## A nanostructure quantum simulator

Phase transitions occurring at absolute zero temperature, or quantum phase transitions (QPTs), can be grouped into broad categories called universality classes. The classification is based on the properties of the transition rather than the microscopic details of the underlying system. Iftikhar *et al.* exploited this fact to study QPTs in clean, tunable nanostructures, rather than in complex materials, where they most often occur. Within a single nanostructure, two different classes of QPTs with profoundly different characters were studied and comprehensively characterized.

*Science*, this issue p. 1315

## Abstract

Quantum phase transitions (QPTs) are ubiquitous in strongly correlated materials. However, the microscopic complexity of these systems impedes the quantitative understanding of QPTs. We observed and thoroughly analyzed the rich strongly correlated physics in two profoundly dissimilar regimes of quantum criticality. With a circuit implementing a quantum simulator for the three-channel Kondo model, we reveal the universal scalings toward different low-temperature fixed points and along the multiple crossovers from quantum criticality. An unanticipated violation of the maximum conductance for ballistic free electrons is uncovered. The present charge pseudospin implementation of a Kondo impurity opens access to a broad variety of strongly correlated phenomena.

Continuous second-order quantum phase transitions (QPTs)—which take place at absolute zero temperature as a control parameter such as the magnetic field is tuned—are accompanied by the development of a highly correlated quantum critical state. With increasing temperature, this state extends over a broadening range of parameters further away from the critical point. In this regime of quantum criticality, the properties of the system obey scaling laws determined by the QPT universality class and do not depend on microscopic details. Although QPTs are ubiquitous in contemporary theoretical physics and have been observed in a multitude of highly correlated materials (*1*), it remains challenging to realize them in simple, well-controlled experimental systems.

Tunable nanostructures provide a path to a microscopic understanding of QPTs that circumvents the complexity of real-world highly correlated materials. So far, however, the rare examples that exhibit a second-order QPT (*2*–*6*) demonstrate only a single quantum critical point (associated with the two-channel Kondo effect, described below); although it is a non-Fermi liquid, this critical point can be treated with a perturbative approach in the low-temperature limit (*7*, *8*). By contrast, we realized and characterized completely a circuit that embodies the three-channel Kondo model, with three fully tunable channels connected to a magnetic impurity emulated by the charge states of a metallic island. Within the same nanostructure, this gives us access to two universality classes of quantum criticality (associated with the two-channel and three-channel Kondo effects) that manifest profoundly dissimilar physics. For instance, the quantum critical point for two symmetric Kondo channels can be understood in terms of free electrons and Majorana fermions (*7*, *8*), whereas for three symmetric channels, it involves () parafermions with irreducibly strong interactions (*9*). The demonstrated high-precision implementation qualifies our device as an analog quantum simulator, providing quantitative experimental solutions for the three-channel Kondo model.

## The multichannel Kondo model

The multichannel Kondo model, which is a generalization of the original (one channel) Kondo model, gives rise to archetypal QPTs and collective, non-Fermi liquid behaviors from a minimal Hamiltonian. Although introduced to account for the different atomic orbitals in metals (*10*–*12*), it has developed over the years into a central testing ground for strongly correlated and quantum critical physics and is a benchmark for many-body theoretical methods (*7*, *11*–*19*). The model describes a local Kondo spin *S* (of 1/2 here) coupled antiferromagnetically to *N* independent free-electron continua (Fig. 1A, *N* = 3)
(1)where *H*_{NCK} is the *N*-channel Kondo Hamiltonian, **s _{i}** is the local spin density of electron continuum (channel)

*i*at the Kondo spin

**S**location,

*J*> 0 is the coupling strengths (here assumed isotropic), and

_{i}*H*

_{continua}is the free-electron continua Hamiltonian. The conventional single-channel model (

*N*= 1) exhibits universal scaling but no second-order QPT or non-Fermi liquid physics. As the temperature

*T*is reduced, the electrons progressively screen the Kondo spin, resulting for

*T*→ 0 in an idle spin-singlet (

*11*). By contrast, for

*N*≥ 2, there is a competition between channels to screen the

*S*= 1/2 Kondo impurity, which develops into second-order QPTs. Each number of identical channels corresponds to a different class of quantum criticality (

*16*), with specific non-Fermi liquid physics (

*12*) and collective excitations revealed by, for example, a divergent specific heat coefficient

*c*/

*T*as

*T*→ 0. The marginal two-channel case corresponds to a logarithmic

*c*/

*T*divergence (

*12*), whereas power law

*c*/

*T*divergences are predicted for

*N*≥ 3 (

*12*).

