Recently, product states have been identified as simple-structured eigenstates of XXZ Heisenberg spin models in arbitrary dimensions, occurring at anisotropy values corresponding to certain roots of unity. Yet, the product states typically only span parts of a larger degenerate eigenspace. Here, we classify this eigenspace in the one-dimensional periodic XXZ chain at all roots of unity $q$, where $q^2$ is an $\ell$-th primitive root of unity. For commensurate chain lengths $N$ with $q^N=1$, we prove that the minimal degeneracy is $2^{N/\ell}\ell$ using the representation theory of the affine Temperley-Lieb (aTL) algebra. For the incommensurate case, we derive analogous exponential lower bounds of $2^{2\lfloor\frac{N}{2\ell}\rfloor+1}$ if $N$ is even and $2^{2\lfloor \frac{N}{2\ell}+\frac{1}{2}\rfloor}$ if $N$ is odd and $q^\ell=1$. Our proof employs the morphisms between aTL modules discovered by Pinet and Saint-Aubin and emphasizes the importance of exact sequences and hidden twisted boundary condition sectors that mediate the degeneracy. In the case of commensurate chain lengths, we connect to the Fabricius-McCoy string construction of all Bethe roots of the degenerate subspace, which previously uncovered parts of our results. We corroborate our results numerically and demonstrate that the lower bound is saturated for chain lengths $N\leq20$. Our work demonstrates for a concrete system how the interplay of the Bethe ansatz, aTL representation theory, and twisted boundary conditions explains degeneracy connected to long-lived product states, stimulating research towards generalization to higher dimensions. Exponential degeneracy could boost applications of spin chains as quantum sensors.
Movable seams in root-of-unity XXZ chains: Relative flux classes and antiunitary spectral pairing
School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院）；Shanghai Key Laboratory of Particle Physics and Cosmology（上海粒子物理与宇宙学重点实验室）；School of Physics, East China University of Science and Technology（华东理工大学物理系）
Finite size scaling for a first order transition where a continuous symmetry is broken: The spin-flop transition in the 3D XXZ Heisenberg antiferromagnet
利用高斯概率分布的一种近似方法，并引入一个唯象的“简并”因子，发展了连续对称性被破坏的一阶相变的有限尺寸标度。为了研究$L \times L \times L$简单立方晶格上类伊辛反铁磁态和倾斜类XY反铁磁态之间的一阶“自旋翻转”跃迁的有限尺寸行为，将预测结果与三维XXZ海森堡反铁磁场的蒙特卡罗模拟数据进行了比较。我们的理论预测，对于大的线性维数$L$的所有的序参量的时刻，以及四阶累积量的字段依赖表现出普遍的交叉。对前导顺序的更正应与反向体积成比例。在自旋翻转转变点处的这些交点的值可以用表征有序相的相对简并性的因子$q$来表示。我们的理论产量$q=π$，我们提出的数值证据，这是与此预测兼容。理论和模拟之间的协议意味着迄今未知的普遍性，可以调用一阶相变。
Exact ballistic energy transport and emergent XXZ dynamics in an integrable three-state chain
Nanophotonics and Biophotonics Key Laboratory of Jilin Province, School of Physics, Changchun University of Science and Technology（吉林省纳米光子学与生物光子学重点实验室，长春理工大学物理学院）
Rational Q-systems at Root of Unity I. Closed Chains
School of physics & Shing-Tung Yau Center, Southeast University（东南大学物理学院与邵逸夫中心）；Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo（东京大学宇宙物理数学研究所（WPI）、东京大学高等研究院）
Construction of some missing eigenvectors of the XYZ spin chain at the discrete coupling constants and the exponentially large spectral degeneracy of the transfer matrix
讨论了在离散耦合参数下构造八顶点模型传递矩阵特征向量的代数方法。本文考虑椭圆量子群$E_{τ, η}(sl_2)$的代数Bethe不等式，其中对任意整数$N$，$m_1$和$m_2$，参数$η$满足$2 N η= m_1 + m_2 τ$.当$m_1$或$m_2$是奇数时，这样获得的本征向量先前没有讨论过。此外，我们还构造了XYZ自旋链的简并本征矢族，其中一些本征矢与XXZ自旋链的$sl_2$圈代数对称性有关。证明了当$L/N$为偶数时，$L$格点上XYZ自旋链的某些简并本征空间的维数由$N 2^{L/N}$给出.本文还讨论了一些相关的IRF模型的传递矩阵的特征向量的构造。
