AI 中文总结
该研究构建了源自𝔰𝔬(3)代数的SO(3)对称自旋模型，分析其在不同边界条件下的基态简并度，还将相关分析推广到SO(d)等李群，为小型父哈密顿量的基态特性提供了新实例。
AI 中文摘要
我们引入一种源自李代数𝔰𝔬(3)代数结构的SO(3)对称自旋模型：它是由李代数𝔰𝔬(3)的生成元与单位矩阵构建的矩阵乘积态（MPS）的最近邻父哈密顿量，因此编码了该李代数的二次关系。我们表征了所得模型的基态空间结构，结果表明：对于开边界条件（OBC），它呈现二次基态简并度，由每个奇数维（整数自旋）的一个不可约表示（irrep）构成；对于周期边界条件（PBC），它具有线性基态简并度，包含一个单重态（即该模型所基于的MPS）和一个铁磁态（具有最大自旋的irrep），因此表现出无能隙激发。我们还将OBC基态的构造与分析推广到SO(d)及其他紧致单李群。该模型所基于的MPS是内射MPS，因此其3-site父哈密顿量具有唯一基态空间且是有隙的。因此，观测到的临界行为源于所考虑的小型父哈密顿量。该模型的代数基态空间标度提供了一个与Schuch等人（2025，arXiv:2503.10767）所报道结果不同的例子，该研究表明内射MPS的小型父哈密顿量通常具有唯一基态，并给出了指数基态简并度的例子。
英文摘要
We introduce an $\mathrm{SO}(3)$-symmetric spin model derived from the algebraic structure of $\mathfrak{so}(3)$: It is obtained as the nearest-neighbor parent Hamiltonian of a Matrix Product State (MPS) built from the generators of the Lie algebra $\mathfrak{so}(3)$ and the identity matrix, and therefore encodes the quadratic relations of the Lie algebra. We characterize the ground space structure of the resulting model and show that for open boundary conditions (OBC), it exhibits a quadratic ground space degeneracy, given by one irreducible representation (irrep) of each odd dimension (integer spin). For periodic boundary conditions (PBC), it has a linear ground space degeneracy, consisting of a singlet --- namely, the MPS underlying the model --- and a ferromagnet (the irrep with maximal spin), and thus exhibits gapless excitations. We also generalize the construction and the analysis of the OBC ground space to $\mathrm{SO}(d)$ and other compact simple Lie groups. The MPS on which the model is based is an injective MPS, and therefore, its 3-site parent Hamiltonian has a unique ground space and is gapped. The observed critical behavior thus has its origin in the small parent Hamiltonian considered. The model's algebraic ground space scaling thus provides an example distinct from that reported in (Schuch et al., 2025, arXiv:2503.10767), where it was shown that small parent Hamiltonians of injective MPS generically have unique ground states, and examples with an exponential ground space degeneracy were given.
