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Spectrum of the Transfer Matrices of the Spin Chains Associated with the \(A^{(2)}_3\) Lie Algebra

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Abstract

We study the exact solution of quantum integrable system associated with the \(A^{(2)}_3\) twist Lie algebra, where the boundary reflection matrices have non-diagonal elements thus the U(1) symmetry is broken. With the help of the fusion technique, we obtain the closed recursive relations of the fused transfer matrices. Based on them, together with the asymptotic behaviors and the values at special points, we obtain the eigenvalues and Bethe ansatz equations of the system. We also show that the method is universal and valid for the periodic boundary condition where the U(1) symmetry is reserved. The results in this paper can be applied to studying the exact solution of the \(A^{(2)}_n\)-related integrable models with arbitrary n.

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Notes

  1. If \(m\le 0\), we have \(0\le L_1\le N\). When \(m\ge 0\), we have \(N\le L_1\le 2N\).

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Acknowledgements

We would like to thank Professor Y. Wang for his valuable discussions and continuous encouragement. The financial supports from National Key R &D Program of China (Grant No. 2021YFA1402104), the National Natural Science Foundation of China (Grant Nos. 12175180, 12105221, 12074410, 12047502, 12075177, 11934015, 11975183, 11947301, 91536115, 12275214, 12205235 and 12105221), Major Basic Research Program of Natural Science of Shaanxi Province (Grant Nos. 2021JCW-19, 2017KCT-12 and 2017ZDJC-32), the Scientific Research Program Funded by Shaanxi Provincial Education Department (Grant No. 21JK0946), Australian Research Council (Grant No. DP 190101529), Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB33000000), Beijing National Laboratory for Condensed Matter Physics (Grant No. 202162100001), Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, Xi’an Jiaotong University, and the Double First-Class University Construction Project of Northwest University are gratefully acknowledged.

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Authors and Affiliations

  1. Ministry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter, School of Physics, Xi’an Jiaotong University, Xi’an, 710049, China

    Guang-Liang Li

  2. Peng Huanwu Center for Fundamental Theory, Xi’an, 710127, China

    Guang-Liang Li, Junpeng Cao, Kun Hao, Xiaotian Xu, Tao Yang & Wen-Li Yang

  3. Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China

    Junpeng Cao

  4. School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Junpeng Cao

  5. Songshan Lake Materials Laboratory, Dongguan, Guangdong, 523808, China

    Junpeng Cao

  6. Institute of Modern Physics, Northwest University, Xi’an, 710127, China

    Kun Hao, Xiaotian Xu, Tao Yang & Wen-Li Yang

  7. Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi’an, 710127, China

    Kun Hao, Pei Sun, Xiaotian Xu, Tao Yang & Wen-Li Yang

  8. School of Physics, Northwest University, Xi’an, 710127, China

    Pei Sun & Wen-Li Yang

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  1. Guang-Liang Li
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Appendix A: Expression of the R-Matrix \(\varvec{R}_{{\overline{1}}{2}}(u)\)

Appendix A: Expression of the R-Matrix \(\varvec{R}_{{\overline{1}}{2}}(u)\)

In this appendix, we give the explicit expression of the R-matrix \(R_{{\bar{1}}{2}}(u)\) define in (3.7) as

