We present analysis of a single channel interacting quantum wire problem inthe presence of spin-orbit interaction. The spin-orbit coupling breaks thespin-rotational symmetry from SU(2) to U(1) and breaks inversion symmetry. Thelow-energy theory is then a two band model with a difference of Fermivelocities $\delta v$. Using bosonization and a two-loop renormalization groupprocedure we show that electron-electron interactions can open a gap in thespin sector of the theory when the interaction strength $U$ is smaller than$\delta v$ in appropriate units. For repulsive interactions, the resultingstrong coupling phase is of the spin-density-wave type. We show that this phasehas peculiar emergent topological properties. The gapped spin sector behaves asa topological insulator, with zero-energy edge modes with fractional spin. Onthe other hand, the charge sector remains critical, meaning the entire systemis metallic. However, this bulk electron liquid as a whole exhibits propertiescommonly associated with the one-dimensional edge states of two-dimensionalspin-Hall insulators, in particular, the conduction of $2e^2/h$ is robustagainst nonmagnetic impurities.
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机译：我们提出了自旋轨道相互作用下单通道相互作用量子线问题的分析。自旋轨道耦合破坏了从SU（2）到U（1）的自旋旋转对称性，并破坏了反演对称性。于是，低能理论是一个两能带模型，具有费米能位置$ \ delta v $的差异。使用玻色化和两环重归一化组过程，我们表明当相互作用强度$ U $小于$ \ delta v $时，电子-电子相互作用可以在理论的旋转扇区中打开一个缺口。对于排斥相互作用，产生的强耦合相为自旋密度波类型。我们表明，该阶段具有特殊的紧急拓扑特性。带隙的自旋扇区表现为拓扑绝缘体，具有零能量边沿模式和分数自旋。另一方面，电荷扇区仍然很关键，这意味着整个系统都是金属的。然而，这种整体电子液体整体上具有与二维自旋霍尔绝缘子的一维边缘状态共同相关的特性，特别是，$ 2e ^ 2 / h $的导电性强于非磁性杂质。
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