A Heisenberg spin-$s$ chain with alternating ferromagnetic ($-J_1^F<0$) and antiferromagnetic ($J_1^A>0$) nearest-neighbor (NN) interactions, exhibits the Dimer and spin-$2s$ Haldane phases in the limits $J_1^F/J_1^A \rightarrow 0$ and $J_1^F/J_1^A \rightarrow \infty$ respectively. These two phases are understood to be topologically equivalent. Induction of the frustration through the next nearest-neighbor ferromagnetic interaction ($-J_2^F<0$) produces a very rich quantum phase diagram. With frustration, the whole phase diagram is divided into a ferromagnetic (FM) and a nonmagnetic (NM) phase. For $s=1/2$, the full NM phase is seen to be of Haldane-Dimer type, but for $s>1/2$, a spiral phase comes between the FM and the Haldane-Dimer phases. The study of a suitably defined string-order parameter and spin-gap at the phase boundary indicates that the Haldane-Dimer and spiral phases have different topological characters. We also find that, along the $J_2^F=\frac 12 J_1^F$ line in the NM phase, an NN dimer state is the {\it exact} groundstate, provided $J_1^A>J_C=\kappa J_1^F$ where $\kappa \le s + h$ for applied magnetic field $h$. Without magnetic field, the position of $J_C$ is on the FM-NM phase boundary when $s=1/2$, but for $s>1/2$, the location of $J_C$ is on the phase separation line between the Haldane-Dimer and spiral phases.

中文翻译：

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受挫自旋链中的霍尔丹和二聚体相：精确的基态和相关的拓扑相变


海森堡自旋 $s$ 链具有交替的铁磁 ($-J_1^F<0$) 和反铁磁 ($J_1^A>0$) 最近邻 (NN) 相互作用，展示了二聚体和自旋 $2s$ Haldane分别在限制 $J_1^F/J_1^A \rightarrow 0$ 和 $J_1^F/J_1^A \rightarrow \infty$ 中。这两个阶段被认为在拓扑上是等效的。通过下一个最近邻铁磁相互作用 ($-J_2^F<0$) 感应挫败会产生非常丰富的量子相图。令人沮丧的是，整个相图被分为铁磁（FM）相和非磁（NM）相。对于 $s=1/2$，完整的 NM 相被认为是 Haldane-Dimer 类型，但对于 $s>1/2$，螺旋相出现在 FM 和 Haldane-Dimer 相之间。对适当定义的弦序参数和相边界自旋间隙的研究表明，Haldane-二聚体和螺旋相具有不同的拓扑特征。我们还发现，沿着 NM 相中的 $J_2^F=\frac 12 J_1^F$ 线，NN 二聚体状态是{\确切}的基态，前提是 $J_1^A>J_C=\kappa J_1^ F$ 其中$\kappa \le s + h$ 表示施加的磁场$h$。在没有磁场的情况下，当$s=1/2$时，$J_C$的位置位于FM-NM相界上，但对于$s>1/2$，$J_C$的位置位于FM-NM相界线上Haldane-二聚体和螺旋相。