Exact physical quantities of the $D_2^{(2)}$ spin chain model with generic open boundary conditions

TL;DR

This work computes exact physical quantities for the $D_2^{(2)}$ spin chain under generic open boundaries by applying the $t$-$W$ method, leveraging a factorization of the transfer matrix into two staggered XXZ chains. The authors derive homogeneous Bethe ansatz equations for the zeros of the transfer-matrix eigenvalues and classify zero-patterns across six boundary-regimes to obtain explicit densities, surface energies, and elementary excitations. Key results include closed-form expressions for the surface energy and a regime-dependent description of boundary excitations, with the bulk and boundary properties showing clear independence in the thermodynamic limit. The approach provides exact benchmarks for twisted-affine integrable models and suggests avenues for exploring helical boundary states and current-carrying modes, as well as extensions to finite temperature and quench dynamics.

Abstract

We study the quantum integrable spin chain model associated with the twisted $D_2^{(2)}$ algebra (or simply the $D_2^{(2)}$ 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 $D_2^{(2)}$ model in different regimes of boundary parameters.

Figures (7)

• Figure 1: The distribution of $\bar{z}$-zeros for the ground state according to the boundary parameters.
• Figure 2: Pattern of $\bar{z}$-zeros with certain model parameters for the ground state. (a) The boundary parameters are chosen in regime I. (b) The boundary parameters are chosen in regime II. The blue asterisks indicate $\bar{z}$-zeros for $\{\bar{\theta}_j=0|j=1,\cdots,2N\}$ and the red circles specify $\bar{z}$-zeros with the inhomogeneity parameters $\{\bar{\theta}_j=0.1 j|j=1,\cdots,2N\}$.
• Figure 3: Patterns of $\bar{z}$-zeros with certain model parameters for $\{\bar{\theta}_j= 0|j=1,\cdots,2N\}$ at the ground state. The boundary parameters in $(a)$-$(d)$ are chosen in the regimes III-VI, respectively.
• Figure 4: Surface energy $E_b$ versus the boundary parameter $r={\rm Re}(s_1)$. The line indicates the analytic results and the squares indicate the DMRG results for $N=100$. They are consistent with each other very well.
• Figure 5: The distribution of $\bar{z}$-zeros for $\{\bar{\theta}_j= 0|j=1,\cdots,2N\}$ at the excited states. (a) The 19th excited state with $n=2$. (b) The 39th excited state with $n=3$.