Majorana zero modes are well studied in the gapped phases of topological systems. We investigate Majorana zero modes at the topological quantum criticality in one dimensional topological superconducting model with longer range interaction. We identify stable localized Majorana zero modes appearing at criticality under certain conditions. Topological invariant number for these non-trivial criticalities is obtained from zeros of a complex function associated with the Hamiltonian. Behavior of parametric curve at criticalities validate the invariant obtained and account for the appearance of Majorana zero modes at criticality. Trivial and non-trivial topological nature of criticality due to the presence of multicritical point cause an unusual topological transition along the critical line. We observe and investigate this unique transition in terms of eigenvalue spectrum. Appearance of MZMs at criticality demands integer value of topological invariant number in order to validate the concept of bulk-boundary correspondence. Hence we propose a scheme to separate the invariant number into fractional and integer contribution to establish bulk-boundary correspondence at criticality.

中文翻译：

---

量子临界状态下的 Majorana 零模和体边界对应

Majorana 零模式在拓扑系统的间隙相中得到了很好的研究。我们在具有更远距离相互作用的一维拓扑超导模型中研究了拓扑量子临界状态下的 Majorana 零模式。我们确定了在某些条件下出现在临界状态的稳定局部马约拉纳零模式。这些非平凡临界性的拓扑不变数是从与哈密顿量相关的复杂函数的零点获得的。临界参数曲线的行为验证了所获得的不变量，并解释了临界状态下 Majorana 零模态的出现。由于多临界点的存在，临界的平凡和非平凡拓扑性质导致沿临界线的异常拓扑转变。我们在特征值谱方面观察并研究了这种独特的转变。临界状态下 MZM 的出现需要拓扑不变数的整数值，以验证体边界对应的概念。因此，我们提出了一种方案，将不变数分成小数和整数贡献，以在临界点建立批量边界对应。