Emergent SU(4) Kondo physics in a spin charge-entangled double quantum dot

Transcription

1 Emergent SU(4) Kondo physics in a spin charge-entangled double quantum dot A. J. Keller 1, S. Amasha 1,, I. Weymann 2, C. P. Moca 3,4, I. G. Rau 1,, J. A. Katine 5, Hadas Shtrikman 6, G. Zaránd 3, and D. Goldhaber-Gordon 1,* 1 Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA 94305, USA 2 Faculty of Physics, Adam Mickiewicz University, Poznań, Poland 3 BME-MTA Exotic Quantum Phases Lendület Group, Institute of Physics, Budapest University of Technology and Economics, H-1521 Budapest, Hungary 4 Department of Physics, University of Oradea, , Romania 5 HGST, San Jose, CA 95135, USA 6 Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 96100, Israel Present address: MIT Lincoln Laboratory, Lexington, MA 02420, USA Present address: IBM Research Almaden, San Jose, CA 95120, USA * Corresponding author; goldhaber-gordon@stanford.edu Contents S1 Full LBTP survey 2 S2 Summary of NRG calculations 8 S2.1 NRG calculations S2.2 Choosing NRG parameters S3 Extracting LBTP cuts from 2D data sets 11 S4 G(ɛ 1,ɛ 2 ) for all measured T 12 S5 Temperature dependence details 17 S6 Empirical Kondo forms 17 NATURE PHYSICS 1

2 S7 Bias spectroscopy 22 S7.1 N e = 3 LBTP, zero magnetic field S7.2 N e = 1 LBTP, T field S8 Technical details 26 S8.1 Electronics S8.2 Electron temperature calibration S8.3 Magnetic field calibration S8.4 g-factor calibration S8.5 Bias spectroscopy NATURE PHYSICS

3 SUPPLEMENTARY INFORMATION S1 Full LBTP survey The data presented in Fig. 1b are only a subset of the full survey of conductance around lines between triple points (LBTPs). The full survey, shown in Fig. S1, demonstrates that 11/12 of N e = 1 or N e = 3 LBTPs exhibit higher conductance towards the adjacent (1,1) hexagon. In addition, twelve (1,1)/(2,0) or (0,2)/(1,1) LBTPs were surveyed: these should possess a five-fold degeneracy assuming the (2,0) ground state is a singlet rather than triplet. The N e = 1 and N e = 3 LBTPs differ qualitatively from the (1,1)/(2,0) and (0,2)/(1,1) LBTPs in that the latter class of LBTPs do not exhibit a simple pattern of which end of the LBTP has higher conductance. Experimental parameters Γ 1,Γ 2 and peak conductances are extracted from each data set and summarized in Table S1. Because we claim that the N e = 1 and N e = 3 LBTP data reflect the particle-hole symmetry of a four-fold degenerate state, it is natural to expect that the pattern is destroyed when the four-fold degeneracy is broken. Fig. S2 shows the N e = 1 and N e = 3 LBTPs surveyed again in an in-plane magnetic field of 2.0 T, corresponding to E Z = gµ B B =51 mev for g =4. Here, E Z > Γ 1,Γ 2 for all of the surveyed LBTPs. With the Zeeman splitting having broken the spin degeneracy at the LBTPs, a periodic pattern is no longer discernible. Table S2 summarizes the extracted parameters for each data set, as in Table S1. Fig. S3 shows how a small but finite V SD affects the observed asymmetry at an N e = 1 LBTP. The LBTP measured here corresponds to the same absolute electron occupation numbers as data set 553 shown in Fig. S1. Only for negative V SD approaching 10 µv does the conductance near (0,0) exceed that near (1,1). For positive V SD, the pattern of higher conductance nearer to (1,1) than (0,0) is actually exaggerated. The effect of finite V SD is similar regardless of whether it is applied to dot 1 or 2. Input offset voltages from current amplifiers could obscure our observed pattern, were it not for our ability to stabilize these voltages to within 1 µv (see section S8.1). NATURE PHYSICS 3