## Kondo “charge” pseudospin implementation

Experimentally, Kondo nanostructures are usually small quantum dots (*20*–*23*), in which coherent electron cotunneling merges the distinct electrical contacts into one Kondo channel (*24*, *25*) [except in the two-channel devices in (*2*, *5*, *26*, *27*)]. By contrast, in the recently demonstrated (*6*) “charge” Kondo approach (*14*, *28*, *29*), the charge Kondo impurity **S** is not a magnetic spin but a pseudospin-1/2 (Fig. 1B, red arrow) built from the macroscopic quantum states describing the overall charge *Q* of a small metallic island (Fig. 1B, red disk). We extended this concept to three independent Kondo channels. In the most straightforward case of a weakly connected island whose charge is well quantized (*30*), the Kondo spin *S* = {↓, ↑} directly maps on the island’s two charge states of lowest energy {*Q*, *Q* + *e*}. All the other charge configurations are indeed frozen out and can be ignored at low temperatures (*E*_{C} = *e*^{2}/2*C* is the charging energy, *e* is the electron charge, *C* is the island geometric capacitance, and *k*_{B} is the Boltzmann constant). The charge pseudospin energy degeneracy is obtained by tuning (with a gate voltage *V*_{g}) the device at the degeneracy point between the charge states *Q* and *Q* + *e*. Detuning *V*_{g} away from charge degeneracy is completely analogous to applying a magnetic field on usual magnetic Kondo impurities (*28*). The island charge Kondo pseudospin *S* is, however, not coupled to the real spin of electrons. Instead, it is flipped by transferring electrons in and out of the island, through the connected electrical channels (Fig. 1B, red dashed lines). This mechanism takes the form of a Kondo (pseudo-) spin-exchange coupling: Introducing an electron pseudospin *s* (Fig. 1B, blue arrow), which corresponds to the electron localization inside (*s* = ↓) or outside (*s* = ↑) of the island, the tunneling of an electron flips both its localization pseudospin *s* as well as the island overall charge pseudospin *S*. A well-developed Kondo effect requires a continuum of electronic states for both localization pseudospins. This implies a continuous density of states in the metallic island (in contrast with small quantum dots). The different conduction channels constitute here separate Kondo channels. Also, the same physics is predicted at for arbitrary connection strengths (*14*, *31*, *32*), except perfectly ballistic contacts, despite the coexistence of many charge states in a quantum superposition near the ballistic limit (*30*). In practice, we found from numerical renormalization group (NRG) calculations performed over a broad range of coupling strengths that ensures negligible deviations from universal Kondo physics (*32*).

Each of the Kondo/conduction channels passes through a different quantum point contact (QPC) individually formed and tuned by means of field effect in a high-mobility Ga(Al)As two-dimensional electron gas (*32*). Single channels, polarized in real electron spin, are obtained by immersing the device into a large magnetic field ( T, corresponding to the regime of the integer quantum Hall effect at filling factor ν = 3). The Kondo channel couplings *J _{i}* (

*i*∈ {1, 2, 3}) are individually characterized by the “intrinsic” (unrenormalized by Kondo or Coulomb effects) transmission probability τ

*across the single open transport channel of QPC*

_{i}*. The micrometer-scale separation between QPCs enables independent fine tuning (~0.1%) and high-precision characterization [, with a large dc bias voltage suppressing Kondo/Coulomb renormalization (*

_{i}*32*)] of the Kondo channels, over the full range τ

*∈ [0, 1]. Such fine tuning of the connected channels to identical couplings is crucial for approaching the frustrated, symmetric Kondo critical points. The two-channel Kondo (2CK) configurations are implemented by setting and τ*

_{i}_{2}= 0, whereas for the three-channel Kondo (3CK) configurations . With the charging energy K (separately obtained from Coulomb diamond measurements) and high-precision shot-noise thermometry (

*33*), the device is completely characterized. The knowledge of these parameters allows for a full quantitative microscopic understanding (

*19*,

*28*,

*29*). In practice, Kondo physics is observed through the renormalized QPC conductances

*G*measured in situ. Because the symmetry between channels is found preserved by renormalization (at an experimental accuracy of ~0.003

_{i}*e*

^{2}/

*h*), we generally display the averages and when investigating the symmetric 2CK and 3CK configurations, respectively.

