标题： 格罗诺夫斯基-杰丹诺夫XYZ自旋链的伤痕：现代代数观点及在高维晶格中的实现 显示英文标题

标题： Granovskii-Zhedanov Scar of XYZ Spin-chain: Modern Algebraic Perspectives and Realization in Higher Dimensional Lattices

摘要： 在Granovskii和Zhedanov的一项工作中，发现了一种表现出零纠缠和长周期性的令人惊讶的疤痕态，值得注意的是，这比多体疤痕概念成为活跃研究主题早了将近三十年。 在本研究中，我们在现代量子多体疤痕理解的框架内揭示了Granovskii-Zhedanov（GZ）疤痕的起源。 我们证明，可以通过标准谱生成代数（SGA）框架以及哈密顿量的群论表述来有效描述该疤痕子空间。 然而，这种描述仅适用于XXZ极限，在此情况下，疤痕子空间内存在准$U(1)$对称性。 相反，对于GZ疤痕子空间缺乏这种准$U(1)$对称性，限制了这些标准表述的适用性。 我们提出了两种替代技术：近似SGA和广义SGA，它们在XYZ情况下构建并描述疤痕子空间。 使用这些方法，我们可以表征疤痕子空间。 我们进一步探讨了在具有中心对称性的高维均匀自旋交换系统中构建与晶格无关的GZ疤痕的可能性，使用为GZ疤痕构建开发的图形规则。 我们的结果表明，具有奇数配位数的均匀晶格或具有奇数边数的晶胞无法支持与晶格无关的GZ疤痕，而具有偶数配位数和偶数边数晶胞的均匀晶格在特定情况下可以容纳此类与晶格无关的疤痕。 值得注意的是，如果某些键保留自旋交换相互作用的完整$SU(2)$对称性，从而破坏晶格的空间均匀性，那么在具有奇数配位数或奇数边数晶胞的系统中，仍可以出现与晶格无关的GZ疤痕。 显示英文摘要

摘要： In a work by Granovskii and Zhedanov, a surprising scar state exhibiting zero entanglement and long periodicity was discovered in the XYZ spin chains; remarkably, nearly three decades before the concept of many-body scars became a subject of active research. In this study, we uncover the origin of the Granovskii-Zhedanov (GZ) scar within the framework of the modern understanding of quantum many-body scars. We demonstrate that the scar subspace can be effectively described using the standard spectrum-generating algebra (SGA) framework and through a group-theoretical formulation of the Hamiltonian. This description, however, is applicable only in the XXZ limit, where a quasi-$U(1)$ symmetry exists within the scar subspace. In contrast, the absence of such quasi-$U(1)$ symmetry for the GZ scar subspace restricts the applicability of these standard formulations. We propose two alternative techniques: approximated SGA and generalized SGA, which construct and describe the scar subspace in the XYZ case. Using these approaches, we can characterize the scar subspaces. We further explore the possibility of constructing lattice-independent GZ scars in higher-dimensional uniform spin-exchange systems with centrosymmetry, using graphical rules developed for GZ scar construction. Our results indicate that lattice-independent GZ scars cannot be supported on uniform lattices with odd coordination numbers or plaquettes with an odd number of edges, while uniform lattices featuring even coordination numbers and even-edged plaquettes can host such lattice-independent scars in specific scenarios. Remarkably, if certain bonds retain the full $SU(2)$ symmetry of the spin-exchange interaction, thereby breaking the spatial uniformity of the lattice, lattice-independent GZ scars can still emerge in systems with odd coordination numbers or plaquettes with an odd number of edges.