A quantum spin Hall insulator is a two-dimensional state of matter consisting of an insulating bulk and one-dimensional helical edge states. While these edge states are topologically protected against elastic backscattering in the presence of disorder, interaction-induced inelastic terms may yield a finite conductivity. By using a kinetic equation approach, we find the backscattering rate $\tau^{-1}$ and the semiclassical conductivity in the regimes of high ($\omega \gg \tau^{-1}$) and low ($\omega \ll \tau^{-1}$) frequency. By comparing the two limits, we find that the parametric dependence of conductivity is described by the Drude formula for the case of a disordered edge. On the other hand, in the clean case where the resistance originates from umklapp interactions, the conductivity takes a non-Drude form with a parametric suppression of scattering in the dc limit as compared to the ac case. This behavior is due to the peculiarity of umklapp scattering processes involving necessarily the state at the "Dirac point". In order to take into account Luttinger liquid effects, we complement the kinetic equation analysis by treating interactions exactly in bosonization and calculating conductivity using the Kubo formula. In this way, we obtain the frequency and temperature dependence of conductivity over a wide range of parameters. We find the temperature and frequency dependence of the transport scattering time in a disordered system as $\tau \sim [\max{(\omega,T)}]^{-2K-2}$, for $K>2/3$ and $\tau \sim [\max{(\omega,T)}]^{-8K+2}$ for $K <2/3$.Comment: 28 pages, 6 figure

We present analysis of a single channel interacting quantum wire problem in the presence of spin-orbit interaction. The spin-orbit coupling breaks the spin-rotational symmetry from SU(2) to U(1) and breaks inversion symmetry. The low-energy theory is then a two band model with a difference of Fermi velocities δv. Using bosonization and a two-loop renormalization group procedure we show that electron-electron interactions can open a gap in the spin sector of the theory when the interaction strength U is smaller than δv in appropriate units. For repulsive interactions, the resulting strong coupling phase is of the spin-density-wave type. We show that this phase has peculiar emergent topological properties. The gapped spin sector behaves as a topological insulator, with zero-energy edge modes with fractional spin. On the other hand, the charge sector remains critical, meaning the entire system is metallic. However, this bulk electron liquid as a whole exhibits properties commonly associated with the one-dimensional edge states of two-dimensional spin-Hall insulators, in particular, the conduction of 2e 2 /h is robust against nonmagnetic impurities. arXiv:1504.05016v2 [cond-mat.mes-hall]

We discuss two one-dimensional model systems -the first is a single channel quantum wire with Ising anisotropy, while the second is two coupled helical edge states. We show that the two models are governed by the same low energy effective field theory, and interactions drive both systems to exhibit phases which are metallic, but with all single particle excitations gapped. We show that such states may be either topological or trivial; in the former case, the system demonstrates gapless end states, and insensitivity to disorder.

We study theoretically the transport of the one-dimensional single-channel interacting electron gas through a strong potential barrier in the parameter regime where the spin sector of the low-energy Luttinger liquid theory is gapped by interaction. This phase is of particular interest since it exhibits non-trivial interaction-induced topological properties. Using bosonization and an expansion in the tunneling strength, we calculate the conductance through the barrier as a function of the temperature as well as the local density of states (LDOS) at the barrier. Our main result concerns the mechanism of bound-state mediated tunneling. The characteristic feature of the topological phase is the emergence of protected zero-energy bound states with fractional spin located at the impurity position. By flipping the fractional spin the edge states can absorb or emit spinons and thus enable single electron tunneling across the impurity even though the bulk spectrum for these excitations is gapped. This results in a finite LDOS below the bulk gap and in a non-monotonic behavior of the conductance. The system represents an important physical example of an interacting symmetry-protected topological phase-which combines features of a topological spin insulator and a topological charge metal-in which the topology can be probed by measuring transport properties. arXiv:1709.08965v1 [cond-mat.str-el]

We theoretically study Coulomb drag between two helical edges with broken spin-rotational symmetry, such as would occur in two capacitively coupled quantum spin Hall insulators. For the helical edges, Coulomb drag is particularly interesting because it specifically probes the inelastic interactions that break the conductance quantization for a single edge. Using the kinetic equation formalism, supplemented by bosonization, we find that the drag resistivity ρD exhibits a nonmonotonic dependence on the temperature T . In the limit of low T , ρD vanishes with decreasing T as a power law if intraedge interactions are not too strong. This is in stark contrast to Coulomb drag in conventional quantum wires, where ρD diverges at T → 0 irrespective of the strength of repulsive interactions. Another unusual property of Coulomb drag between the helical edges concerns higher T for which, unlike in the Luttinger liquid model, drag is mediated by plasmons. The special type of plasmon-mediated drag can be viewed as a distinguishing feature of the helical liquid-because it requires peculiar Umklapp scattering only available in the presence of a Dirac point in the electron spectrum.