$$\begin{aligned} R_{\bar{1}2}(u)&= \begin{pmatrix} \begin{array}{cccc|cccc|cccc|cccc|cccc|cccc} r_1&{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{} \\ &{}r_1&{}&{}&{}&{} &{}&{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{} \\ &{}&{}r_2&{} &{}&{}r_7 &{}&{} &{}r_8&{}&{}&{} &{}r_{10}&{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{} \\ &{}&{}&{}r_{2} &{}&{} &{}&{} &{}&{}r_9&{}&{} &{}&{}r_{11}&{}&{} &{}r_{12}&{} &{}&{}&{}&{}&{}&{} \\ \hline &{}&{}&{}&{}r_{1}&{} &{}&{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{} \\ &{}&{}r_7&{} &{}&{}r_{2} &{}&{} &{}r_8&{}&{}&{} &{}-r_{10}&{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{} \\ &{}&{}&{} &{}&{} &{}r_{1}&{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{} \\ &{}&{}&{} &{}&{} &{}&{}r_{2} &{} &{}&{}r_9&{} &{}&{}&{}-r_{11}&{} &{}&{} &{}&{} &{}r_{12}&{}&{}&{} \\ \hline &{}&{}r_{13}&{} &{}r_{13}&{} &{}&{}&{}r_{3}&{}&{}&{} &{}&{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{} \\ &{}&{}&{}r_{14} &{}&{} &{}&{} &{}&{}r_{4}&{}&{} &{}&{}r_{15}&{}&{} &{}r_9&{} &{}&{}&{}&{}&{}&{} \\ &{}&{}&{} &{}&{} &{}&{}r_{14} &{}&{}&{}r_{4}&{} &{}&{}&{}-r_{15}&{} &{}&{} &{}&{} &{}r_{9}&{}&{}&{} \\ &{}&{}&{} &{}&{} &{}&{} &{}&{}&{}&{}r_{3} &{}&{}&{}&{} &{}&{} &{}r_8&{} &{}&{}r_8&{}&{} \\ \hline &{}&{}r_{16}&{} &{}-r_{16}&{} &{} &{} &{}&{}&{}&{} &{}r_{5}&{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{} \\ &{}&{}&{}r_{17} &{}&{} &{}&{} &{}&{}-r_{15}&{}&{} &{}&{}r_{6}&{}&{} &{}-r_{11}&{} &{}&{}&{}&{}&{}&{} \\ &{}&{}&{} &{}&{} &{}&{}-r_{17} &{}&{}&{}r_{15}&{} &{}&{}&{}r_{6}&{} &{}&{} &{}&{}&{}r_{11}&{}&{}&{} \\ &{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{} &{}&{}&{}&{}r_{5}&{}&{} &{}-r_{10}&{} &{}&{}r_{10}&{}&{} \\ \hline &{}&{}&{}r_{18} &{}&{} &{}&{} &{}&{}r_{14}&{}&{} &{}&{}-r_{17}&{}&{}&{}r_{2}&{} &{}&{}&{}&{}&{}&{} \\ &{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{}r_{1} &{}&{}&{}&{}&{}&{} \\ &{}&{}&{} &{}&{} &{}&{} &{}&{}&{}&{}r_{13} &{}&{}&{}&{}-r_{16}&{}&{} &{}r_{2}&{} &{}&{}r_7&{}&{} \\ &{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{} &{}&{}r_{1}&{}&{}&{}&{} \\ \hline &{}&{}&{} &{}&{} &{}&{}r_{18} &{}&{}&{}r_{14}&{} &{}&{}&{}r_{17}&{}&{}&{} &{}&{}&{}r_{2}&{}&{}&{} \\ &{}&{}&{} &{}&{} &{}&{} &{}&{}&{}&{}r_{13} &{}&{}&{}&{}r_{16}&{}&{} &{}r_7&{}&{}&{}r_{2}&{}&{} \\ &{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{} &{}&{}&{}&{}&{}r_{1}&{} \\ &{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{} &{}&{}&{}&{}&{}&{}r_{1} \end{array} \end{pmatrix}, \end{aligned}$$
(A.1)
$$\begin{aligned} r_1=&\,2\sinh (u-3\eta ),\ r_2=2\sinh (u-\eta ),\ r_3=4\sinh \frac{1}{2}(u-3\eta )\cosh \frac{1}{2}(u-\eta ),\nonumber \\ r_4=&\,2(\sinh (u-2\eta )+\sinh 2\eta \sinh \eta ),\ r_5=4\sinh \frac{1}{2}(u-\eta )\cosh \frac{1}{2}(u-3\eta ),\nonumber \\ r_6=&\,2(\sinh (u-2\eta )-\sinh 2\eta \sinh \eta ),\ r_7=-2\sinh 2\eta ,\nonumber \\ r_8=&\,-4e^{-\frac{u}{2}}\sinh \eta \sqrt{\cosh \eta }\sinh \frac{1}{2}(u-3\eta ),\nonumber \\ r_9=&\,-4e^{-\frac{u}{2}+\eta }\sinh \eta \sqrt{\cosh \eta }\cosh \frac{1}{2}(u-\eta ),\nonumber \\ r_{10}=&\,4e^{-\frac{u}{2}}\sinh \eta \sqrt{\cosh \eta }\cosh \frac{1}{2}(u-3\eta ), \nonumber \\ r_{11}=&\,4e^{-\frac{u}{2}+\eta }\sinh \eta \sqrt{\cosh \eta }\sinh \frac{1}{2}(u-\eta ),\nonumber \\ r_{12}=&\,2e^{-\eta }\sinh 2\eta ,\ r_{13}=-e^ur_8,\ r_{14}=e^{u-2\eta }r_9, \ r_{15}=-2\sinh \eta \sinh 2\eta ,\nonumber \\ r_{16}=&\,e^ur_{10},\ r_{17}=-e^{u-2\eta }r_{11},\ r_{18}=2e^{\eta }\sinh 2\eta . \end{aligned}$$
(A.2)

The above expression allows us to derive the very properties (3.9) of the resulting fused R-matrix \(R_{\bar{1}2}(u)\).

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Li, GL., Cao, J., Hao, K. et al. Spectrum of the Transfer Matrices of the Spin Chains Associated with the \(A^{(2)}_3\) Lie Algebra. Commun. Math. Phys. 399, 651–672 (2023). https://doi.org/10.1007/s00220-022-04566-9

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