Vertex operator approach to semi-infinite spin chain : recent progress

TL;DR

This paper reviews recent advances in the vertex operator approach to semi-infinite spin chains, including boundary condition generalizations and symmetry extensions, with explicit bosonization and summation formulas for boundary magnetization.

Contribution

It introduces new bosonization techniques for boundary states and extends the approach to supersymmetric and generalized boundary conditions in spin chains.

Findings

Bosonization of boundary vacuum states achieved.

Derived summation formulas for boundary magnetization.

Extended vertex operator methods to supersymmetric and generalized boundary conditions.

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 2013, Baseilhac and Kojima 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 $U_q(\widehat{sl}(M|N))$ spin chain with a diagonal boundary [Kojima 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-Kojima 2000, Yang-Zhang 2001, Kojima 2011, Miwa-Weston 1997, Kojima 2011], where bosonizations of vertex operators are realized by "monomial" . However the vertex operator for $U_q(\widehat{sl}(M|N))$ is realized by "sum", a bosonization of boundary vacuum state is realized by "monomial".