4 (e,e) -173 (o,o) (e,e) -179 (o,o) (e,e) -184 (o,o) _743 _737 _732 _729 _722 _ (o,o) -309 (o,o) (e,e) -308 (e,e) _ _ (o,o) (e,e) -219 (o,o) (e,e) (o,o) (e,e) _678 _688 _695 _704 _ (e,e) (o,o) _ _ (e,e) (o,o) _ _ (e,e) (o,o) _ (o,o) (e,e) _754 _773 _787 _758 _766 _ Fraction of max G (adjusted per plot) Figure S1: Experimental zero bias conductance G = G1 + G2 for a survey of 24 LBTPs, at zero magnetic field and at T = 20 mk. Electron occupation numbers are labeled here by their parity (e = even, o = odd). Each square of measured data corresponds to a region spanning 3 mv in VP 1 and VP 2. The color scales for each square have been individually set so that only data between 75% and 100% of the maximum conductance are visible. Each data set is identified by a number in the bottom-left of each plot. Set 766 (marked by triangle) corresponds to the bottom-left plot in Fig. 1d. All sets appearing in Fig. 1d have a gray background. Of the four other odd parity data sets, only one (672) does not show the expected asymmetry; it has no clear asymmetry at all. 4 NATURE PHYSICS

5 SUPPLEMENTARY INFORMATION Data set Γ 1 Γ 2 γ 1 γ 2 Data set Γ 1 Γ 2 γ 1 γ Table S1: For each data set shown in Fig. S1, experimentally controllable parameters Γ 1,Γ 2, γ 1, and γ 2 are extracted by fitting a Lorentzian lineshape to a Coulomb blockade (CB) peak neighboring the LBTP. Γ 1(2) corresponds to the FWHM of the CB peak in dot 1 (2), in units of µev. The width in gate voltage is converted to an energy using conversion factors derived from bias spectroscopy, taken near each LBTP. γ 1(2) are defined to equal the conductance at the CB peak of dot 1 (2) in e 2 /h. For these data it is not known whether the source or drain lead is more coupled for either dot. In all cases, the electron temperature T e = 20 mk. NATURE PHYSICS 5

6 _ _ _ _ _ _ _ _ _ _ _ _ Fraction of max G (adjusted per plot) Figure S2: Experimental zero source-drain bias conductance G = G1 + G2 for a survey of twelve Ne = 1 and Ne =3 LBTPs, in an in-plane magnetic field of 2.0 T at T = 20 mk. The LBTPs surveyed correspond to the same absolute electron occupations as the LBTPs of Fig. S1. The data are presented as described in the caption of Fig. S1. Set 1135 (marked by triangle) corresponds to the bottom-left plot in Fig. 1b, and set 766 in Fig. S1. 6 NATURE PHYSICS

7 SUPPLEMENTARY INFORMATION Data set Γ 1 Γ 2 γ 1 γ Table S2: For each data set shown in Fig. S2, experimentally controllable parameters Γ 1,Γ 2, γ 1, and γ 2 are extracted and reported as in Table S1. NATURE PHYSICS 7

8 V SD (1) = -10 µv V SD (1) = -5 µv V SD (1) = +5 µv V SD (1) = +10 µv V V (1,1) V V V V V V V V SD (2) = -10 µv V SD (2) = -5 µv (0,0) Zero bias V V SD (2) = +5 µv V SD (2) = +10 µv V V V V Figure S3: Experimental conductance G = G1 + G2 measured at an Ne = 1 LBTP, with small but finite VSD. All color scales show from 0.70 to 1.12 e 2 /h, emphasizing the conductance along the LBTP near (0,0) and (1,1). Center: VSD =0 for each dot. This data set was taken immediately after checking the input offset voltage of both current amplifiers. Top row: Finite VSD is applied across dot 1 only. Bottom row: Finite VSD is applied across dot 2 only. 8 NATURE PHYSICS

