标题： 量子自旋链中的不对称价键固态：奇自旋与偶自旋的区别 显示英文标题

标题： The Asymmetric Valence-Bond-Solid States in Quantum Spin Chains: The Difference Between Odd and Even Spins

摘要： 自旋$S$量子反铁磁链在低能性质上的定性差异，由哈代恩发现的整数$S$和半奇数$S$的差异，可以通过阿弗雷克、肯尼迪、利布和塔斯基提出的价键图解直观地理解。 在这里，我们提供一种类似的直观图解，说明奇数$S$和偶数$S$链之间的定性差异，这是对称保护拓扑（SPT）相理论的核心。 更准确地说，我们定义了一族单参数状态，我们称之为非对称价键固体（VBS）状态，这些状态在自旋为$S=1$和 2 的量子自旋链中，连续地在阿弗雷克-肯尼迪-利布-塔斯基（AKLT）态和平凡零态之间变化。 非对称 VBS 态是通过对 AKLT 态进行系统修改得到的。 它始终具有指数衰减的截断关联函数，并且是短程哈密顿量的唯一能隙基态。 我们还观察到，非对称的VBS态在$\mathbb{Z}_2\times\mathbb{Z}_2$时具有时间反演、$S=2$和键中心反演对称性，但在$S=1$时不具有这些对称性。 这与已知的事实一致，即当$S=2$时，AKLT模型属于平凡的SPT相，而当$S=1$时属于非平凡的SPT相。 尽管这种无序态的插值族早已为人所知，但我们的构造是统一的，并基于一个简单的物理图像。 它还可以扩展到具有任意整数$S$的自旋链，并为我们提供了对奇数自旋和偶数自旋模型之间本质区别的直观解释。 显示英文摘要

摘要： The qualitative difference in low-energy properties of spin $S$ quantum antiferromagnetic chains with integer $S$ and half-odd-integer $S$ discovered by Haldane can be intuitively understood in terms of the valence-bond picture proposed by Affleck, Kennedy, Lieb, and Tasaki. Here we develop a similarly intuitive diagrammatic explanation of the qualitative difference between chains with odd $S$ and even $S$, which is at the heart of the theory of symmetry-protected topological (SPT) phases. More precisely, we define one-parameter families of states, which we call the asymmetric valence-bond solid (VBS) states, that continuously interpolate between the Affleck-Kennedy-Lieb-Tasaki (AKLT) state and the trivial zero state in quantum spin chains with $S=1$ and 2. The asymmetric VBS state is obtained by systematically modifying the AKLT state. It always has exponentially decaying truncated correlation functions and is a unique gapped ground state of a short-ranged Hamiltonian. We also observe that the asymmetric VBS state possesses the time-reversal, the $\mathbb{Z}_2\times\mathbb{Z}_2$, and the bond-centered inversion symmetries for $S=2$, but not for $S=1$. This is consistent with the known fact that the AKLT model belongs to the trivial SPT phase if $S=2$ and to a nontrivial SPT phase if $S=1$. Although such interpolating families of disordered states were already known, our construction is unified and is based on a simple physical picture. It also extends to spin chains with general integer $S$ and provides us with an intuitive explanation of the essential difference between models with odd and even spins.