本文利用 Onsager 代数为可积 N 态 chiral-clock 模型引入枢轴过程，揭示 D_{2N} 对称性导致的偶数 N SPT 与奇数 N RSPT 序，并分析其基态、纠缠谱及丰富相图。
摘要：
Onsager 代数最初为求解二维 Ising 模型而提出，可用于为一族可积的 N 态 chiral-clock 模型构造守恒荷。我们展示它如何自然地为这一族手征 Hamilton 量产生一种“枢轴”过程。这些 Hamilton 量具有反幺正 CPT 对称性，当它与通常的 Z_N clock 对称性结合时，会给出非阿贝尔二面体对称群 D_{2N}。我们证明，该对称性使这一家族在所有偶数 N 下产生对称性保护拓扑（SPT）序，并在所有奇数 N 下产生表示-SPT（RSPT）物理。最简单的例子是一条次近邻链，它推广了 spin-1/2 cluster 模型，属于一种 SPT 物相。我们推导了其不动点基态的矩阵乘积态表示，以及随之而来的纠缠谱和对称分数化。我们分析了将该模型与 Onsager 可积 chiral Potts 链结合所得的丰富相图，发现了平庸、对称性破缺和（R）SPT 序，以及扩展的无隙区域。对于奇数 N，从 SPT 的观点看，这些相变“不必要地”具有临界性。
英文摘要：
The Onsager algebra, invented to solve the two-dimensional Ising model, can be used to construct conserved charges for a family of integrable $N$-state chiral clock models. We show how it naturally gives rise to a "pivot" procedure for this family of chiral Hamiltonians. These Hamiltonians have an anti-unitary CPT symmetry that when combined with the usual $\mathbb{Z}_N$ clock symmetry gives a non-abelian dihedral symmetry group $D_{2N}$. We show that this symmetry gives rise to symmetry-protected topological (SPT) order in this family for all even $N$, and representation-SPT (RSPT) physics for all odd $N$. The simplest such example is a next-nearest-neighbour chain generalising the spin-1/2 cluster model, an SPT phase of matter. We derive a matrix-product state representation of its fixed-point ground state along with the ensuing entanglement spectrum and symmetry fractionalisation. We analyse a rich phase diagram combining this model with the Onsager-integrable chiral Potts chain, and find trivial, symmetry-breaking and (R)SPT orders, as well as extended gapless regions. For odd $N$, the phase transitions are "unnecessarily" critical from the SPT point of view.
Charge pumps, pivot Hamiltonians and symmetry-protected topological phases
St John’s College and Mathematical Institute, University of Oxford（牛津大学圣约翰学院和数学研究所）；Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy, University of California, Los Angeles（加州大学洛杉矶分校物理与天文学系曼尼·L·鲍姆克理论物理研究所）；Pritzker School of Molecular Engineering, University of Chicago（芝加哥大学普利策分子工程学院）；Harish-Chandra Research Institute（哈里什-昌德拉研究所）
