Graded Off-diagonal Bethe ansatz solution of the $SU(2|2)$ spin chain model with generic integrable boundaries

The graded off-diagonal Bethe ansatz method is proposed to study supersymmetric quantum integrable models (i.e., quantum integrable models associated with superalgebras). As an example, the exact solutions of the $SU(2|2)$ vertex model with both periodic and generic open boundary conditions are constructed. By generalizing the fusion techniques to the supersymmetric case, a closed set of operator product identities about the transfer matrices are derived, which allows us to give the eigenvalues in terms of homogeneous or inhomogeneous $T-Q$ relations. The method and results provided in this paper can be generalized to other high rank supersymmetric quantum integrable models.


Introduction
Quantum integrable models [1] play important roles in fields of theoretical physics, condensed matter physics, field theory and mathematical physics, since exact solutions of those models may provide useful benchmarks to understand a variety of many-body problems. During the past several decades, much attention has been paid to obtain exact solutions of integrable systems with unusual boundary conditions. With the development of topological physics and string theory, study on off-diagonal boundaries becomes an interesting issue. Many interesting phenomena such as edge states, Majorana zero modes, and topological excitations have been found.
A remarkable one is the off-diagonal Bethe ansatz (ODBA) [16,17], which allow us to construct the exact spectrum systematically. The nested ODBA has also been developed to deal with the models with different Lie algebras such as A n [22,23], A 2 [24], B 2 [25], C 2 [26] and D (1) 3 [27]. Nevertheless, there exists another kind of high rank integrable models which are related to superalgebras [28] such as the SU(m|n) model, the Hubbard model, and the supersymmetric t − J model. The SU(m|n) model has many applications in AdS/CFT correspondence [29,30], while the Hubbard and t − J model have many applications in the strongly correlated electronic theory. These models with U(1) symmetry have been studied extensively [31,32,33,34,35,36,37,38,39,40,41]. A general method to approach such kind of models with off-diagonal boundaries is still missing.
In this paper, we develop a graded version of nested ODBA to study supersymmetric integrable models (integrable models associated with superalgebras). As an example, the SU(2|2) model with both periodic and off-diagonal boundaries is studied. The structure of the paper is as follows. In section 2, we study the SU(2|2) model with periodic boundary condition. A closed set of operator identities is constructed by using the fusion procedure.
These identities allow us to characterize the eigenvalues of the transfer matrices in terms of homogeneous T − Q relation. In section 3, we study the model with generic open boundary conditions. It is demonstrated that similar identities can be constructed and the spectrum can be expressed in terms of inhomogeneous T − Q relation. Section 4 is attributed to concluding remarks. Some technical details can be found in the appendices.

The system
Let V denote a 4-dimensional graded linear space with an orthogonal basis {|i |i = 1, · · · , 4}, where the Grassmann parities are p(1) = 0, p(2) = 0, p(3) = 1 and p(4) = 1, which endows the fundamental representation of the SU(2|2) Lie superalgebra. For the matrix A j ∈ End(V j ), A j is a super embedding operator in the Z 2 -graded tensor space V 1 ⊗ V 2 ⊗ · · ·, which acts as A on the j-th space and as identity on the other factor spaces. For the matrix R ij ∈ End(V i ⊗ V j ), R ij is a super embedding operator in the Z 2 graded tensor space, which acts as identity on the factor spaces except for the i-th and j-th ones. The super tensor product of two operators is the graded one satisfying the rule (A ⊗ B) αγ βδ = (−1) [p(α)+p(β)]p(γ) A α β B γ δ [42]. The supersymmetric SU(2|2) model is described by the 16 × 16 R-matrix where u is the spectral parameter and η is the crossing parameter. The R-matrix (2.1) enjoys the following properties regularity : R 12 (0) = ηP 12 , unitarity : R 12 (u)R 21 (−u) = ρ 1 (u) × id, crossing − unitarity : where P 12 is the Z 2 -graded permutation operator with the definition

