ABSTRACT We consider the interacting helical liquid system at the one-dimensional edge of a two-dimensional topological insulator, coupled to an external magnetic field and s-wave superconductor and map it to an XYZ spin chain system. This model undergoes quantum Berezinskii–Kosterlitz–Thouless (BKT) transition with two limiting conditions. We derive the renormalization group (RG) equations explicitly and also present the flow lines behavior. We also present the behavior of RG flow lines based on the exact solution. We observe that the physics of Majorana fermion zero modes and the gaped Ising-ferromagnetic phase, which appears in a different context. We observe that the evidence of gapless helical Luttinger liquid phase as a common non-topological quantum phase for both quantum BKT transitions. We explain analytically and physically that there is no Majorana-Ising transition. In the presence of chemical potential, the system shows the commensurate to incommensurate transition.

中文翻译：

---

拓扑绝缘体的量子 Berezinskii-Kosterltz-Thouless 跃迁

摘要我们考虑在二维拓扑绝缘体的一维边缘相互作用的螺旋液体系统，耦合到外部磁场和 s 波超导体，并将其映射到 XYZ 自旋链系统。该模型经历了具有两个限制条件的量子 Berezinskii-Kosterlitz-Thouless (BKT) 跃迁。我们明确地推导出重整化群 (RG) 方程，并呈现流线行为。我们还展示了基于精确解的 RG 流线的行为。我们观察到马约拉纳费米子零模和带隙 Ising-铁磁相的物理特性，它们出现在不同的上下文中。我们观察到无间隙螺旋 Luttinger 液相作为两种量子 BKT 跃迁的常见非拓扑量子相的证据。我们从分析和物理上解释不存在 Majorana-Ising 过渡。在存在化学势的情况下，系统显示出相称到不相称的转变。