Research Article

Tunable quantum criticality and super-ballistic transport in a “charge” Kondo circuit

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Science  22 Jun 2018:
Vol. 360, Issue 6395, pp. 1315-1320
DOI: 10.1126/science.aan5592
  • Fig. 1 Multichannel Kondo model and charge implementation.

    (A) In the Kondo model, a local spin (red arrow) is antiferromagnetically coupled to the spin of electrons (blue arrows). Each Kondo channel corresponds to one distinct electron continuum (three continua are shown here). (B) Sample schematic realizing the charge pseudospin implementation of the three-channel Kondo model. A micrometer-scale metallic island (red disk) is connected to large electrodes (small gray disks) through three QPCs (green split gates), each set to a single (spin-polarized) conduction channel (red dashed lines) indexed by i ∈ {1, 2, 3}. (C and D) Quantum channels conductance measured versus gate voltage Vg is displayed over half a Coulomb oscillation period Embedded Image mV (several sweeps including different consecutive peaks are averaged). Measurements at Embedded Image and 29 mK are shown, respectively, as open and full symbols for two (C) or three (D) symmetric channels. The squares correspond to an “intrinsic,” unrenormalized transmission probability across the connected QPCs of Embedded Image, and triangles to that of Embedded Image. The red continuous line (C) displays the T = 7.9 mK prediction for two channels both set to τ = 0.90 (32). Green arrows indicate the direction of conductance change at δVg = 0 as temperature is reduced.

  • Fig. 2 Quantum critical fixed points.

    (A and B) The conductance of (A) two or (B) three symmetric channels measured at the charge degeneracy point (δVg = 0) is plotted as symbols versus temperature on a logarithmic scale. Each set of identical symbols connected by dashed lines corresponds to the same device setting (τ). The predicted (A) 2CK and (B) 3CK low-temperature fixed points for the conductance per channel in the present charge Kondo implementation are shown as horizontal continuous lines [G2CK = e2/h, G3CK = 2sin2(π/5)e2/h].

  • Fig. 3 Non-Fermi liquid scaling exponents.

    The absolute difference between symmetric channels conductance at charge degeneracy and predicted Kondo fixed point (Embedded Image and Embedded Image) is plotted as symbols (open and solid for 2CK and 3CK, respectively) versus T/TK in a log-log scale for T ∈ {7.9, 9.5, 12, 18, 29} mK. Statistical error bars are shown when larger than symbols. The red and green continuous straight lines display the predicted power-law scaling at Embedded Image for the conductance per channel in the present charge 2CK and 3CK implementations, respectively. The scaling Kondo temperature TK is adjusted separately for each tuning τ of the symmetric channels (insets, corresponding symbols). This is done by matching the lowest-temperature data point Embedded Image with the corresponding displayed power law. Continuous lines in insets show the predicted power-law divergences of TK versus τ for 2CK (bottom right inset) and 3CK (top left inset).

  • Fig. 4 Universal renormalization flow to quantum criticality.

    The measured conductance of the two or three connected, symmetric channels is shown as symbols (open and solid for 2CK and 3CK, respectively) for a broad range of settings τ. (A and B) Data (T ∈ {7.9, 9.5, 12, 18} mK) and predictions are plotted versus T/TK in log scale. The corresponding experimental TK are shown in insets as symbols versus τ, together with theoretical predictions for tunnel contacts Embedded Image (light-blue continuous lines) and for very large TK at Embedded Image [respectively, red and green continuous lines for 2CK and 3CK in insets of (A) and (B)]. (C) Direct data-theory comparison (no T/TK rescaling) with ∂G1,3/∂log(T) plotted versus G1,3 and ∂G1,2,3/∂log(T) plotted versus G1,2,3. The discrete experimental differentiation is performed with measurements at T ∈ {7.9, 12, 18} mK. Kondo fixed points are indicated by arrows. Black continuous lines are NRG calculations of the universal renormalization flows [2CK in (A) and (C) and 3CK in (B) and (C)]. Colorized dashed lines shown at low T/TK [(A) and (B)], and close to the Kondo fixed points (C), display the predicted low-temperature power laws for 2CK [red in (A) and (C)] and 3CK [green in (B) and (C)]. Light blue dashed lines shown at large T/TK [(A) and (B)], and for small channels conductance (C), represent the predicted high-temperature logarithmic scaling proportional to Embedded Image, with the slightly different 2CK and 3CK prefactors and γ here used as fit parameters.

