Dynamics of One‐Dimensional Electrons With Broken Spin–Charge Separation

Abstract

Spin-charge separation is known to be broken in many physically interesting one-dimensional (1D) and quasi-1D systems with spin-orbit interaction because of which spin and charge degrees of freedom are mixed in collective excitations. Mixed spin-charge modes carry an electric charge and therefore can be investigated by electrical means. We explore this possibility by studying the dynamic conductance of a 1D electron system with imagepotential-induced spin-orbit interaction. The real part of the admittance reveals an oscillatory behavior versus frequency that reflects the collective excitation resonances for both modes at their respective transit frequencies. By analyzing the frequency dependence of the conductance the mode velocities can be found and their spin-charge structure can be determined quantitatively.Spin-orbit interaction (SOI) causes a range of non-trivial effects in low-dimensional electron systems, especially if combined with electron-electron (e-e) interaction.[1] Below we investigate one yet little studied aspect of SOI in one-dimensional (1D) and quasi-1D systems. Owing to the e-e interaction 1D electrons form a strongly correlated state known as the TomonagaLuttinger liquid, the hallmark of which is a spin-charge separation (SCS).[2] The SCS was studied in detail in systems without SOI. In the presence of SOI the SCS is still respected in strictly 1D systems. However, the SCS is violated in realistic quasi-1D structures with transverse quantization sub-bands since the spin is no longer a good quantum number there, resulting in new collective excitations modes, in which spin and charge degrees of freedom are mixed. [3] Even more interesting effects accompanied by the SCS violation appear in 1D electron systems with the spin-dependent e-e interaction. This happens when a 1D electron system is placed close to a metallic gate. The electric field of the image charges that electrons induce on the gate gives rise to the imagepotential-induced spin-orbit interaction (iSOI), which produces a spin-dependent contribution to the e-e interaction Hamiltonian. The iSOI not only breaks the SCS, but also leads the system to the instability for sufficiently strong interaction. [4] The SCS can also be violated in 1D edge states of two-dimensional topological insulators. These states are known to have a helical structure with the spin locked to the electron momentum. In the simplest commonly studied case of the S z symmetry when the spin orientation depends only on the momentum direction but not on its magnitude, the SCS is respected.[5] However, the S z symmetry is not an inherent property of the topological insulator. Generally, the S z symmetry is violated by the SOI.[6-9] The single-particle states are then modeled as the Kramers pair of 1D states with the spin orientation depending on the momentum magnitude rather then on its direction alone. [10,11] The packet composed of such states does not possess a definite spin and, which is particularly interesting, the e-e interaction becomes effectively spin-dependen...

Pair spin–orbit interaction in low-dimensional electron systems

“…e form of the PSOI Hamiltonian in a speci c low-dimensional structure is determined by the con guration of the Coulomb elds therein. In a 1D quantum wire the only source of the PSOI is the Coulomb eld of e-e interaction directed normally to the wire, which appears if the system is not symmetric in the radial direction [4][5][6][7]. Such asymmetry can be created, for example, by a proximate metallic gate.…”

“…It is clear that the behavior of the many-electron system depends on both e ects, the relative role of which is yet to be determined. In this section we focus on the PSOI-driven instability of the strongly correlated 1D electron liquids and on the PSOI signatures in electron transport in quantum wires [5,6]. e instability of the many-electron system is due to a qualitatively new property of PSOI, as compared to the single-electron Rashba SOI.…”

“…In the presence of the single-particle Rashba SOI, the SCS is respected in strictly 1D systems. However, the PSOI violates the SCS in 1D single-mode quantum wires, which leads to the formation of new collective excitations with intertwined spin and charge degrees of freedom [4][5][6]. e spin-charge composition of the collective mode is quantitatively characterized by a spincharge separation parameter ξ that is de ned via the relative weight of the electron densities with spin up n ↑ and down n ↓ in each mode, and can be directly expressed via the mode phase velocity…”

“…e PSOI breaks the spin-charge separation, because of which the spin and charge degrees of freedom are mixed in the collective excitations. e PSOI signatures in the spin-charge structure of the collective excitations and in their renormalized velocities can be quite simply identi ed by the Fabry-Pérot resonances in the frequency-dependent conductance of the quantum wire coupled to leads [6].…”

Dynamic Transport in a Quantum Wire Driven by Spin–Orbit Interaction

“…[15,16]. Here we would like to stress that in the presence of iSOI (E q = 0) both v ± and ξ ± can be controlled via the mean electric field Eby tuning the gate potential.…”