Quantitative validation of the charge fractionalization fraction measured in non-chiral quantum wires

The 2008 experiment of Steinberg and co-workers provided the first experimental evidence for charge fractionalization in a one-dimensional quantum wire: electrons injected with a well-defined momentum into the bulk of an interacting cleaved-edge-overgrowth GaAs wire split into right- and left-moving charge pulses carrying fractions $f e$ and $(1-f)e$, with $f\approx0.7$–$0.75$ inferred from the empirical equality of the two-terminal drain conductance $G_{2\mathrm{T}}$ and the drain-current asymmetry $\mathrm{AS}$, combined with a single-mode Luttinger-liquid model and an externally estimated interaction parameter $g_{\mathrm{c}}\approx0.4$–$0.5$. The inference rests on unverified assumptions (single occupied sub-band, symmetric drain junctions, a phenomenological drain-coupling parameter $\beta$), on an unexplained empirical law ($G_{2\mathrm{T}}$ approximately linear in lower-wire density with an extrapolated zero at $20\ \mu\mathrm{m}^{-1}$), and on an interaction parameter not independently validated in the same geometry. The problem is to turn that measurement into a quantitatively robust, model-independent determination of the fractionalization fraction in a non-chiral quantum wire. No follow-up experiment on the same platform and no microscopic explanation of the density law are known to date.

Background

Interacting electrons confined to one dimension do not form a Fermi liquid of particle-like quasiparticles: the low-energy theory is the Tomonaga–Luttinger liquid (TLL), in which an injected electron fractionalizes into independent collective charge and spin density waves. The charge sector is characterized by a dimensionless interaction parameter $g_{\mathrm{c}}$ (equal to one for non-interacting electrons and smaller than one for repulsive Coulomb interactions). In a uniform, momentum-conserving one-dimensional (1D) wire, theory predicts that an electron injected at a single Fermi point decomposes into a right-moving and a left-moving charge pulse carrying $f e$ and $(1-f)e$ of the electron charge, with $f=(1+g_{\mathrm{c}})/2$ — charge fractionalization, predicted by Pham, Gabay and Lederer and by Imura and co-workers. Unlike the fractionally charged quasiparticles of the fractional quantum Hall effect, these fractions are not quantized: they are set continuously by the interaction strength.

Observing this fractionalization in a non-chiral quantum wire (one in which excitations can move both left and right, unlike quantum-Hall edge channels) is difficult. The two-terminal d.c. conductance of a wire adiabatically coupled to Fermi-liquid leads is pinned to the universal value $G_0=2e^2/h$ independent of interactions (results of Maslov and Stone and of Safi and Schulz), and low-frequency shot noise in an ideal non-chiral wire probes the contacts rather than the bulk, as shown by Trauzettel, Safi, Dolcini and Grabert. Resolving the internal fractional pulses directly in time or frequency requires probes at frequencies of order $v_{\mathrm{F}}/(g_{\mathrm{c}}L)$, which are experimentally challenging.