9 SUPPLEMENTARY INFORMATION S2 Summary of NRG calculations S2.1 NRG calculations In our numerical calculations the double quantum dot (DQD) system is modeled by the following Hamiltonian H = H DQD + H Tun + H Leads, (1) where H DQD = jσ ɛ j n jσ + j U j n j n j + U σσ n 1σ n 2σ + gµ B B z S z, (2) describes the two dots, with n jσ = d jσ d jσ the occupation number operator of dot j =1, 2 for spin σ, ɛ jσ the energy of a spin-σ electron residing on dot j. U j (U ) denotes the intradot (interdot) Coulomb correlations, while B z is the magnetic field applied along the z-direction and S z is the z-component of the double dot s spin. The tunneling Hamiltonian H Tun reads H Tun = t αj (c αjkσ d jσ + d jσ c αjkσ), (3) αk jσ where c αjkσ is the creation operator of an electron in lead α = L, R coupled to dot j, with momentum k and spin σ of energy ɛ αjk. Tunneling processes between the dots and leads are described by hopping matrix elements t αj. Tunneling between the two dots is suppressed by tuning gates in our experiment, and hence is omitted from the model. The leads are described by noninteracting quasiparticles H Leads = ɛ αjk c αjkσ c αjkσ. (4) αjkσ Due to the coupling to external leads, the dots levels acquire a width described by αj = πρ αj t αj 2, with ρ αj the density of states of lead α coupled to dot j. We performed the full density-matrix numerical renormalization group calculations (fdm-nrg) [1, 2, 3, 4], employing the Budapest Flexible DM-NRG code [5]. For efficient calculations, we used the charge U(1) and the spin SU(2) symmetries in each NATURE PHYSICS 9

10 channel, resulting in four symmetries altogether. When considering the effect of external magnetic field B z, the spin invariance is reduced to the U(1) symmetry for the spin z-component in each channel. In our computations we retained states at each iteration depending on the exploited symmetries and used the discretization parameter Λ = 2. We calculated the linear conductance through dot j using the following formula G j = e2 h α ( j j dω πa jσ (ω) f(ω) ), (5) ω σ where f(ω) is the Fermi-Dirac distribution function and α j = 4 Lj Rj /( Lj + Rj ) 2 is the left-right asymmetry factor for dot j, with j = Lj + Rj. A jσ (ω) denotes the spectral function of the j-th dot level for spin σ, A jσ (ω) = 1 π ImGR jσ(ω), with G R jσ(ω) the Fourier transform of the retarded Green s function, G R jσ(t) = iθ(t) {d jσ (t),d jσ (0)}. To improve the quality of the spectral functions and reduce the effects related with broadening of Dirac delta functions, we also used the z-averaging trick [6]. S2.2 Choosing NRG parameters Most of the parameters used in NRG calculations may be extracted from routine measurements of the two dots. To a good approximation, a small decrement in the dot level is proportional to a small increment in gate voltage. The proportionality constant, as well as the charging energies U, U 1, and U 2, are measured directly by routine bias spectroscopy. U may be extracted from the change in ɛ 1 of dot 1 s Coulomb blockade peak position as an electron is added to dot 2, or vice versa. U 1 and U 2 are determined from Coulomb blockade diamonds taken over a wider range of energy. Results of the conductance calculations around the LBTP are, however, largely insensitive to values of U 1 and U 2 as they are much greater than U. Therefore, in addition to U extracted from the stability diagram, we need to determine four parameters as precisely as possible to characterize conductance around the LBTP: the coupling strengths 1 and 2 (or linewidths) for dot 1 and 2 in the underlying Anderson model and the asymmetry parameters α 1 and α 2. 1 may be extracted by taking cuts away from the LBTP on a mixed valence peak of dot 1 (side of charge stability hexagon). There, for large intradot interactions U 1 and U 2, the FWHM of the conductance curve Γ 1, divided by T, must be a universal 10 NATURE PHYSICS

