On the inhomogeneous T-Q relation for quantum integrable models

The off-diagonal Bethe Ansatz method [1] is used to revisit the periodic XXX Heisenberg spin2 chain. It is found that the spectrum of the transfer matrix can be characterized by an inhomogeneous T-Q relation, a natural but nontrivial extension of Baxter’s T-Q relation [2]. One of Baxter’s important discoveries is the T-Q relation [2], which provides a convenient and universal parametrization for eigenvalues of transfer matrices of most quantum integrable models. Nevertheless, the Q operator does not allow polynomial solutions for some integrable models such as the XYZ quantum spin chain with an odd number of sites and generic coupling constants, the chiral Potts model and the quantum spin chains with non-diagonal boundaries, despite the fact that the transfer matrix is a polynomial operator. It is obvious that the T-Q parametrization for transfer matrix is not the unique one because there are many ways to characterize a polynomial function, e.g., with its roots or with its coefficients. In a recent series of works (see ref.[1] and the references therein), a generalization of the T-Q relation with an extra inhomogeneous term, i.e., the inhomogeneous T-Q relation, was proposed and used in solving some integrable models without U(1) symmetry. This generalization seems to be a universal solution of the Hirota type equations (recursive inversion identities) and can account for the boundary conditions self-consistently without losing a polynomial Q operator. In this note, we show that the inhomogeneous T-Q relation can also characterize the spectrum of the ordinary integrable models that can be characterized by Baxter’s T-Q relation and can be solved with the ordinary Bethe Ansatz methods. Let us consider the periodic XXX spin2 chain. The corresponding R-matrix reads R0,j(u) = u+ ηP0,j = u+ 1 2 η(1 + ~σ0 · ~σj), (1) where η is the crossing parameter (we put η = 1 in this case), ~σj = (σ x j , σ y j , σ z j ) are the Pauli matrices, and Pi,j is the permutation operator possessing the properties: Pi,jOj = OiPi,j, P 2 i,j = id, trjPi,j = triPi,j = id, (2) for arbitrary operator O defined in the corresponding tensor space. This R-matrix satisfies the Yang-Baxter equation R1,2(u− v)R1,3(u)R2,3(v) = R2,3(v)R1,3(u)R1,2(u− v). (3) It is easy to show that the R-matrix (1) also satisfies the following relations: Initial condition : R1,2(0) = P1,2, (4) Unitary relation : R1,2(u)R2,1(−u) = −u(u− 1)× id, (5) Crossing relation : R1,2(u) = −σ y 1R t1 1,2(−u− 1)σ y 1 . (6) 2 The monodromy matrix and the corresponding transfer matrix of the periodic XXX spin2 chain are respectively defined as T0(u) = R0,N(u− θN) · · ·R0,1(u− θ1) = ( A(u) B(u) C(u) D(u) ) , (7) t(u) = tr0T0(u) = A(u) +D(u), (8) with {θj |j = 1, · · · , N} being some generic site-dependent inhomogeneity parameters. With the Yang-Baxter equation we can show that [t(u), t(v)] = 0. The Hamiltonian of the XXX spin2 chain is thus expressed as