The high-precision implementation/quantum simulation of the charge Kondo model is validated in Fig. 1, C and D, and the different two-channel and three-channel Kondo behaviors are qualitatively illustrated. The renormalized conductance across channels tuned to “intrinsic” (Fig. 1, C and D, squares) or 0.68 (Fig. 1, C and D, triangles) is displayed for and 29 mK while sweeping the gate voltage *V*_{g}. The charge degeneracy point is identified as the conductance peak (δ*V*_{g} = 0). The good match, without any fit parameters, between the conductance data and the quantitative predictions of the charge Kondo model derived analytically for two near ballistic channels at low temperature (Fig. 1C, continuous line) (*29*, *32*) attests to the accurate device characterization and to its precise implementation of the model for arbitrary Kondo pseudospin energy splitting (*19*, *32*). At large δ*V*_{g}, the conductance is systematically reduced upon lowering *T* as usually expected from plain charge quantization. At δ*V*_{g} = 0 and for two or three symmetric channels set to , we observed instead a conductance increase with *T* owing to the Kondo renormalization of weakly connected channels. At the larger , 2CK and 3CK exhibit qualitatively different conductance renormalizations at δ*V*_{g} = 0, with opposite signs.

## Observation of an intermediate nontrivial fixed point

The above findings corroborate the theoretical expectations for the different 2CK and 3CK low-temperature conductance fixed points (*29*, *34*). Both 2CK and 3CK quantum critical fixed points are associated with an intermediate value of the renormalized Kondo coupling (*10*, *12*). In previous experiments on small quantum dots (*2*, *5*), the 2CK intermediate coupling could not be established. Indeed, *T* was not low enough with respect to the scaling Kondo temperature *T*_{K} to show a saturation; furthermore, asymmetries between electrical channels (*15*) can lead to a trivial intermediate asymptotic value of the measured conductance, which therefore does not necessarily imply an intermediate coupling in these spin Kondo devices. Moreover, the intermediate coupling character of the 2CK fixed point is not entirely invariable but depends on the choice of representation (*7*, *8*, *14*, *29*). In particular, the 2CK fixed point can be described as a noninteracting system involving two Majorana modes [one free, one in the strong coupling limit (*7*)]. This dual strong-coupling character of the 2CK fixed point also materializes in the present charge Kondo implementation: Here, *G*_{1,3} constitutes an alternative probe of the coupling between electrons and charge Kondo impurity, which flows not toward an intermediate value per electrical channel but toward the maximum free-electron quantum limit *G*_{2CK} = *e*^{2}/*h* (*29*). By contrast, the genuinely intermediate character of the interacting 3CK fixed point is predicted to show up directly in charge Kondo circuits, as a flow of the conductance per channel *G*_{1,2,3} toward the nontrivial intermediate universal conductance (*34*).

The precise 2CK and 3CK low-temperature universal conductance fixed points are experimentally established by measuring the temperature evolution of *G*_{1,3} and *G*_{1,2,3}, respectively, for a broad range of symmetric channel settings [τ ∈ (0.56, 0.985)]. For this purpose, and until explicitly specified otherwise, the device is tuned at charge degeneracy (δ*V*_{g} = 0), where Kondo effect is expected. Measurements of *G*_{1,3} and *G*_{1,2,3} versus *T* in logarithmic scale are shown as symbols in Fig. 2. In the 2CK configuration (Fig. 2A), whatever the setting τ, we found that *G*_{1,3} always grows as *T* is reduced. This observation validates the predicted *e*^{2}/*h* Kondo fixed point (Fig. 2A, horizontal red line), at an experimental accuracy of 0.006*e*^{2}/*h* (*6*). Upon lowering *T* in the 3CK configuration (Fig. 2B), *G*_{1,2,3} systematically grows when below 0.68*e*^{2}/*h* (and for *T* ≤ 40 mK) and decreases when above 0.70*e*^{2}/*h*. This validates the predicted 3CK universal conductance fixed point (horizontal green line) at an experimental accuracy of ±0.01*e*^{2}/*h*. This constitutes direct experimental evidence of an intermediate non-Fermi liquid fixed point.