11 SUPPLEMENTARY INFORMATION function of 1 /T, and likewise for dot 2. We produced a first estimate for 1(2) by computing these universal curves with NRG, but managed to determine 1(2) only with an accuracy of about 5 10%, as noise and other effects (perhaps Fano interference at zero magnetic field, or neglected internal states of the dots, etc.) also affect the widths of the measured peaks. Although this accuracy seems to be very good, it is not sufficient: combined with the 5% accuracy of U it amounts in a 15% error in the ratios 1(2) /U, which can give rise to a factor of 1.5 difference in the Kondo temperature along the LBTP, the latter being exponentially sensitive to this ratio. We therefore fine-tuned these values of 1(2) further by applying a complex fitting procedure (for details, see Section S3), where we computed full two-dimensional conductance plots at a fixed temperature (T = 40 mk) and fixed 1(2) by analyzing various cuts of these. In this way, the value of the ratios 1(2) /U could be fixed with a1 3% accuracy. The asymmetry parameters α 1 and α 2 were selected such that the calculations reproduce the experimentally observed height of the mixed valence peaks of dot 1 and 2, as well as the temperature dependent conductance in other regions of parameter space. We estimate the accuracy of these parameters to be around 5%. Since we do not use any broadening procedure in the conductance calculations but calculate G(T ) directly from the spectral peaks, the only possible source of error in the NRG calculations is due to the finite number of kept states. We checked, however, that the number of kept states in our calculations was sufficient and changing it did not influence the accuracy of the computed conductance curves. The NRG conductance curves can thus be considered as numerically exact. For the 2D conductance plots, we kept N = 2500 states, and for the G(T ) traces we kept N = 8000 states. The spectral function calculations, however, contain an additional broadening parameter, which typically reduces the accuracy of the computations at high and intermediate frequencies. Therefore, curves presented in Figs. S11c and S11e are less accurate and should not be considered as quantitative regarding the precise width and shape of the predicted (and observed) high-energy features. In Fig. 4, most of the parameters used for the spectral function calculation were unchanged from those used in NRG calculations earlier in the paper. However, in the calculation we set α 1 = α 2 = 1 for simplicity, as it would only contribute a scale factor otherwise. For each value of experimental E PZ, the values of ɛ 1,ɛ 2 used in the calculation are shown in Table S3. NATURE PHYSICS 11

12 E PZ (mev) ɛ 1 (mev) ɛ 2 (mev) Table S3: Parameters ɛ 1 and ɛ 2 used for each value of experimental E PZ in Fig. 4. For the data of Fig. 4, the precise values of 1 and 2 were not determined, as the tuning of the device was different from when the data of Figs. 2 and 3 were taken. Nonetheless, the values should be similar and the spectral functions describe the data remarkably well. S3 Extracting LBTP cuts from 2D data sets The zero-detuning cuts presented in Fig. 2c and 2d were extracted numerically from 2D data sets. The cuts are highly sensitive to cut direction such that adjusting the endpoints by even a few µev can result in significantly different conductances along the cut. With experimental data alone, this poses a significant problem, since the line of zero detuning cannot be exactly identified. Moreover, it is difficult to control for shifts of the LBTP unrelated to renormalization as the temperature is varied. Physically meaningful shifts of the mixed-valence peaks with temperature are to be expected, but undesirable shifts, predominantly from random charge transitions in the donor layer of the heterostructure, may also contribute. To address these concerns, for fixed NRG parameters we compare the 2D experimental data sets to the 2D NRG calculations, at each measured temperature. The pseudospin-resolved conductances from the experimental data and from NRG were fit to Lorentzians to find the peak positions. The experimental data were then offset such that the peak positions matched those in the NRG data. Some manual shifts of 05 mev or less were used following the fitting procedure to provide best agreement along the LBTP cuts. Note that the scale factor between gate voltage and energy is experimentally determined, and only the offsets of the axes are adjusted. 12 NATURE PHYSICS

