Abstract
Interacting quantum manybody systems constitute a fascinating research field because they form quantum liquids with remarkable properties and universal behaviour^{1}. In fermionic systems, such quantum liquids are realized in helium3 liquid, heavy fermion systems^{1}, neutron stars and cold gases^{2}. Their properties in the linearresponse regime have been successfully described by the theory of Fermi liquids^{1}. The idea is that they behave as an ensemble of noninteracting ‘quasiparticles’. However, nonequilibrium properties have still to be established and remain a key issue of manybody physics. Here, we show a precise experimental demonstration of Landau Fermi liquid theory extended to the nonequilibrium regime in a zerodimensional system. Combining transport and ultrasensitive current noise measurements, we have unambiguously identified the SU(2) (ref. 3) and SU(4) (refs 4,5,6,7,8) symmetries of a quantum liquid in a carbon nanotube tuned in the universal Kondo regime. Whereas the free quasiparticle picture is found valid around equilibrium^{9}, an enhancement of the current fluctuations is detected out of equilibrium and perfectly explained by an effective charge induced by the residual interaction between quasiparticles^{8,10,11,12,13,14,15,16,17}. Moreover, an asyetunknown scaling law for the effective charge is discovered, suggesting a new nonequilibrium universality. Our method paves a new way to explore the exotic nature of quantum liquids out of equilibrium through their fluctuations in a wide variety of physical systems^{18}.
Main
The Kondo effect^{19} is a typical example of a quantum manybody effect, where a localized spin is screened by the surrounding conduction electrons at low temperature to form a unique correlated ground state. The Kondo state is described well by the Fermi liquid theory at equilibrium^{1,9}, which makes it an ideal testbed to go beyond equilibrium. To unveil the universal behaviour of nonequilibrium Fermi liquid^{11}, we have used the current fluctuations or shot noise in a Kondocorrelated nanotube quantum dot^{18}.
When electrons are transmitted through this system, the scattering induces the shot noise, which sensitively reflects the nature of the quasiparticles^{20}, as shown in the upper panel of Fig. 1a. A remarkable prediction of the nonequilibrium Fermi liquid theory is that the residual interaction between quasiparticles creates an additional scattering of two quasiparticles which enhances the noise (see the lower panel of Fig. 1a)^{10,12,13,14,15}. This twoparticle scattering is characterized by an effective charge e^{∗} larger than e (electron charge). This value, closely related to the Wilson ratio, is universal for the Fermi liquid in the Kondo regime as it depends only on the symmetry group of the system^{13,14,15}. Although some aspects of Kondoassociated noise have been reported^{8,16,17}, a rigorous, selfconsistent treatment in a regime where universal results apply is at the core of the present work. Actually, by investigating the same nanotube quantum dot in the spin degenerate SU(2) Kondo regime and in the spin–orbit degenerate SU(4) regime, the noise is proved to contain distinct signatures of these two symmetries, confirming theoretical developments of Fermi liquid theory out of equilibrium.
In our experiment, we measured the conductance and current noise through a carbon nanotube quantum dot grown by chemical vapour deposition^{21}. Iron catalyst was deposited on an oxidized undoped silicon wafer and exposed to 10 mbar of acetylene for 9 s at 900 °C. The nanotube was connected with a Pd(6 nm)/Al(70 nm) bilayer deposited by egun evaporation. The distance between the contacts is 400 nm and a side gate electrode is deposited to tune the potential of the quantum dot (see Fig. 1b). A magnetic field of 0.08 T is applied to suppress superconductivity of the contacts. To measure accurately the shot noise, our sample is connected to a resonant (2.58 MHz) LC circuit thermalized at the mixing chamber of the dilution fridge. The signal across this resonator is amplified with an inhousebuilt cryogenic lownoise amplifier^{22} thermalized on the 1 K pot and again at room temperature. The power spectral density of the noise is obtained by fast Fourier transformation of the timedomain signal acquired by a digitizer (National Instruments PCI5922). The current noise of the sample is extracted from the fit of the shape of the resonance in the frequency domain.
Figure 2a presents the image plot of the differential conductance of the sample (G) at temperature T = 16 mK as a function of source–drain voltage (V_{sd}) and gate voltage (V_{g}). This stability diagram shows the fourfolddegenerated Coulomb diamonds specific to carbon nanotubes. The spectrum consists of successive fourelectron shells. We denote by N = 0,1,2,3 the number of electrons in the last shell. Remarkably, the SU(2) and SU(4) Kondo ridges^{6} emerge as horizontal bright regions (high conductance) at V_{sd} = 0.
For the moment, we concentrate on the SU(2) region. A cut of the conductance at V_{sd} = 0 is represented in the upper panel of Fig. 2b. Two Kondo ridges appear as plateaux where G is maximum for the fillings N = 1 (ridge A) and N = 3 (ridge B), whereas G decreases to almost zero for even N. In addition, the ridge B is flat and the unitary limit is achieved: the conductance reaches the quantum of conductance G_{Q} = 2e^{2}/h, which is a signature of the perfect Kondo effect in a dot with symmetric coupling to the leads. The Kondo temperature (T_{K}) is 1.6 ± 0.05 K in the centre of this ridge (see Supplementary Information).
