Surveying the quantum group symmetries of integrable open spin chains

TL;DR

This paper surveys the quantum group symmetries in integrable open spin chains derived from affine Lie algebras, revealing duality and $Z_2$ symmetries that explain transfer matrix degeneracies.

Contribution

It constructs new families of integrable open quantum spin chains with specific quantum group symmetries and uncovers additional duality and $Z_2$ symmetries.

Findings

Transfer matrices exhibit quantum group invariance.

Duality symmetry is present for certain affine Lie algebra cases.

Additional $Z_2$ symmetries relate complex representations to conjugates.

Abstract

Using anisotropic R-matrices associated with affine Lie algebras $\hat g$ (specifically, $A_{2n}^{(2)}, A_{2n-1}^{(2)}, B_n^{(1)}, C_n^{(1)}, D_n^{(1)}$) and suitable corresponding K-matrices, we construct families of integrable open quantum spin chains of finite length, whose transfer matrices are invariant under the quantum group corresponding to removing one node from the Dynkin diagram of $\hat g$. We show that these transfer matrices also have a duality symmetry (for the cases $C_n^{(1)}$ and $D_n^{(1)}$) and additional $Z_2$ symmetries that map complex representations to their conjugates (for the cases $A_{2n-1}^{(2)}, B_n^{(1)}, D_n^{(1)}$). A key simplification is achieved by working in a certain "unitary" gauge, in which only the unbroken symmetry generators appear. The proofs of these symmetries rely on some new properties of the R-matrices. We use these symmetries to explain the degeneracies of the transfer matrices.