13 SUPPLEMENTARY INFORMATION S4 G(ɛ 1,ɛ 2 ) for all measured T Fig. 2a and 2b show respectively the measured and calculated conductance G = G 1 +G 2 as a function of ɛ 1 and ɛ 2 at T = 40 mk. We reiterate that the scale factor converting gate voltage to energy is experimentally determined, with an offset determined for each axis by comparing with NRG calculations. The validity of this assumption using NRG to determine offsets for ɛ 1 and ɛ 2 is borne out by the impressive agreement throughout the 2D conductance maps for T 40 mk. Fig. S4 shows experimental measurements and NRG calculations of G(ɛ 1,ɛ 2,T). Cuts through each plot for both fixed ɛ 1 and fixed ɛ 2 are shown in Figs. S5 (ɛ 1,ɛ 2 = 0 mev), S6 (ɛ 1,ɛ 2 = -5 mev), and S7 (ɛ 1,ɛ 2 = -0.1 mev). Apart from the 22 mk and 30 mk data sets, we find agreement between theory and experiment over a wide range of gate voltages and temperatures. NATURE PHYSICS 13

14 22 mk 55 mk 99 mk ε2 (mev) -ε2 (mev) -ε2 (mev) mk 65 mk 112 mk ε2 (mev) -ε2 (mev) -ε2 (mev) mk 77 mk 147 mk ε2 (mev) ε2 (mev) mk 87 mk T (mk) ε2 (mev) ε2 (mev) -ε2 (mev) % % Figure S4: Conductance G = G1+G2 through the dots as a function of temperature, ɛ1, and ɛ2. For each temperature, experimental conductance (left) is paired with NRG-calculated conductance (right). Each pair shares an individually-set color scale. The color scales all range from 0 e 2 /h to the value indicated in the table. 14 NATURE PHYSICS

15 SUPPLEMENTARY INFORMATION 22 mk 55 mk 99 mk 30 mk 65 mk 112 mk 40 mk 77 mk 147 mk 87 mk 49 mk Figure S5: Cuts through the experimental and calculated G of Fig. S4. Experimental data are denoted by red crosses, and NRG calculations by solid black lines. Each pair corresponds to a temperature stated within the plot. In the left (right) plot of each pair, ɛ 2(1) = 0 mev. 15 NATURE PHYSICS 15

16 mk 55 mk 99 mk 30 mk 65 mk 112 mk 40 mk 77 mk 147 mk 87 mk 49 mk Figure S6: Cuts through the experimental and calculated G of Fig. S4. Experimental data are denoted by red crosses, and NRG calculations by solid black lines. Each pair corresponds to a temperature stated within the plot. In the left (right) plot of each pair, ɛ 2(1) =5 mev NATURE PHYSICS

17 SUPPLEMENTARY INFORMATION mk 55 mk 99 mk mk 65 mk 112 mk mk 77 mk 147 mk mk 49 mk Figure S7: Cuts through the experimental and calculated G of Fig. S4. Experimental data are denoted by red crosses, and NRG calculations by solid black lines. Each pair corresponds to a temperature stated within the plot. In the left (right) plot of each pair, ɛ 2(1) =0.1 mev. 17 NATURE PHYSICS 17