The current noise S_{i} and G are plotted in Fig. 2c, d for N = 2 and N = 3, respectively, as a function of the source–drain current I_{sd}. Outside the Kondo ridge (N = 2) S_{i} is linear with I_{sd}, whereas on the ridge the shot noise is flat around I_{sd} = 0 and enhanced at high current when the energy of incoming electrons approaches a fraction of T_{K}. At high voltage (eV_{sd} ≫ k_{B}T_{K}), a linear behaviour is recovered, with S_{i} = 2eI_{sd}. To analyse the lowenergy properties, we have extracted the Fano factor (F), which is defined as S_{i} = 2eFI_{sd}, from a linear fit at low current. F varies from approximately zero on the Kondo ridge to F = 1 outside, as shown in the bottom panel of Fig. 2b. Indeed, at very low energy, as the free quasiparticle picture of the Fermi liquid theory teaches us, the conductance and the noise for a multichannel conductor can be written as a function of transmission T_{i} for each channel i = 1,2,3… (ref. 20): For the SU(2) symmetry, transport occurs through one single channel, yielding G = G_{Q}T_{1} and F = 1 − T_{1}. On the Kondo ridge, the conductance is G = G_{Q}, yielding T_{1} = 1 and F = 1 − T_{1} = 0. This is a direct signature of the Kondo resonance which allows a perfect transmission and thus no partition of quasiparticles near equilibrium. In the Coulomb blockade regime, for even N, the transport is blocked (T_{1} ≪ 1), yielding F ≈ 1, indicating that transport occurs through tunnelling events resulting in conventional Poissonian noise.
At higher voltages (0 < eV_{sd} ≤ k_{B}T_{K}/2) we concentrate only on the nonlinear terms for current and noise by subtracting the linear part. We defined S_{K} = S_{i} − 2eFI_{sd} and the backscattered current I_{K} = G(0)V_{sd} − I_{sd} (refs 10,13). These quantities are related through the effective charge^{10,12,13,23}e^{∗}: This effective charge does not imply an exotic charge, as in the quantum Hall regime, but is related to the probability that one particle or two particles are backscattered in the Fermi liquid (see Supplementary). Figure 3a, b shows the evolution of the conductance on ridge B as a function of magnetic field B and the corresponding noise S_{K} as a function of I_{K}, respectively. The temperature dependence is analysed in detail in the Supplementary Information. The effective charge is directly given by the slope at low current I_{K} (eV_{sd} < k_{B}T_{K}), yielding for the lowest field and temperature e^{∗}/e = 1.7 ± 0.1. This result is in good agreement with theory^{10,12}, which predicts e^{∗}/e = 5/3 ≈ 1.67, corresponding to equal probabilities for one or twoparticle scattering. Figure 3c represents the evolution of e^{∗} with magnetic field and temperature. On this graph e^{∗} is represented as a function of the reduced scales T/T_{K} for temperature or gμ_{B}B/2k_{B}T_{K} for the field, where g = 2 is the Landé factor and μ_{B} the Bohr magneton (see Supplementary Information). All the data points seem to fall on the same curve, suggesting that e^{∗} obeys a logarithmic scaling law which has not yet been predicted.
The Wilson ratio R has been extracted from the formula^{12,15} where n characterizes the symmetry group SU(n) of the Kondo state. This number R is directly related to the ratio U/Γ, with U the charging energy of the quantum dot and Γ the coupling to the electrodes. It reflects the strength of the interaction in the Fermi liquid and is the only parameter to characterize the system, going from R = 1 in the noninteracting case (U = 0) to R = 2 in the strong SU(2) Kondo limit (U → ∞) (see Fig. 3d). Our value for e^{∗} yields R = 1.95 ± 0.1, ensuring strong interactions and thus a universal regime.
For consistency, R was independently extracted by fitting the evolution of conductance with B, V_{sd} and T at low excitation (see Supplementary Information) without any assumption on T_{K}. This result, R = 1.95 ± 0.1, agrees perfectly with the value extracted from e^{∗}. Finally, from the dependence of T_{K} with gate voltage, we have independently extracted the values U = 6 ± 0.5 meV and Γ = 1.8 ± 0.2 meV. This consistency is illustrated in Fig. 3d, where the two independent values for R and U/Γ cross on the theoretical curve.
The effect of asymmetric lead–dot coupling was tested on the ridge A, where G = 0.85 G_{Q}. Asymmetry is defined by the factor δ such that G(0) = (1 − δ)G_{Q}. We have measured an effective charge e^{∗}/e = 1.2 ± 0.08, in good agreement with ref. 14, which predicts e^{∗}/e = 5/3 − (8/3)δ = 1.26.