## Universal scalings toward quantum criticality

First, we characterized the power-law exponents when approaching the 2CK and 3CK low-temperature fixed points and found them to be different from the characteristic *T*^{2} for Fermi liquids. For this purpose, the distances Δ*G* between measured *G*_{1,3} and *G*_{1,2,3} and, respectively, the theoretically predicted fixed points *G*_{2CK} and *G*_{3CK} are plotted in Fig. 3 versus *T*/*T*_{K}. The continuous straight lines show the universal power-law scalings asymptotically predicted at low *T*/*T*_{K} for the conductance in the present charge Kondo implementation: Δ*G* ∝ *T* for 2CK (*19*, *29*, *35*) and Δ*G* ∝ *T*^{2/5} for 3CK (*12*, *13*, *32*) [further discussion is provided in (*32*)]. Comparing with the data requires us to, for each τ, fix the corresponding scaling Kondo temperature *T*_{K}(τ). Symbols in the Fig. 3 insets represent the experimentally extracted values of *T*_{K} versus τ, which were obtained for each tuning τ by matching the lowest-temperature data point with the displayed theoretical power law. The data-theory comparison in the main Fig. 3 panel is therefore in the conductance evolution as temperature is increased. We found that sufficiently close to the fixed points (), the experiment is consistent with predictions. The precision is here limited by the increasing relative experimental uncertainty as Δ*G* is reduced. A direct extraction of the temperature exponents from the Δ*G* < 0.1*e*^{2}/*h* data at *T* ∈ {7.9, 12} mK (satisfying the NRG universality criteria mK) gives α_{2CK} = 0.83 ± 0.08 for 2CK and α_{3CK} = 0.42 ± 0.17 for 3CK.

We then investigated the full 2CK and 3CK universal renormalization flows. Measurements (symbols) are now compared in Fig. 4, A to C, with NRG calculations spanning the whole range of *T*/*T*_{K} (Fig. 4, continuous black lines) (*32*). In Fig. 4, A and B, respectively, *G*_{1,3} and *G*_{1,2,3} are plotted versus log(*T*/*T*_{K}). Following standard procedures, the theoretical scaling Kondo temperature *T*_{K} was normalized so that the NRG universal conductance takes a value equal to half that of the Kondo fixed point at *T* = *T*_{K}. As in Fig. 3, the experimental *T*_{K}(τ) (symbols in Fig. 4 insets) are adjusted by matching data with theory at mK. These *T*_{K}(τ) remain therefore identical to those in the insets of Fig. 3 as long as NRG calculations and asymptotic power laws are indistinguishable (for mK). We observed a quantitative agreement between the data and the universal NRG prediction over six (2CK) or eight (3CK) orders of magnitude in *T*/*T*_{K}. A direct comparison of the same measurements and predictions is shown in Fig. 4C in a scale-invariant representation that does not involve rescaling the temperature in units of *T*_{K}, by displaying ∂*G*_{1,3}/∂log(*T*) versus *G*_{1,3} and ∂*G*_{1,2,3}/∂log(*T*) versus *G*_{1,2,3}. In this representation, data points correspond to experimental measurements of the so-called β-function that determines the corresponding 2CK or 3CK renormalization group equation for the conductance. In Fig. 4C, the straight dashed lines near 2CK and 3CK fixed points (Fig. 4C, arrows) represent the predicted non-Fermi liquid power-law behaviors discussed in the previous paragraph. Comparing with the experimental slope therefore complements the approach in Fig. 3. Also shown is the experimental “analog quantum simulation” of the universal 3CK β-function at *G*_{1,2,3} > *G*_{3CK}, out of reach of NRG calculations.

