Exact surface energies and boundary excitations of the Izergin-Korepin model with generic boundary fields

TL;DR

This work extends the $t$-$W$ method to the Izergin-Korepin model with generic (non-diagonal) boundary fields, leveraging twisted affine symmetry $A_2^{(2)}$ to parameterize transfer-matrix eigenvalues by zeroes. By deriving homogeneous zeroes BAEs and analyzing their ground-state zero patterns, it yields exact, regime-dependent surface energies and characterizes boundary excitations, uncovering correlation effects between the two boundary fields in certain parameter regions. The authors show that the leading-order surface energy is independent of the boundary twist parameters $ar\sigma$ and $ar\sigma'$, while providing a complete description of non-conjugate zero patterns that distinguish this twisted-model from untwisted ones. The results provide a precise thermodynamic and boundary-phenomenology framework for a non-$U(1)$-symmetric integrable system and pave the way for extending these methods to other twisted-affine algebras and related dynamical studies.

Abstract

The Izergin-Korepin model is an integrable model with the simplest twisted quantum affine algebra $U_q(A_2^{(2)})$ symmetry. Applying the $t-W$ method, we derive the homogeneous zero roots Bethe ansatz equations and the corresponding zero root patterns of the Izergin-Korepin model with generic integrable boundaries. Based on these results, we analytically compute the surface energies and boundary excitations in different regimes of boundary parameters of the model. It is shown that in some regimes, correlation effect appears between two boundary fields.

Figures (5)

• Figure 1: The distribution of $\bar{z}$-zeroes for the ground state in the $\chi_+$-$\chi_-$ plane.
• Figure 2: Pattern of $\bar{z}$-zeroes 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 $\bar{z}$-zeroes for $\{\bar{\theta}_j=0|j=1,\cdots,N\}$ and the red circles specify $\bar{z}$-zeroes with the inhomogeneity parameters $\{\bar{\theta}_j=0.1 j|j=1,\cdots,N\}$. The zeroes on the dashed thick lines, solid thick lines, dashed thin lines, and solid thin lines are classified as bulk pairs, free open boundary zeroes, boundary pairs, and extra pairs, respectively.
• Figure 3: Patterns of $\bar{z}$-zeroes with certain model parameters for $\{\bar{\theta}_j= 0|j=1,\cdots,N\}$ at the ground state. The boundary parameters in $(a)$-$(d)$ are chosen in the regimes III-VI, respectively. The zeroes on the dashed thick lines, solid thick lines, dashed thin lines, and solid thin lines are classified as bulk pairs, free open boundary zeroes, boundary pairs, and extra pairs, respectively.
• Figure 4: Surface energies versus the boundary parameters $\varepsilon$ and $\varepsilon^{\prime}$ for $\eta=0.5$. The lines indicate the analytic results and the squares indicate the DMRG results for $N=180$.
• Figure 5: (a) The distribution of $\bar{z}$-zeroes for $\{\bar{\theta}_j= 0|j=1,\cdots,N\}$ with $N=8$, $\eta=0.35$, $\varsigma=0.6$, $\bar{\varsigma}^{\prime}=0.7$, $\varepsilon=2$ and $\varepsilon^{\prime}=0.3$. Here the blue asterisks represent the pattern of zeroes at the ground state and the red squares denote those at the 32nd excited state with boundary pairs $\pi+i(2\eta-\chi_{-})$, $-i(2\eta-\chi_{-})$, $\pi-i(4\eta-\chi_{-})$, $i(4\eta-\chi_{-})$. (b) The excitation energy versus $\varepsilon$ in the thermodynamic limit.