We study the exactly solvable one-dimensional model: the dimerized $XY$ chain with uniform and staggered transverse fields, equivalent upon fermionization to the noninteracting dimerized Kitaev-Majorana chain with modulation. The model has three known gapped phases with local and nonlocal (string) orders, along with the gapless incommensurate (IC) phase in the $U\left(1\right)$ limit. The criticality is controlled by the properties of zeros of model's partition function, analytically continued onto the complex wave numbers. In the ground state they become complex zeros of the spectrum of the Hamiltonian. The analysis of those roots yields the phase diagram which contains continuous quantum phase transitions and weaker singularities known as disorder lines (DLs) or modulation transitions. The latter are shown to occur in two types: DLs of the first kind with continuous appearance of the IC oscillations, and DLs of the second kind corresponding to a jump of the wave number of oscillations. The salient property of zeros of the spectrum is that the ground state is shown to be separable (factorized), and the model is disentangled on a subset of the DLs. From analysis of those zeros we also find the Majorana edge states and their wave functions.

中文翻译：

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可解量子链中的解缠结、无序线和马约拉纳边缘状态

我们研究了完全可解的一维模型：二聚体$X是$具有均匀和交错横向场的链，在费米化后等效于具有调制的非相互作用二聚化 Kitaev-Majorana 链。该模型具有三个已知的具有局部和非局部（串）阶的间隙相，以及无间隙不相称（IC）相$ü\left(1\right)$限制。临界由模型的配分函数的零点的性质控制，分析地延续到复波数上。在基态，它们成为哈密顿量谱的复零点。对这些根的分析产生了包含连续量子相变和称为无序线 (DL) 或调制跃迁的较弱奇点的相图。后者显示为两种类型：第一种 DLs 具有 IC 振荡的连续出现，第二种 DLs 对应于振荡波数的跳跃。频谱零点的显着特性是基态被证明是可分离的（分解），并且模型在 DL 的子集上解开。