Last, we explored the quantitative relationship between scaling Kondo temperature *T*_{K} and microscopic model parameter τ (Figs. 3 and 4, insets). At small , the same expected exponential behavior is observed for 2CK and 3CK (*29*). At larger τ, *T*_{K} appears to diverge at a specific setting τ_{c}, with extracted *T*_{K} values orders of magnitude above the high-energy cutoff mK (in which case, universal Kondo physics can only be probed at ). For 2CK, theory predicts a divergence at τ_{c} = 1 as , which is displayed by the identical continuous red lines in the insets of Figs. 3 and 4A (*29*) [(*31*, *36*), the prediction of a peaked *T*_{K}(*J*)]. For 3CK, the observed value is higher than . This is caused by the conductance suppression by Coulomb interaction at temperatures , before the development of universal Kondo physics at low temperatures. Assuming theoretically that *T*_{K} diverges at τ_{c}, we generally find (*32*) that a low-temperature conductance power law Δ*G* ∝ *T*^{α} corresponds to a power-law divergence as . The observed close agreement between experimental in 2CK and 3CK configurations with, respectively, (Figs. 3 and 4A, insets, red lines) and (Figs. 3 and 4B, insets, green lines) therefore further establishes the predicted non-Fermi liquid Kondo exponents for two (α_{2CK} = 1) and three (α_{3CK} = 2/5) symmetric channels.

## Crossover from quantum criticality

As the temperature is increased [up to some limit; here, ], the quantum critical regime is generally expected to span over a larger range of system parameters, away from the *T* = 0 quantum critical point (Fig. 5A). The so-called crossover temperature *T*_{co} delimits quantum criticality from below, with the critical point itself corresponding to *T*_{co} = 0. Generically, the crossover from quantum criticality as temperature is lowered should follow universal curves versus the reduced parameter *T*/*T*_{co}. Indeed, *T*_{co} is the only relevant temperature scale, encapsulating all microscopic details, provided that the high-energy cutoff for quantum criticality is much higher. In tunable circuits, the crossover from 2CK quantum criticality was explored versus Kondo channels asymmetry (*5*, *6*) and, in the different implementation of a spin-polarized quantum dot embedded into a dissipative circuit, versus the difference between resonant dot level and Fermi energy (*4*). These experiments corroborate the existence of a universal *T*/*T*_{co} scaling, as well as the predicted quadratic increase of *T*_{co} for small deviations from the 2CK critical point (*12*, *15*, *18*). We explored the disparate universal and exotic behaviors along the different crossovers induced by breaking the Kondo (pseudo)spin degeneracy or the channel symmetry, observed the development of the quantum phase transition across the symmetric 3CK quantum critical point, and demonstrated “super-ballistic” conductances.

In a first step, we investigated the so-far-unexplored crossover from 2CK and 3CK quantum criticality induced by breaking the energy degeneracy of the Kondo impurity, with the connected channels remaining symmetric. We established (i) the different 2CK and 3CK power-law dependence for small energy splitting of the charge pseudospin , with mV being the gate voltage period; (ii) a generalized expression of *T*_{co} for arbitrary Δ*E*; and (iii) the theoretical universal crossover curves and , obtained analytically in (*19*, *29*) for 2CK and by NRG here for 3CK.

The crossover temperature *T*_{co} is defined so that the conductance is halfway between the quantum critical regime (≈*G*_{2CK} or ≈*G*_{3CK}, at ) and the Fermi liquid regime (≈0, at )—that is, *G*_{1,3}(Δ*E*, *T* = *T*_{co}) ≡ *G*_{2CK}/2 or *G*_{1,2,3}(Δ*E*, *T* = *T*_{co}) ≡ *G*_{3CK}/2. In practice, we fixed the electronic temperature *T* and adjusted the energy splitting Δ*E* ∝ δ*V*_{g} in order to obtain this midway conductance value, if possible. In Fig. 5, B and C, this corresponds to the crossings between continuous and horizontal dashed lines, where the experimentally extracted crossover temperature directly reads . Symbols in Fig. 5, D and E, display versus Δ*E* for the settings τ where *T*_{co} ∝ Δ*E*^{γ} is expected (*32*). The predicted corresponding power laws are shown as continuous lines (*T*_{co}∝Δ*E*^{2} for 2CK, *T*_{co} ∝ Δ*E*^{5/3} for 3CK) (*12*). Fitting separately, for each τ, the mK data (fulfilling the universality NRG criteria) yield the values of γ displayed as symbols in the insets of Fig. 5, D and E. A statistical analysis of these values gives γ_{2CK} = 2.01 ± 0.04 and γ_{3CK} = 1.69 ± 0.02 for the crossovers from 2CK and 3CK, respectively, which is in close agreement with theory.

