Fractional Wigner Crystal in the Helical Luttinger Liquid

Abstract

The properties of the strongly interacting edge states of two dimensional topological insulators in the presence of two particle backscattering are investigated. We find an anomalous behavior of the density-density correlation functions, which show oscillations that are neither of Friedel nor of Wigner type: they instead represent a Wigner crystal of fermions of fractional charge e/2, with e the electron charge. By studying the Fermi operator, we show that the state characterized by such fractional oscillations still bears the signatures of spin momentum locking. Finally, we compare the spin-spin correlation functions and the density-density correlation functions to argue that the fractional Wigner crystal is characterized by a non trivial spin texture. [7][8][9] (2DTI). Particular emphasis has been devoted to the investigation of the transport properties of 2DTI: topological protection of the edge states facilitates the observation of conductance quantization [5,6,10] in short samples. Long edges on the other hand are characterized by a reduced conductance. Even though a comprehensive theoretical understanding of the scattering sources causing the reduction of the conductance of the edge is still lacking, the role of electron-phonon interactions[11], of magnetic [12] and nonmagnetic impurities [13,14], in the presence of random Rashba disorder [15,16], of breaking of axial symmetry in combination with electronelectron interactions and impurities [17,18], of tunneling among the edges and charge puddles in the bulk of the 2DTI [19], and of the coupling between opposite edges [20,21] has been theoretically elucidated. The mathematical tool allowing for most of such calculations is bosonization [22,23], a procedure that enables us to recast the Hamiltonian of the interacting electrons on the edges into a Hamiltonian of free bosonic excitations, representing charge density waves, and to express the Fermi operator in terms of the creation and annihilation bosonic operators. The physical meaning of the bosonization technique can be understood within the framework of Luttinger liquid theory [24], that is the one dimensional counterpart of the Fermi liquid theory for one dimensional gapless systems. More precisely, the exactly solvable Luttinger model [25][26][27], a strictly linear theory of interacting one dimensional fermions with infinite bandwidth in the single particle dispersion is usually employed. The validity of the Luttinger model as a basis for the description of interacting electrons has a number of experimental demonstrations, ranging from spin charge separation [28], to charge fractionalization [29,30], and to anomalous tunneling [31,32]. On the other hand, the Luttinger model alone fails in predicting a reasonable behavior of local observables [33][34][35][36], such as the electron density and the density-density correlation functions when electron-electron interactions are strong. In particular, the Luttinger model is not able to capture the transition between a weakly correlated state dominated by Fri...

Dynamic response functions and helical gaps in interacting Rashba nanowires with and without magnetic fields

“…The resulting system then hosts fractionalized quasiparticles which have attracted a lot of interest in the past years. [13][14][15][16][17][18][19][20] Therefore, in an interacting quasi-1D Rashba wire, helical gaps can in principle be created by either a magnetic field or by spin-umklapp scattering. This prompts the question about whether the two possible underlying causes can be distinguished experimentally.…”

Out-of-equilibrium density dynamics of a quenched fermionic system

“…The validity of LLs as low energy theories for 1D systems of fermions, bosons and spin, has been demonstrated experimentally by means of anomalous tunneling effects 32,33 , or by observing spin-charge separation [34][35][36] . Moreover, the LL theory represents a very useful tool for the study of a wide range of 1D systems, including integer 37 and fractional 38 quantum Hall effects and two dimensional topological insulators [39][40][41][42] , weakly interacting quantum wires 32 , even in the presence of spin-orbit coupling [43][44][45][46][47][48] , carbon nanotubes 33,49,50 , eventually including electron phonon coupling 51,52 , spin chains 53,54 and, complemented with its spin incoherent version 55,56 , Wigner crystals [57][58][59][60][61][62][63] . The validity of the LL picture as a low energy theory for 1D Hamiltonians is however limited to the low energy excitations of gapless phases 29,31 .…”

Fractional charge oscillations in quantum dots with quantum spin Hall effect

“…A different type of charge fractionalization is known to take place in strongly interacting condensed matter systems. Apart from the well established fractional quantum Hall effect in two spatial dimensions [30], in one dimension, the interplay between strong spin-orbit coupling and electron-electron interactions is predicted to lead to charge fractionalization [31][32][33][34][35] and, in the presence of superconductors, to parafermions [36][37][38][39].…”