Exact surface energy of the $D^{(1)}_2$ spin chain with generic non-diagonal boundary reflections

TL;DR

This paper derives the exact solution for the $D^{(1)}_2$ quantum spin chain with non-diagonal boundaries, revealing its eigenvalues, Bethe ansatz equations, and surface energies, providing a basis for studying higher-rank models.

Contribution

It introduces a novel factorization approach and solves the model exactly, including eigenvalues, Bethe equations, and surface energies, for the first time.

Findings

Eigenvalues and Bethe ansatz equations obtained

Root distribution patterns identified

Surface energies calculated in the thermodynamic limit

Abstract

The exact solution of the $D^{(1)}_2$ quantum spin chain with generic non-diagonal boundary reflections is obtained. It is found that the generating functional of conserved quantities of the system can be factorized as the product of transfer matrices of two anisotropic $XXZ$ spin chains with open boundary conditions. By using the factorization identities and the fusion technique, the eigenvalues and the Bethe ansatz equations of the model are obtained. The eigenvalues are also parameterized by the zero roots of the transfer matrix, and the patterns of root distributions are obtained. Based on them, ground states energy and the surface energies induced by the twisted boundary magnetic fields in the thermodynamic limit are obtained. These results are checked by the numerical calculations. The corresponding isotropic limit is also discussed. The results given in this paper are the foundation to study the exact physical properties of high rank $D^{(1)}_{n}$ model by using the nested processes.