The theoretically predicted universal crossover curves and , shown as thick dashed lines in Fig. 5, B and C, right hand sides, are compared with conductance data. Continuous lines in Fig. 5, B and C, left hand sides represent the conductance measured at different temperatures versus gate voltage for (Fig. 5B) and (Fig. 5C). These settings correspond to well-developed quantum critical regimes (small Δ*G*), a necessary condition to investigate and down to small *T*_{co}/*T*. In Fig. 5, B and C, right hand sides, the gate voltage sweeps at different temperatures (continuous lines) fall on top of one another when plotted versus calculated *T*_{co}/*T*, demonstrating the predicted universal character of the crossover from quantum criticality. Moreover, we found a precise match between experimental universal curves and theoretical predictions and . *T*_{co} is obtained from experimental parameters by using generalized expressions that remain valid for arbitrary gate voltage beyond the power law at small detuning. For the charge 2CK device with near ballistic channels, the full quantitative expression derived in (*29*) was used in Fig. 5B: . The data- comparison in Fig. 5B is therefore without any fit parameter. For 3CK, we expect from NRG calculations the similar generalization *T*_{co} = λ_{3CK}sin^{5/3}(πδ*V*_{g}/Δ) (*32*), which was used in Fig. 5C. Because the prefactor λ_{3CK}(τ, *E*_{C}) is not known, the value λ_{3CK} = 36 mK was freely adjusted in the data- comparison shown in Fig. 5C.

In a second step, the development of the 3CK QPT driven by the channels’ competition to screen the Kondo spin is plainly observed through the conductance renormalization flow of asymmetric channels upon lowering temperature (Fig. 6). The Kondo charge pseudospin is energy degenerate (δ*V*_{g} = 0), QPC_{1,3} are tuned symmetric (), and τ_{2} is adjusted separately. Displayed in Fig. 6 as colored lines with arrowheads is the temperature evolution of the measured conductances *G*_{2} (vertical axis) and *G*_{1,3} (horizontal axis) from 55 to 7.9 mK (arrowhead at lowest *T*); each line corresponds to a different device setting. In total, 15 × 14 settings of were measured, with τ_{1,2,3} picked among 14 fixed values ranging from 0.1 to 0.985 (*32*) and including also τ_{2} = 0. The data closest to the Fig. 6 diagonal gray line correspond to three channels tuned symmetric (). Below the diagonal, where , the data flow toward the predicted 2CK fixed point (Fig. 6, red disk, at *G*_{1,3} = *e*^{2}/*h* and *G*_{2} = 0). Above the diagonal, where so that a flow toward the 1CK fixed point involving QPC_{2} is expected (blue disk, at *G*_{1,2,3} = 0), we observed a monotonous decrease of the conductance *G*_{1,3} across the less strongly coupled QPCs. By contrast, *G*_{2} first rises, markedly oversteps the free-electron quantum limit *e*^{2}/*h* (up to +25%), and then decreases toward the zero conductance 1CK fixed point as *T* is further reduced.

