Integrable XYZ Spin Chain with Boundaries

We consider a general class of boundary terms of the open XYZ spin-1/2 chain compatible with integrability. We have obtained the general elliptic solution of K-matrix obeying the boundary Yang-Baxter equation using the R-matrix of the eight vertex model and derived the associated integrable spin-chain Hamiltonian. 1+1 dimensional integrable models with boundaries find interesting applications in particle physics as well as condensed matter systems. In view of this, attempts have been made recently at the integrable extension of conformal field theories, [1,2] both massive and massless integrable quantum field theories [3∼6] and solvable lattice models [7∼13] to those with boundary terms. In the case of lattice models, relying on the earlier work by Cherednik, [8] Sklyanin has given [9] a general framework which enables us to treat this problem on an algebraic footing. Especially, the general solution of integrable boundary terms has been found in the XXZ and the XXX Heisenberg spin chain system [12] based on his framework. In this letter, we will work out the general solution of boundary interactions in the case of XYZ Heisenberg spin chain system. The Hamiltonian of the XYZ spin-1/2 chain is given by the transfer matrix of the eight vertex model. [14] The eight vertex model is defined in terms of the Boltzmann weights given by the elliptic solution R(u) of the Yang-Baxter (Y-B) equation R12(u− u ′)R13(u)R23(u ) =R23(u ′)R13(u)R12(u− u ). (1) Here we regard R(u) as linear operators acting on the tensor product of vector spaces V ⊗ V with V = Cv+ ⊕ Cv− and R12 ≡ R ⊗ 1, R23 ≡ 1 ⊗ R, etc. as those acting on V1 ⊗ V2 ⊗ V3, where Vi ∼= V, i = 1, 2, 3. Setting R(u)vε1 ⊗ vε2 = ∑ ε1,ε2 vε1 ⊗ vε2R(u) ε′1ε ′ 2 ε1ε2 and arranging the elements of R in the order (ε1, ε2) = (++), (+−), (−+), (−−), one can express the eight vertex R-matrix as follows.