Now, what can we learn from the SU(4) symmetry emerging on the righthand side of Fig. 2a? A zoom is plotted in the upper panel of Fig. 4a and a crosssection is shown in the middle panel. Because spin and orbital degrees of freedom are degenerated, two channels contribute to transport, and Kondo resonance emerges at every filling factor, N = 1, 2 and 3 electrons^{4,5,6,7}. At odd filling, the channels are half transmitted (T_{1} = T_{2} = 0.5), yielding the same conductance G_{Q} as in the SU(2) symmetry. However, for N = 2, current is transmitted through two perfect channels (T_{1} = T_{2} = 1), increasing the conductance to G = 2 G_{Q}. In this region the conductance hardly depends on temperature up to 800 mK, reflecting a large T_{K}, as expected for the SU(4) symmetry^{24}. This is confirmed by the full width of the curve G(V_{sd}), which gives T_{K} ≈ 11 K for N = 2 and T_{K} ≈ 17 K for N = 1 or 3.
The first result to emphasize is that the linear part of the current noise is qualitatively different from SU(2) and is a powerful experimental tool to distinguish the two symmetries^{14,18}. The upper part of Fig. 4b represents the conductance at N = 3 for SU(2) and SU(4) as a function of the rescaled voltage eV_{sd}/k_{B}T_{K}. The two curves are barely distinguishable. However, the current noise shown on the lower part of Fig. 4b is qualitatively different as it is almost zero for the SU(2) symmetry, whereas it is linear with I_{sd} for SU(4). The linear noise is one order of magnitude stronger for SU(4) than for SU(2). Indeed, the Fano factor for two channels such that T_{1} = T_{2} = 0.5 is F = 0.5 for the SU(4) symmetry, whereas for SU(2) a single channel with T_{1} = 1 yields F = 0. The complete evolution of F is summarized in the lower part of Fig. 4a. It changes from F ≈ 1 outside the Kondo ridge to F = 0.5 for N = 1 and 3, reaching F = 0.07 for N = 2. This confirms that in the N = 2 SU(4) case, transport takes place through two almost perfect channels (T_{1} = T_{2} = 1) without partition, yielding F ≈ 0.
Finally we discuss e^{∗} and R for the SU(4) symmetry at half filling (N = 2). S_{K} and I_{K} have been computed by using the same procedure as explained for SU(2). The result, shown in Fig. 4c, gives e^{∗}/e = 1.45 ± 0.1 and R = 1.35 ± 0.1 from (equation (3)). The decreasing value for R and e^{∗} reflects the increasing number of degenerated states. When degeneracy increases, electrons correlations become weaker and the noninteracting value is recovered for SU(n) when n → ∞. The values for e^{∗} and R are in good agreement with refs 14,15 (also see Fig. 3d), which predict e^{∗} = 1.5 and R = 4/3, confirming that nonequilibrium Fermi liquid theory can be extended to more exotic classes of Fermi liquids.
Our experimental results emphasize three important points for the theory of Fermi liquids. First, in the linear regime they can be described as free quasiparticles, as the spirit of the theory teaches us. This allows one to clearly distinguish the different symmetry classes of Fermi liquids through shot noise measurements, whereas it is hardly possible from the conductance. Second, by probing the nonlinear noise, we have shown that out of equilibrium the residual interaction between quasiparticles shows up and creates a peculiar twoparticle scattering, which bears the signature of the correlated nature of quantum liquids. Finally, the newly discovered outofequilibrium scaling law should trigger theoretical developments to deepen our understanding of universal behaviours in these liquids.
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Acknowledgements
We appreciate discussions with H. Bouchiat and R. Yoshii. This work was partially supported by a GrantinAid for Scientific Research (S) (No. 26220711), JSPS KAKENHI (No. 26400319, 25800174 and 15K17680), Invitation Fellowships for Research in Japan from JSPS, GrantinAid for Scientific Research on Innovative Areas ‘Fluctuation & Structure’ (No. 25103003) and ‘Topological Materials Science’ (KAKENHI Grant No. 15H05854), the Program for Promoting the Enhancement of Research Universities from MEXT, and Yazaki Memorial Foundation for Science and Technology, and the French programmes ANR DYMESYS (ANR2011IS0400101) and ANR MASH (ANR12BS040016). K.K. acknowledges the stimulating discussions in the meeting of the Cooperative Research Project of RIEC, Tohoku University.
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Affiliations
Department of Physics, Graduate School of Science, Osaka University, 11 Machikaneyama, Toyonaka, 5600043 Osaka, Japan
 Meydi Ferrier
 , Tomonori Arakawa
 , Tokuro Hata
 , Ryo Fujiwara
 & Kensuke Kobayashi
Laboratoire de Physique des Solides, CNRS, Univ. ParisSud, Université Paris Saclay, 91405 Orsay Cedex, France
 Meydi Ferrier
 , Raphaëlle Delagrange
 , Raphaël Weil
 & Richard Deblock
The Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 2778581, Japan
 Rui Sakano
Department of Physics, Osaka City University, Sumiyoshiku, Osaka 5588585, Japan
 Akira Oguri
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Contributions
All the authors contributed to the analysis of the data and commented on the manuscript. M.F., R. Delagrange., R.W. and R. Deblock designed the sample. M.F., T.A., T.H. and R.F. performed the noise experiments and analysed the data. M.F. and K.K. planned and supervised the research.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Meydi Ferrier or Kensuke Kobayashi.
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