Vertex Operator Approach to Semi-infiniteSpin Chain: Recent Progress

Abstract

Vertex operator approach is a powerful method to study exactly solvable models. We review recent progress of vertex operator approach to semi-infinite spin chain. (1) The first progress is a generalization of boundary condition. We study \(U_{q}(\widehat{sl}(2))\) spin chain with a triangular boundary, which gives a generalization of diagonal boundary (Baseilhac and Belliard, Nucl Phys B873:550–583, 2013; Baseilhac and Kojima, Nucl Phys B880:378–413, 2014). We give a bosonization of the boundary vacuum state. As an application, we derive a summation formulae of boundary magnetization. (2) The second progress is a generalization of hidden symmetry. We study supersymmetry \(U_{q}(\widehat{sl}(M\vert N))\) spin chain with a diagonal boundary (Kojima, J Math Phys 54(043507):40 pp., 2013). By now we have studied spin chain with a boundary, associated with symmetry \(U_{q}(\widehat{sl}(N))\), \(U_{q}(A_{2}^{(2)})\) and \(U_{q,p}(\widehat{sl}(N))\) (Furutsu and Kojima, J Math Phys 41:4413–4436, 2000; Yang and Zhang, Nucl Phys B596:495–512, 2001; Kojima, Int J Mod Phys A26:1973–1989, 2011; Miwa and Weston, Nucl Phys B486:517–545, 1997; Kojima, J Math Phys 52(01351):26 pp., 2011), where bosonizations of vertex operators are realized by “monomial”. However the vertex operator for \(U_{q}(\widehat{sl}(M\vert N))\) is realized by “sum”, a bosonization of boundary vacuum state is realized by “monomial”.