Charge Fractionalization in nonchiral Luttinger systems

TL;DR

The paper analyzes charge fractionalization in nonchiral Luttinger liquids and proposes a three-terminal, momentum-resolved tunneling geometry to observe it. By decomposing the charge sector into chiral modes and analyzing unidirectional injection, it shows that injected electrons split into counterpropagating excitations carrying fractions $fe$ and $(1-f)e$ with $f=(1+g)/2$, where $g$ is the Luttinger parameter. A central result is a universal ratio, $A_S(2e^2/h)/G_2=1$, linking the asymmetry of drain currents to the two-terminal conductance in the near-equilibrium limit, which aligns with recent experiments. The work also characterizes the maximal transmission scenario, derives the weak-tunneling regime, and discusses how the Luttinger parameter $g$ can be extracted from tunneling measurements, highlighting the potential to observe fractional charges beyond simple averages and to synthesize excitations with arbitrary charge through inhomogeneous interaction profiles.

Abstract

One-dimensional metals, such as quantum wires or carbon nanotubes, can carry charge in arbitrary units, smaller or larger than a single electron charge. However, according to Luttinger theory, which describes the low-energy excitations of such systems, when a single electron is injected by tunneling into the middle of such a wire, it will tend to break up into separate charge pulses, moving in opposite directions, which carry definite fractions $f$ and $(1-f)$ of the electron charge, determined by a parameter $g$ that measures the strength of charge interactions in the wire. (The injected electron will also produce a spin excitation, which will travel at a different velocity than the charge excitations.) Observing charge fractionalization physics in an experiment is a challenge in those (nonchiral) low-dimensional systems which are adiabatically coupled to Fermi liquid leads. We theoretically discuss a first important step towards the observation of charge fractionalization in quantum wires based on momentum-resolved tunneling and multi-terminal geometries, and explain the recent experimental results of H. Steinberg {\it et al.}, Nature Physics {\bf 4}, 116 (2008).

Figures (5)

• Figure 1: Wire tunnel-coupled to an extended 1D probe (upper wire) allowing momentum-resolved tunneling; because of the uniformity of the barrier, momentum along the wire is conserved during tunneling. A transverse magnetic field is applied so that the band structures of the two wires obey $k_{F2} + k_F = q_B = 2\pi Bd/\phi_0\gg 0$; $k_{F2}$ and $k_F$ are the Fermi wavectors of the probe and of the wire, $B$ embodies the magnetic field applied perpendicular to the plane ($q_B\approx 2k_F$ is the resulting momentum boost), $d$ is the distance between the two wires, and $\phi_0$ is the flux quantum.
• Figure 2: Three-lead geometry, where the upper wires inject unidirectional electrons, which allows to study the asymmetry parameter $A_S=(I_L-I_R)/I_S$; within our conventions, $I_L$ and $I_R$ represent the left and right currents in the system.
• Figure 3: General Scattering Matrix formulation for the interacting case.
• Figure 4: Experimental results of Ref. Amir which confirm that $A_S(2e^2/h)/G_2=1$ for different density $n_L$ in the lower wire. (The two-terminal conductance $G_2$ has been normalized to $2e^2/h$.) Different samples correspond to different colors.
• Figure 5: Electron spectral function in the Luttinger theory (for an electron from the right Fermi branch) as a function of frequency $\omega$ for different wavevectors $q$ measured from $k_F$ ($\Lambda/\hbar=1$ and $u=1$). Temperature is zero, $g=1/2$, and for simplicity, $L_F\rightarrow +\infty$ (a finite $L_F$ produces the broadening of the different peak structures). The $\omega<0$-part has been multiplied by 10 for clarity. Far from the Fermi "surface" (point), the electron spectral function reveals two peak features associated with the spin and right-moving charge mode (the counterpropagating charge mode also gives some spectral weight at negative $\omega$). One can determine $g$ from these two peaks.