Fractional charge and quantized current in the quantum spin Hall state

Abstract

A profound manifestation of topologically non-trivial states of matter is the occurrence of fractionally charged elementary excitations. The quantum spin Hall insulator state is a fundamentally novel quantum state of matter that exists at zero external magnetic field. In this work, we show that a magnetic domain wall at the edge of the quantum spin Hall insulator carries one half of the unit of electron charge, and we propose an experiment to directly measure this fractional charge on an individual basis. We also show that as an additional consequence, a rotating magnetic field can induce a quantized dc electric current, and vice versa. 1Soon after the theoretical proposal of the intrinsic spin Hall effect [1,2] in doped semiconductors, the concept of a time-reversal invariant spin Hall insulator [3] was introduced. In the extreme quantum limit, a quantum spin Hall (QSH) insulator state has been proposed for various systems [4,5,6]. The QSH insulators are time-reversal invariant and have a bulk charge-excitation gap. However, this system also possesses topologically protected gapless edge states that lie inside the bulk insulating gap. The edge states of the QSH insulator state differ from the quantum Hall effect and have a distinct helical property: two states with opposite spin-polarization counter-propagate at a given edge [4,7,8]. The edge states come in Kramers' doublets, and time reversal symmetry ensures the crossing of their energy levels at special points in the Brillouin zone. Because of this level crossing, the spectrum of a QSH insulator cannot be adiabatically deformed into a topologically trivial insulating state; therefore, in this precise sense, the QSH insulators represent a topologically distinct new state of matter.Recently, the QSH effect has been theoretically proposed [6] and experimentally observed [9] in HgTe quantum wells. In this experiment, an applied gate voltage can tune the carrier type from n-type doping to p-type doping, passing through a nominally insulating state. A residual charge conductance approaching 2e 2 /h has been measured in this insulating regime. Furthermore, the residual charge conductance is independent of the width of the sample, indicating that it is due to the helical edge state channels of the QSH insulator. liquid at a given edge of the QSH insulator has only two: a spin up (or down) right mover and a spin down (or up) left mover. Therefore, the helical liquid has half the degrees of freedom of a conventional one-dimensional system, and thus avoids the doubling problem.Because of this fundamental topological property of the helical liquid, a domain wall carries charge e/2. We propose a Coulomb blockade experiment to observe this fractional charge.As a temporal analog of the fractional charge effect, the pumping of a quantized charge current during each periodic rotation of a magnetic field is also proposed. This provides a direct realization of Thouless's topological pumping [14].Theoretical description. In the absence of time-reversal symmetry (TRS) br...

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