Renormalization group approach for the scattering off a single Rashba impurity in a helical liquid

Abstract

The occurrence of two-particle inelastic backscattering has been conjectured in helical edge states of topological insulators and is expected to alter transport. Here, by using a renormalization group approach, we provide a microscopic derivation of this process, in the presence of a time-reversal invariant Rashba impurity potential. We are able to prove that such an effect only occurs in the presence of electron-electron interactions. Furthermore, we find that the linear conductance as a function of temperature exhibits a crossover between two scaling behaviors, T 4K for K > 1/2 and T 8K−2 for K < 1/2, with K the Luttinger parameter. PACS number(s): 72.15.Nj, 72.25.−b, 85.75.−d Introduction. Since the prediction of the quantum spin Hall phase 1,2 in HgTe quantum wells, 3 transport measurements on these compounds have shown evidence of a quantized edge conductance G = 2e 2 /h, thereby paving the way for nonlocal dissipationless transport in semiconductors at zero external magnetic field. 4-6 In the simplest case of quantum wells with inversion symmetry, transport occurs through two counterpropagating edge channels that carry opposite spin-1/2 quantum numbers. Such helical liquids form a new class of one-dimensional (1D) quantum liquids in the sense that they are protected by time-reversal symmetry against single-particle elastic backscattering. 2,7,8 However, deviations from the quantized conductance arise in various situations, involving either a breaking of time-reversal symmetry-by a magnetic impurity, for instance-or the interplay between a time-reversal invariant (TRI) external potential and a source of inelastic scattering. Inelastic single-particle backscattering 9,10 and two-particle backscattering 2,7,8,11 are two examples of the latter. In this Rapid Communication, we focus on twoparticle backscattering off a TRI impurity and report results regarding the temperature scaling of conductance corrections. Our purpose is to derive the Hamiltonian for such a process starting with a minimal model of an interacting helical liquid coupled to a TRI potential. In particular, we focus on a Rashba spin-orbit potential, 9,11 which can originate from fluctuations of an electric field perpendicular to the two-dimensional (2D) electron gas, 12 and acts as a TRI effective magnetic field that couples right and left movers. In the recent literature, inelastic two-particle backscattering off an impurity was mostly studied phenomenologically, by postulating the generic form of the Hamiltonian due to symmetry considerations, namely, TRI and the Pauli principle, 2,7,8,13

Hyperfine interactions in two-dimensional HgTe topological insulators

“…10 However, also deviations from perfect conductance have been observed in longer HgTe devices, 6,7,11,12 which could stem from, e.g., inelastic scattering mechanisms. [13][14][15][16][17][18] The effect of external magnetic fields have also been considered. 6,8,[19][20][21][22][23][24] The TI state in HgTe QWs was predicted by Bernevig, Hughes, and Zhang (BHZ) 25 by using a simplified k · p model containing states with S-and P -like symmetries, respectively.…”

Shot noise in the edge states of 2D topological insulators

“…The spin-dependent chemical potentials of electrons in the puddle may be obtained either by solving Eqs. (32) and (34) or by making use of the elastic distribution function (13) and integrating the difference F σ (x) − (−ε) over the energy.…”

“…[12], the authors suggested that the backscattering could result from a dephasing of electrons captured from the edge state into a quantum puddle with chaotically arranged scatterers, but they could not draw a definite conclusion about the relevance of this mechanism to actual experiments. Two-particle scattering on a defect with localized spin-orbit coupling in a presence of electron-electron interaction was also considered [13]. This process results in a backscattering of electrons in the edge states and a weak temperature dependence of the conductance only if the Luttinger parameter describing the electron interaction in the edge states is very close to the value K = 1/4.…”

Charge and spin density in the helical Luttinger liquid

“…The interest in such systems is motivated by potential applications in spintronics [31][32][33] and quantum computation [34,35]. The properties under consideration are most often related to transport, partially because the explanation of the weak temperature dependence of the non-perfectly quantized conductance in long samples still represents an open and challenging issue, even though several scattering mechanisms have been inspected [36][37][38][39][40][41][42][43][44][45][46]. Much less emphasis has, on the other hand, been devoted to its local properties [47].…”