2)
R 21 (u) = P 12 R 12 (u)P 12 , st i denotes the super transposition in the i-th space , and the functions ρ 1 (u) and ρ 2 (u) are given by The R-matrix (2.1) satisfies the graded Yang-Baxter equation (GYBE) [43] In terms of the matrix entries, GYBE (2.4) reads For the periodic boundary condition, we introduce the "row-to-row" (or one-row) monodromy where the subscript 0 means the auxiliary space V 0 , the other tensor space V ⊗N is the physical or quantum space, N is the number of sites and {θ j |j = 1, · · · , N} are the inhomogeneous parameters. In the auxiliary space, the monodromy matrix (2.6) can be written as a 4 × 4 matrix with operator-valued elements acting on V ⊗N . The explicit forms of the elements of monodromy matrix (2.6) are The monodromy matrix T 0 (u) satisfies the graded Yang-Baxter relation The transfer matrix t p (u) of the system is defined as the super partial trace of the monodromy matrix in the auxiliary space (2.10)

Fusion
One of the wonderful properties of R-matrix is that it may degenerate to the projection operators at some special points, which makes it possible to do the fusion procedure [44,45,46,47,48]. It is easy to check that the R-matrix (2.1) has two degenerate points. The first one is u = η. At which, we have where P 12 is a 8-dimensional supersymmetric projector (2.12) and the corresponding basis vectors are with the corresponding parities The operator P (8) 12 projects the original 16-dimensional tensor space V 1 ⊗ V 2 into a new 8-dimensional projected space spanned by {|f i |i = 1, · · · , 8}. Taking the fusion by the operator (2.12), we construct the fused R-matrices where P 21 can be obtained from P 12 by exchanging V 1 and V 2 . For simplicity, we denote the projected space as V1 = V 12 = V 21 . The fused R-matrix R1 2 (u) is a 32 × 32 matrix defined in the tensor space V1 ⊗ V 2 and has the properties From GYBE (2.4), one can prove that the following fused graded Yang-Baxter equations hold It is easy to check that the elements of fused R-matrices R1 2 (u) and R 21 (u) are degree one polynomials of u.
The operator P Taking the fusion by the projector P 12 , we obtain another new fused R-matrix where P (20) 21 can be obtained from P by exchanging V1 and V 2 . For simplicity, we denote the projected subspace as V1 = V 1 2 = V 21 . The fused R-matrix R1 2 (u) is a 80 × 80 matrix defined in the tensor space V1 ⊗ V 2 and satisfies following graded Yang-Baxter equations The elements of fused R-matrix R1 2 (u) are also degree one polynomials of u.
Taking the fusion by the projectorP (8) 12 , we obtain the fused R-matrices For simplicity, we denote the projected space as V1′ = V 12 ′ = V 21 ′ . The fused R-matrix R1′ 2 (u) is a 32 × 32 matrix defined in the product space V1′ ⊗ V 2 and possesses the properties Now, we consider the fusions of R1′ 2 (u), which include two different cases. One is the fusion in the auxiliary space V1 and the other is the fusion in the quantum space V 2 . Both are necessary to close the fusion processes.
We first introduce the fusion in the auxiliary space. At the point u = 3 2 η, we have where P 1 ′ 2 is a 20-dimensional supersymmetric projector with the form of P (20) 31) and the corresponding vectors are |φ 11 = |g 5 ⊗ |1 , |φ 12 = |g 6 ⊗ |1 ), The parities read The operator P 1 ′ 2 projects the 32-dimensional product space V1′ ⊗ V 2 into a 20-dimensional projected space spanned by {|φ i , i = 1, · · · , 20}. Taking the fusion by the projector P (20) 1 ′ 2 , we obtain the following fused R-matrices For simplicity, we denote the projected space as V1′ = V 1′ 2 = V 21 ′ . The fused R-matrix R1′ 2 (u) is a 80 × 80 one defined in the product spaces V1′ ⊗ V 2 and satisfies following graded Yang-Baxter equation A remarkable fact is that after taking the correspondences the two fused R-matrices R1 2 (u) given by (2.20) and R1′ 2 (u) given by (2.32) are identical, which allows us to close the recursive fusion processe.
The fusion of R1′ 2 (u) in the quantum space is carried out by the projector P 23 , and the resulted fused R-matrix is which is a 64 × 64 matrix defined in the space V1′ ⊗ V2 and satisfies the graded Yang-Baxter which will help us to find the complete set of conserved quantities.

