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arXiv:hep-th/9502060 · Nuclear Physics B

Difference equations in spin chains with a boundary

Michio Jimbo, Rinat Kedem, Hitoshi Konno, Tetsuji Miwa, Robert Weston

Published Aug 1, 1995• 108 citations
arXivHTMLPDF
DOI
10.1016/0550-3213(95)00218-h

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Exact physical quantities of the D2(2) {D}_2^{(2)} D2(2)​ spin chain model with generic open boundary conditions
A bstract We study the quantum integrable spin chain model associated with the twisted D2(2) {D}_2^{(2)} D2(2)​ D 2 2 algebra (or simply the D2(2) {D}_2^{(2)} D2(2)​ D 2 2 model) under generic open boundary conditions. The Hamiltonian of this model can be factorized into the sum of two staggered XXZ spin chains. Applying the t - W method, we derive the homogeneous Bethe ansatz equations for the zeros of the transfer matrix eigenvalues and the patterns of the corresponding zeros of the staggered XXZ spin chain with generic integrable boundaries. Based on these results, we analytically compute the surface energies and excitation energies of the D2(2) {D}_2^{(2)} D2(2)​ D 2 2 model in different regimes of boundary parameters.
The bounded XXZ spin chain is studied. The operator appeared in the q -difference equation discovered by Cherednik is modified into the transfer matrix of the model. Its eigenstate is composed by the vertex operators of the quantum affine algebra. By use of a geometric symmetry called the “turning symmetry” that the spin chain without boundary magnetic fields satisfies, the asymptotic form of the eigenstate is found and the spontaneous magnetization at a boundary is calculated. It is roughly twice as large as that in the half-infinite XXZ spin chain.
Motivated by a study of the crossing symmetry of the asymmetric twin or ‘gemini’ representation of the affine Hecke algebra we give a construction for crossing tensor space representations of ordinary Hecke algebras. These representations build solutions to the Yang–Baxter equation satisfying the crossing condition (that is, integrable quantum spin chains). We show that every crossing representation of the Temperley–Lieb algebra appears in this construction, and in particular that this construction builds new representations. We extend these to new representations of the blob algebra, which build new solutions to the boundary Yang–Baxter equation (i.e. open spin chains with integrable boundary conditions). We prove that the open spin chain Hamiltonian derived from Sklyanin’s commuting transfer matrix using such a solution can always be expressed as the representation of an element of the blob algebra, and determine this element. We determine the representation theory (irreducible content) of the new representations and hence show that all such Hamiltonians have the same spectrum up to multiplicity, for any given value of the algebraic boundary parameter. (A corollary is that our models have the same spectrum as the open XXZ chain with nondiagonal boundary—despite differing from this model in having reference states.) Using these multiplicity data, and other ideas, we investigate the underlying quantum group symmetry of the new Hamiltonians. We derive the form of the spectrum and the Bethe ansatz equations.