Charge-spin duality in nonequilibrium transport of helical liquids

Abstract

Non-equilibrium transport properties of charge and spin sector of two edges of a quantum spin Hall insulator are investigated theoretically in a four-terminal configuration. A simple duality relation between charge and spin sector is found for two helical Tomonaga Luttinger liquids (hTTLs) connected to non-interacting electron reservoirs. If the hTLLs on opposite edges are coupled locally or non-locally, the mixing between them yields interesting physics where spin information can be easily detected by a charge measurement and vice versa. Particularly, we show how a pure spin density in the absence of charge current can be generated in a setup that contains two hTLL and one spinful Tomonaga Luttinger liquid in between.PACS numbers: 71.10. Pm,72.15.Nj, Introduction.-The discovery of topological insulators (TIs) in both two spatial dimensions (2D) and three spatial dimensions (3D) has recently attracted a lot of interest 1-3 . Unlike in normal insulators, there is a gapless mode appearing within the bulk gap at the edge of TIs which originates from strong spin-orbit coupling and is protected by time reversal. In 2D, a TI is also called quantum spin Hall (QSH) insulator since its edge states are one dimensional (1D) counter-propagating modes with opposite spin. These 1D systems have been coined helical liquids or helical Tomonaga Luttinger liquids (hTLLs). Transport properties of hTLLs have been predicted and observed at the edge of HgTe quantum wells 4,5 and proposed to also exist in InAs/GaSb quantum wells 6 as well as Bi 2 Se 3 or Bi 2 Te 3 thin films 7,8 . An important feature of the hTLL is that spin and momentum are locked to each other. Remarkably, one hTLL has only half the degrees of freedom of a spinful Tomonaga Luttinger liquid (sTLL). Thus, two hTLLs, which naturally exist at two opposite edges of a QSH insulator, can recover the degrees of freedom of a sTLL. It is well known and has even been experimentally confirmed that there is spin-charge separation for a 1D sTLL 9 .

Spectral properties of Luttinger liquids: A comparative analysis of regular, helical, and spiral Luttinger liquids

“…It is therefore more convenient to treat the Hamiltonian of the HLL as an effectively spinless LL by introducing the fields Φ = (φ R↑ + φ L↓ )/ √ 2 and Θ = (φ L↓ − φ R↑ )/ √ 2. When including the interactions, the bosonized form of the Hamiltonian of the interacting HLL becomes [30][31][32]70,97,98 …”

Proximity-InducedπJosephson Junctions in Topological Insulators and Kramers Pairs of Majorana Fermions

“…We note that the same setup could be assembled in the framework of strip of stripes models [22][23][24][25][26][27][28] based on an array of coupled one-dimensional channels with spinorbit interaction [27]. As a striking consequence we find that the proposed models for proximity-induced JπJs in a TI provide an alternative approach to engineer Kramers pairs of Majorana fermions (MFs) [29][30][31][32][33][34][35][36][37][38][39] easily movable by gates. Remarkably, no magnetic fields are needed.…”

Robust Transport Signatures of Topological Superconductivity in Topological Insulator Nanowires

“…1(b), similar to the one describing the Majorana interferometer proposed in Refs. 9 and 10 (see also related studies of Majorana interferometry with chiral Majorana modes [45][46][47] and Majorana bound states [48][49][50][51][52][53][54][55][56][57]). In this model, an incoming chiral mode leaving the source is split into two Majorana modes that appear at the interface between the the superconductor and the regions with finite B ⊥ [58].…”