Operator product identities
Now, we are ready to extend the fusion from one site to the whole system. From the fused R-matrices given by (2.13), (2.20), (2.26) and (2.32), we construct the fused monodromy matrices as According to the property that the R-matrices in above equations can degenerate into the projectors P 12 ,P 12 , P 12 , P 1 ′ 2 and using the definitions (2.39), we obtain following fusion relations among the monodromy matrices The fused transfer matrices are define as the super partial traces of fused monodromy matrices in the auxiliary space From Eq.(2.41), we know that these fused transfer matrices with certain spectral difference must satisfy some intrinsic relations. We first consider the quantity 12 +P 12 +P Here we give some remarks. Both V 1 and V 2 are the 4-dimensional auxiliary spaces. From Eq.(2.42), we see that the 16-dimensional auxiliary space V 1 ⊗ V 2 can be projected into two 12 defined in the subspace V 12 ≡ V1, and the other is achieved by the 8dimensional projectorP (8) 12 defined in the subspace V 12 ′ ≡ V1′. The vectors in P (8) 12 and those inP (8) 12 constitute the complete basis of V 1 ⊗ V 2 , and all the vectors are orthogonal, From Eq.(2.42), we also know that the product of two transfer matrices with fixed spectral difference can be written as the summation of two fused transfer matrices t (1) p (u) and t (2) p (u). At the point of u = θ j − η, the coefficient of the fused transfer matrix t (1) p (u) is zero, while at the point of u = θ j , the coefficient of the fused transfer matrix t (2) p (u) is zero. Therefore, at these points, only one of them has the contribution.
Motivated by Eq.(2.41), we also consider the quantities 21 +P 21 +P During the derivation, we have used the relations P 21 +P 21P and a 12-dimensional subspace V 1 2 by the projectorP is the summation of two new fused transfer matricest (1) p (u) andt (1) p (u) with some coefficients. In Eq.(2.44), the 32-dimensional auxiliary space V1′ ⊗ V 2 is projected into a 20-dimensional and a 12-dimensional subspaces by the operators P (20) is the summation of two fused transfer matricest (2) p (u) and t (2) p (u) with some coefficients. At the point of u = θ j −η, the coefficient oft p (u) are omitted because we donot use them.
Combining the above analysis, we obtain the operator product identities of the transfer matrices at the fixed points as From the property (2.36), we obtain that the fused transfer matricest p (u) and t (2) p (u), From the graded Yang-Baxter relations (2.40), the transfer matrices t p (u), t (1) p (u) and t (2) p (u) commutate with each other, namely, Therefore, they have common eigenstates and can be diagonalized simultaneously. Let |Φ be a common eigenstate. Acting the transfer matrices on this eigenstate, we have p (u) and Λ (2) p (u) are the eigenvalues of t p (u), t p (u) and t where j = 1, 2, · · · N. Because the eigenvalues Λ p (u), Λ p (u) and Λ (2) p (u) are the polynomials of u with degree N − 1, the above 3N conditions (2.52) can determine these eigenvalues completely.

T − Q relations
Let us introduce the z-functions , l = 1, 2, where the Q-functions are The regularities of the eigenvalues Λ p (u), Λ p (u) and Λ , j = 1, · · · , L 1 , We have verified that the above BAEs indeed guarantee all the T − Q relations (2.54) are polynomials and satisfy the functional relations (2.52). Therefore, we arrive at the conclusion that Λ p (u), Λ p (u) and Λ (2) p (u) given by (2.54) are indeed the eigenvalues of the transfer matrices t p (u), t p (u), respectively. The eigenvalues of the Hamiltonian (2.10) are while K + (u) satisfies the graded dual RE The general solution of reflection matrix K − 0 (u) defined in the space V 0 satisfying the graded RE (3.1) is and the dual reflection matrix K + (u) can be obtained by the mapping where the ξ,ξ and {c i ,c i |i = 1, · · · , 4} are the boundary parameters which describe the boundary interactions, and the integrability requires For the open case, besides the standard "row-to-row" monodromy matrix T 0 (u) specified by (2.6), one needs to consider the reflecting monodromy matrix T 0 (u) = R N 0 (u + θ N ) · · · R 20 (u + θ 2 )R 10 (u + θ 1 ), (3.5) which satisfies the graded Yang-Baxter relation The transfer matrix t(u) is defined as The graded Yang-Baxter relations (