  • Fig. 5 Crossover from quantum criticality by pseudospin degeneracy breaking.

    (A) Quantum criticality extends as T rises. It is delimited from below by the crossover temperature Tco, which increases as a power law for small parameter-space distances from the critical point (for example, charge pseudospin energy splitting ΔE ∝ δVg, channels asymmetry Δτ). Along the crossover, theory predicts universal T/Tco scalings [for example, Embedded Image]. (B and C) The conductance of (B) two and (C) three symmetric channels set, respectively, to Embedded Image and Embedded Image, are plotted as continuous lines versus Embedded Image (left side) and Tco/T (right side) for T ∈ {7.9, 9.5, 12, 18, 29, 40, 55} mK. Colored thick dashed lines (gray dash-dotted lines) shown in right sides display the theoretical universal crossover curves Embedded Image and Embedded Image (the predicted Embedded Image power laws). The only fit parameter is an unknown fixed prefactor for the 3CK crossover scale Tco [no fit parameters in (B)]. (D and E) Experimental crossover temperatures Embedded Image are plotted as symbols in a log-log scale versus ΔE, for (D) two and (E) three symmetric channels. Each set of symbols connected by dashed lines represents one device setting τ1,3 or τ1,2,3 (insets). Full symbols correspond to Embedded Image mK. Straight continuous lines display the predicted power laws Tco ∝ ΔEγ, with γ = 2 for 2CK and γ = 5/3 for 3CK. Fitting Embedded Image mK separately for each τ yields the values of γ shown as symbols in the insets with the fit standard error.

  • Fig. 6 Three-channel Kondo renormalization flow with super-ballistic conductances.

    Each colored line with an arrowhead displays the measured channels’ conductance at T = 55, 40, 29, 18, 12, and 7.9 mK (arrowhead is shown at lowest T) for a fixed device tuning (τ1 ≈ τ3, τ2) at charge degeneracy (δVg = 0). The lines’ colors reflect the direction (the angle) of the vector connecting lowest- and highest-temperature data points, to improve readability. Because QPC1 and QPC3 are set symmetric [τ1 ≈ τ3 tuned among 14 values from 0.1 to 0.985 (32)], only the renormalized average G1,3 is shown on the horizontal axis. QPC2 is separately adjusted to a coupling τ2 selected among the same 14 values and also τ2 = 0. For the solid lines and solid arrows, the experimental standard error of G2h/e2 and G1,3h/e2 is below 0.05 (usually well below). For the dashed lines and open arrows, the standard error of G2h/e2 is between 0.05 and 0.1. The green, red, and blue disks correspond, respectively, to the predicted 3CK, 2CK, and 1CK low-temperature fixed points. The thick gray lines represent NRG calculations of the universal crossover flows from 3CK (32), with arrows pointing to lower temperatures. The conductance G2 can markedly exceed the maximum free electron limit e2/h.

Supplementary Materials

  • Tunable quantum criticality and super-ballistic transport in a "charge" Kondo circuit

    Z. Iftikhar, A. Anthore, A. K. Mitchell, F. D. Parmentier, U. Gennser, A. Ouerghi, A. Cavanna, C. Mora, P. Simon, F. Pierre

    Materials/Methods, Supplementary Text, Tables, Figures, and/or References

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    • Materials and Methods
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    • Figs. S1 to S5
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