The nonmonotonous behavior of *G*_{2} when higher than *G*_{1,3} might appear counterintuitive. A flow toward the low-temperature 1CK “strong coupling” fixed point is expected, which corresponds to a renormalized Kondo coupling growing monotonously (*J*_{2} → ∞). However, *J*_{2} connects with the tunnel coupling/hopping integral of electrons across QPC_{2} in the charge Kondo mapping, and free-electron theory predicts a nonmonotonous dependence of the conductance with the hopping integral [with a maximum for the value that best preserves translational invariance, and *G*_{2}(*J*_{2} → ∞) = 0]. By contrast, the present measurement of a conductance that exceeds the maximum possible value for noninteracting electrons in the ballistic limit is highly nontrivial and was not anticipated (although reproduced by our NRG calculations). Such a super-ballistic conductance, also in an intermediate temperature range and of similar amplitude, was coincidently observed in clean graphene constrictions (*37*) and explained as a collective viscous flow of the electronic fluid induced by electron-electron collisions (*38*). We speculate that the electron-electron interactions mediated by the Kondo impurity within the electronic channel across QPC_{2}, expected to be particularly strong near the turning point where *G*_{2} is maximum, might also result in such a viscous electronic fluid behavior. One specificity of our system is that the super-ballistic magnitude and the temperature range in which it takes place can be controlled in situ, by separately adjusting the channels.

The experimental findings are compared with NRG calculations of the universal crossover flow from 3CK quantum criticality, induced by an initially minute asymmetry between *G*_{2} and *G*_{1,3} (*32*). These are displayed in Fig. 6 as two thick gray lines originating from the 3CK fixed point, with arrows pointing toward lower temperatures. For *G*_{1,3} > *G*_{2}, NRG predicts a monotonous crossover flow from 3CK to 2CK conductance fixed points that closely matches the nearby data. For *G*_{2} > *G*_{1,3}, the universal NRG crossover flow from 3CK to 1CK reproduces the observed nonmonotonous behavior, confirms the naively expected vanishing of *G*_{2} at the 1CK fixed point, and establishes that a super-ballistic conductance exceeding by ~20% the free-electron maximum limit follows from the 3CK model, which is in quantitative agreement with the experiment. Although experimental and NRG flows point to the same direction near 3CK and 1CK fixed points, clear crossings are also visible in intermediate regimes above the diagonal, including between different experimental device settings. These mostly take place between flows involving opposite renormalization directions of *G*_{2}, as expected from the nonmonotonous relationship between *G*_{2} and Kondo coupling *J*_{2} that specifically shows up above the diagonal.

## Outlook

The observation of super-ballistic conductance opens a research path for low-power electronics. Although the present implementation has no clear application potential, it forms a powerful platform from which to understand the underlying mechanisms of behaviors that arise in diverse clean systems with strong electron-electron interactions. We anticipate that similar metal-semiconductor hybrids will form building blocks for a wide range of investigations of the strongly correlated electron physics, and in particular the emergence of exotic parafermion quasiparticles (*7*, *9*, *35*). Measurements of complementary observables—such as charge susceptibility, fluctuations, and heat current—as well as investigations of the dynamical and out-of-equilibrium responses could unveil yet hidden facets of the exotic underlying physics. Furthermore, direct generalizations of the present charge Kondo implementation should grant access to quantitative investigations of many thus-far-inaccessible strongly correlated phenomena (*16*, *39*), including the nanoengineered competition between Kondo channels, dissipation [specific proposal in (*40*)], fractional quantum Hall effect, and multiple impurities.

## Supplementary Materials

www.sciencemag.org/content/360/6395/1315/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S5

Data File S1

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**We thank I. Affleck, S. Florens, L. Fritz, L. Glazman, C. Kane, K. Matveev, Y. Nazarov, E. Sela, G. Zarand, and F. Zhang for discussions.

**Funding:**This work was supported by the French RENATECH network and the national French program “Investissements d’Avenir” (Labex NanoSaclay, ANR-10-LABX-0035).

**Author contributions:**Z.I. and F.P. performed the experiment; Z.I., A.A., and F.P. analyzed the data; A.K.M. performed the NRG developments; C.M. and P.S. extended the analytical predictions; F.D.P. fabricated the sample with inputs from A.A.; U.G., A.C., and A.O. grew the 2DEG; and F.P. led the project and wrote the manuscript, with inputs from Z.I., A.A., A.K.M., U.G, C.M., and P.S.

**Competing interests:**The authors declare no competing financial interests.

**Data and materials availability:**The displayed experimental data are provided in a separate supplementary file (Data file S1). Correspondence and requests for materials should be addressed to F.P. (frederic.pierre@u-psud.fr).