps are consistently affiniable.

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The dual vectors of these basis vectors are defined using the following rules,

New integrable model of strongly correlated electrons
The new R-matrix (3.9) 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

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From the explicit expression of the R-matrix (3.9) and via a direct computation, we find the two-site local Hamiltonian In view of the grading (3.1) 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 we obtain the same local Hamiltonian with sign(α) replaced by −sign(α

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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 [22]. Of particular interest is to consider twisted or antiperiodic 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 [23]. It is also important to derive explicit bivariate R-matrices associated 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 for 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.