标题： 量子自旋链中的不对称价键固体态：奇偶自旋之间的差异 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： Haldane 发现的自旋$S$整数$S$和半奇数$S$量子反铁磁链在低能性质上的定性差异可以通过 Affleck、Kennedy、Lieb 和 Tasaki 提出的价键图象直观地理解。 在这里，我们发展了一种类似的直观图解解释，用于解释奇数$S$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，我们称之为不对称价键固溶体（VBS）状态，这些状态在具有$S=1$和 2 的量子自旋链中的 Affleck-Kennedy-Lieb-Tasaki (AKLT) 态和平凡零态之间连续插值。 不对称 VBS 态通过系统性修改 AKLT 态获得。 它总是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带隙基态。 我们还观察到，不对称VBS态对于$S=2$具有时间反演、$\mathbb{Z}_2\times\mathbb{Z}_2$和 bond-centered 反演对称性，但不对于$S=1$。 这与已知的事实一致，即如果$S=2$，AKLT模型属于平凡的SPT相；如果$S=1$，则属于非平凡的SPT相。 尽管这种无序态的插值族已经为人所知，但我们的构建是统一的，并基于一个简单的物理图像。 它也扩展到了具有普遍整数$S$的自旋链，并为我们提供了奇偶自旋模型之间本质差异的直观解释。 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.