Fused reflection matrices
In order to solve the eigenvalue problem of the transfer matrix (3.7), we should study the fusion of boundary reflection matrices [51,52].
12 , 12 , 21 . (3.9) By specific calculation, we know that all the fused K-matrices are the 8 × 8 ones and their matric elements are the polynomials of u with maximum degree two. The fused reflection K-matrices (3.9) satisfy the resulting graded reflection equations. We can further use the reflection matrices K ± 1 (u) [or K ± 1 ′ (u)] and K ± 2 (u) to obtain the the 20-dimensional projector P 12 , 21 , which will be used to close the fusion processes with boundary reflections.

Operator production identities
For the model with open boundary condition, besides the fused monodromy matrices (2.39), we also need the fused reflecting monodromy matrices, which are constructed aŝ The fused reflecting monodromy matrices satisfy the graded Yang-Baxter relations The fused transfer matrices are defined as (3.14) Using the method we have used in the periodic case, we can obtain the operator product identities among the fused transfer matrices as The proof of the above operator identities is given in Appendix B.
From the definitions, we know that the transfer matrix t(u) is a operator polynomial of u with degree 2N + 2 while the fused ones t (1) (u) and t (2) (u) are the operator polynomials of u both with degree 2N + 4. Thus they can be completely determined by 6N + 13 independent conditions. The recursive fusion relations (3.15), (3.16) and (3.17) gives 6N constraints and we still need 13 ones, which can be achieved by analyzing the values of transfer matrices at some special points. After some direct calculation, we have Meanwhile, the asymptotic behaviors of t(u), t (1) (u) and t (2) Here we find that the operatorÛ related to the coefficient of transfer matrix t(u) with degree 2N + 1 is given byÛ where M i is given by (3.3),M i is determined by (3.4) and the operatorÛ i iŝ We note thatÛ i is the operator defined in the i-th physical space V i and can be expressed by a diagonal matrix with constant elements. The summation ofÛ i in Eq.(3.20) is the direct summation and the representation matrix of operatorÛ is also a diagonal one with constant elements. Moreover, we find that the operatorQ related to the coefficient of the fused transfer matrix t (1) (u) with degree 2N + 3 is given bŷ where the operatorQ i is defined in i-th physical space V i with the matrix form of Again, the operatorQ i is a diagonal matrix with constant elements and the summation of

Inhomogeneous T − Q relations
For simplicity, we define z (l) (u), x 1 (u) and x 2 (u) functions Here the structure factor α l (u) is defined as , l = 2, 3.
Comparing the right hand sides of Eqs.(A.1) and (A.2), we obtain P (d) Which give the general principle of fusion of the reflection matrices. If we define P (d) 21 as the fused reflection matrix K − 12 (u) ≡ K − 1 (u), where the integrability requires that the inserted R-matrix with determined spectral parameter is necessary, we can prove the the fused K-matrix K − 1 (u) also satisfies the (graded) reflection equation In the derivation, we have used the relation From the dual reflection equation (3.2), we obtain the general construction principle of fused dual reflection matrices P (d) 12 K + 2 (u)R 12 (−2u − α)K + 1 (u + α)P (d) If R 12 (−β) = S 12 P (d) 12 , the corresponding fusion relations are Appendix B: Proof of the operator product identities We introduce the reflection monodromy matriceŝ which satisfy the graded Yang-Baxter equations In order to solve the transfer matrix t(u) (3.7), we still need the fused transfer matrices which are defined ast Similar with periodic case, from the property that above R-matrices can degenerate into the projectors and using the definitions (3.12) and (B.1), we obtain following fusion relations among the reflecting monodromy matrices P 21T 2 (u)T 1 (u + η)P 21T 2 (u)T 1 (u − η)P From the definitions, we see that the auxiliary spaces are erased by taking the super partial traces and the physical spaces are the same. We remark that these transfer matrices are not independent. Substituting Eqs.(2.36) and (3.11) into the definitions (B.3), we obtain that the fused transfer matricest (1) (u) andt (2) (u) are equal Consider the quantity 12 K + 2 (u + η)R 12 (−2u − η)K + 1 (u)P