18 S5 Temperature dependence details As stated in the main text, the point ɛ = -3 mev was chosen for the temperature dependence because it is a point where T K is large compared to experimentally accessible temperatures. However, apart from the saturation observed at T = 40 mk that prevents observation of the low-t rollover, the experimental data are consistent with both SU(4) universal scaling and NRG calculations for our device configuration at other points along the LBTP. In Figs. S8 and S9 we show the temperature dependence at ɛ = -4 mev and ɛ = -5 mev, respectively. Uncertainties in the experimental conductances of Fig. 3 are likely dominated by the uncertainty in maintaining constant ɛ 1 and ɛ 2 between data taken at different temperatures, rather than conductance noise. We extract the conductances from the 2D maps of Figs. 2a and 2b and similar maps at other temperatures. The offsets (but not the scale) of the ɛ 1 and ɛ 2 experimental axes of Figs. 2a and 2b are set using the theoretical calculations, and this considerably reduces this uncertainty. After this alignment procedure, the remaining uncertainty in ɛ 1 and ɛ 2 may be conservatively taken as the pixel spacing of ɛ 1 and ɛ 2 in our 2D conductance maps, approximately 03 mev. In determining error bars, experimental points in the 2D conductance map neighboring ɛ 1 = ɛ 2 = 3 mev are considered to be independent measurements of the conductance at ɛ 1 = ɛ 2 = 3 mev, with a Gaussian weight: w i = exp[ ((ɛ 1 ( 3)) 2 +(ɛ 2 ( 3)) 2 )/σ 2 ], where σ =03 mev. The error bars then reflect the standard deviation of the weighted mean, and are largest at low temperatures where the conductance varies the most rapidly in any direction in ɛ 1 and ɛ 2. The (unbiased) standard deviation of the weighted mean, s, is given by: s 2 = V 1 V 2 1 V 2 Σ N i=1w i (x i µ ) 2 (6) where µ is the weighted mean, V 1 =Σ N i=1w i, and V 2 =Σ N i=1w 2 i. S6 Empirical Kondo forms The empirical Kondo form was introduced by D. Goldhaber-Gordon, et al. [7] and provides a convenient approximation of conductance through a quantum dot in the NATURE PHYSICS

19 SUPPLEMENTARY INFORMATION G(T=T K ) Experimental data NRG calculations of device SU(4) Anderson model SU(2) Anderson model SU(2) Anderson model } T K = 152 mk T K = 182 mk Temperature (mk) Figure S8: Experimental data for the temperature dependence of the conductance (circles) at ɛ 1 = ɛ 2 = 4 mev in Fig. 2d. Experimental data are compared with NRG calculations of the device (solid black line), as well as with the SU(4) and SU(2) Anderson models in the Kondo regime (N e = 1). The blue dash-dotted SU(4) scaling curve and the green dotted SU(2) scaling curve have T K = 152 mk fixed to that identified in the device calculations, where G(T = T K ) = G(T = 0)/2. The red dashed SU(2) scaling curve is for a best-fit T K = 182 mk. Parameters for the NRG computations were: B = 0, U 1 = mev, U 2 =1.5 mev, U = 0.1 mev, 1 = 17 mev, 2 = 148 mev, α 1 = α 2 = 75. These are the same used in Fig. 3. NATURE PHYSICS 19

20 G(T=T K ) Experimental data NRG calculations of device SU(4) Anderson model SU(2) Anderson model SU(2) Anderson model } T K = 113 mk T K = 121 mk Temperature (mk) Figure S9: Experimental data for the temperature dependence of the conductance (circles) at ɛ 1 = ɛ 2 = 5 mev in Fig. 2d. Experimental data are compared with NRG calculations of the device (solid black line), as well as with the SU(4) and SU(2) Anderson models in the Kondo regime (N e = 1). The blue dash-dotted SU(4) scaling curve and the green dotted SU(2) scaling curve have T K = 113 mk fixed to that identified in the device calculations, where G(T = T K ) = G(T = 0)/2. The red dashed SU(2) scaling curve is for a best-fit T K = 121 mk. Parameters for the NRG computations were the same as in Fig. S8. 20 NATURE PHYSICS