Steinberg and co-workers (Nature Physics 4, 116 (2008)) provided the first experimental evidence for 1D charge fractionalization using a cleaved-edge overgrowth (CEO) GaAs device, in which two parallel quantum wires are grown on a cleaved crystal facet and contacted separately. A perpendicular magnetic field $B$ shifts the relative dispersions of the two wires by a momentum $q_B=eBd$ (with $d$ the wire separation), so that momentum-conserving tunneling occurs at fields $B^{\pm}$ where the Fermi momenta of the two wires match; this injects electrons selectively into a single propagation direction at the bulk of the lower, interacting wire. The experiment measured the two-terminal conductance $G_{2\mathrm{T}}$ between the left and right drains and the current asymmetry $\mathrm{AS}=(I_{\mathrm{R}}-I_{\mathrm{L}})/(I_{\mathrm{R}}+I_{\mathrm{L}})$ of the drain currents. In a single-mode TLL model with symmetric drain junctions described by a phenomenological drain-coupling parameter $\beta$ (with $\beta+\gamma=1$; $\beta$ parameterizes how the outgoing charge-mode chemical potential at a drain depends on the incoming one and accounts for backscattering in the wire and coupling to the Fermi-liquid leads), one obtains $G_{2\mathrm{T}}=g_{\mathrm{c}}G_0\beta/(2-\beta)$ and $\mathrm{AS}=(\beta/(2-\beta))(2f-1)$, so that the ratio $G_{2\mathrm{T}}[G_0]/\mathrm{AS}=g_{\mathrm{c}}/(2f-1)$ is independent of the unknown $\beta$. The empirical observation $G_{2\mathrm{T}}=\mathrm{AS}$ in three samples over a range of lower-wire densities then yields $2f-1=g_{\mathrm{c}}$, i.e. $f=(g_{\mathrm{c}}+1)/2\approx0.7$–$0.75$, using $g_{\mathrm{c}}\approx0.4$–$0.5$ estimated in the Supplementary Information from tunneling measurements. The companion theory paper of Le Hur, Halperin and Yacoby derives this universal ratio, treats asymmetric drain couplings (a strongly asymmetric junction drives $\mathrm{AS}\to1$ independently of fractionalization), and cautions that the measurement does not by itself prove that the fractional charges are sharp rather than a quantum-statistical average. Empirically, $G_{2\mathrm{T}}$ measured at $B^{+}$ varies approximately linearly with the lower-wire linear density $n_{\mathrm{L}}$, extrapolating to zero at $n_{\mathrm{L}}=20\ \mu\mathrm{m}^{-1}$, an observation for which no microscopic mechanism has been identified. The quantitative extraction of $f$ therefore rests on assumptions (single-sub-band operation, meaning that only the lowest quantized transverse mode of the wire is occupied, identical drain junctions, negligible drain scattering) that were not independently verified, and on a density-dependent conductance law that is not understood.

Problem Statement

Establish a quantitatively robust and validated determination of the charge fractionalization fraction $f$ in a non-chiral one-dimensional quantum wire, resolving the issues that the 2008 three-terminal d.c. asymmetry measurement left open. Concretely:

(1) Quantify, theoretically and experimentally, how sensitive the inferred $f$ — extracted from the relation $G_{2\mathrm{T}}[G_0]/\mathrm{AS}=g_{\mathrm{c}}/(2f-1)$ together with the empirical equality $G_{2\mathrm{T}}=\mathrm{AS}$ — is to violations of the model assumptions: occupation of more than one 1D sub-band in the lower wire, asymmetric drain junction parameters ($\beta_1\neq\beta_3$, for which theory already shows that strong asymmetry drives $\mathrm{AS}\to1$ independently of fractionalization), and additional microscopic backscattering or non-ideal coupling to the Fermi-liquid drains, including the two-dimensional-electron-gas tunneling background and the degree of source/drain density uniformity required for the method to apply.

(2) Identify the microscopic origin of the approximately linear dependence of the two-terminal conductance $G_{2\mathrm{T}}$ on the lower-wire density $n_{\mathrm{L}}$, including the physical meaning of its extrapolated zero at $n_{\mathrm{L}}=20\ \mu\mathrm{m}^{-1}$, which currently enters the measurement as an unexplained empirical calibration law.

(3) Determine the charge Luttinger parameter $g_{\mathrm{c}}$ independently and more precisely in the same experimental geometry — or devise an extraction of $f$ that does not rely on an externally estimated $g_{\mathrm{c}}$ — so that the fractional charges $f e$ and $(1-f)e$ are obtained with quantified systematic uncertainty.

These three threads serve one common objective: a trustworthy, model-independent experimental test of the Luttinger-liquid prediction $f=(1+g_{\mathrm{c}})/2$ for charge fractionalization in a non-chiral quantum wire. A quantitatively validated determination of $f$ requires the sensitivity analysis in (1) to cover each of the assumption violations listed there, the analysis in (2) to identify the microscopic origin of the density law, and the extraction in (3) to yield $f$ with quantified systematic uncertainty, whether from an independently determined $g_{\mathrm{c}}$ or from a $g_{\mathrm{c}}$-free extraction. The extraction of $f$, in the original momentum-resolved tunneling geometry or in another probe of the same physics in a non-chiral 1D conductor, must be validated against the microscopic drain and sub-band effects above.

References

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