21 SUPPLEMENTARY INFORMATION SU(2) crossover regime as a function of temperature: ( ) n ) s T G(T )=G 0 (1+(2 1/s 1) T K, (7) where s =2, n = 2, G 0 is the conductance attained at zero temperature, and T K is the Kondo temperature. This form is purely phenomenological and was invented to describe succinctly the numerically-calculated spin-1/2 SU(2) universal scaling [8]. With such a formula it is convenient to estimate T K from experimental results using nonlinear regression, however care must be taken in its application. Importantly, for s =2 and n = 2 this formula does not describe the universal SU(4) scaling. Various papers have nonetheless used the empirical SU(2) form (7) to fit data for which the applicability is not clear. In the absence of an alternative, this is a reasonable heuristic since the differences between the SU(4) and SU(2) scaling are subtle, but this procedure is not strictly justified. In particular, the leading-order temperature dependence of (7) is quadratic by design at T T K in order to describe SU(2) Kondo scaling, but conformal field theory predicts the SU(4) Kondo state to have a leading-order cubic temperature dependence at T T K, despite retaining a Fermi liquid character (normally associated with quadratic dependence) [9]. Therefore, both parameters s and n must be changed to expect a nice agreement for T T K, where the empirical form is designed to apply. Fig. S10 shows how s =2, n = 2 describes SU(2) universal scaling in the crossover regime. Changing s alone is seen to be insufficient to describe the SU(4) universal scaling especially for temperatures T <T K, where the fitting is most sensitive. However, a good fit to the SU(4) universal scaling may be obtained with s =0, n = 3. We must emphasize that although (7) provides an accurate fitting in the full crossover region, it fails at temperatures T T K, where it does not reproduce the well-known logarithmic behavior characteristic of the Kondo problem. From our experiences with analyzing the experimental data in this paper, empirical forms must be used with great care and supported by other methods. A blind application to our data would yield spurious conclusions, owing to the saturation at T = 40 mk. Also, as can be seen from the NRG results for our device, there are some expected deviations from the universal scaling, particularly at T >T K, where the empirical forms become less accurate. NATURE PHYSICS 21

22 G/G 0 SU(2) universal scaling SU(4) universal scaling s = 2, n = 2 s = 0.33, n = 2 s = 0, n = T/T K Figure S10: Universal SU(2) (red) and 1/4-filling SU(4) (blue) scaling curves for the conductance as a function of temperature. T KSU(2) and T KSU(4) are both defined such that G/G 0 =0.5. Also shown are empirical fits in the form of (7): s =2, n =2 describes SU(2) (black dotted); s =0.33, n = 2 best approximates the SU(4) form without changing n (solid black); s =0, n = 3 provides a good approximation of the SU(4) form. 22 NATURE PHYSICS

23 SUPPLEMENTARY INFORMATION S7 Bias spectroscopy S7.1 N e =3LBTP, zero magnetic field Fig. S11 shows the orbital state-resolved bias spectroscopy and calculated spectral functions for the N e = 3 LBTP of Fig. 4, but with zero magnetic field (Fig. S11a). As in Fig. 4, dot 2 only exhibits hole-like processes, but since E Z = 0, there is only a single peak expected at ω = E PZ. The NRG calculations (Fig. S11e) corroborate the naive expectation, and the experimental data (Fig. S11d) show a peak evolving into a shoulder that tracks with E PZ. In dot 1, an electron-like process may be expected at ω = E PZ, but owing to the unpaired electron, a peak should also appear at ω = 0. This is clearly seen in the NRG calculations (Fig. S11c), but not evident in the experimental data (Fig. S11b). Rather, a peak near ω = 0 for E PZ = 0 appears to move towards positive ω as E PZ increases. Because the shift is small, the increasing spectral weight at ω>0 together with limitations in measurement resolution may explain this observation. The peak near ω = 30 µev is unexpected but may be due to a low-lying excited state. When E PZ = 0, the width of the peak near ω = 0 should change noticeably as a magnetic field is applied, reflecting an SU(4) to SU(2) crossover with lower T K when the four-fold degeneracy is broken. Experimentally, making such a comparison with confidence is challenging. Maintaining a particular tuning long enough to perform bias spectroscopy with compensation for capacitances (see section S8.5) at both zero magnetic field and finite magnetic field places extreme demands on the stability of the device being measured. In our experiment, we needed to adjust the gate voltages controlling the dot-lead tunnel barriers in between measuring the zero and finite magnetic field data, and thus a direct comparison between Fig. 4 and Fig. S11 is invalid. Additionally, supposing the device were stable indefinitely, there could in principle be an uncontrolled orbital effect from an in-plane magnetic field, owing to the finite thickness of the 2DEG. This would change and therefore T K, regardless of the degeneracy being broken. NATURE PHYSICS 23

24 b 5 4 d 5 4 E PZ E PZ = 0 µev 14 µev 20 µev 30 µev a 1 G 1 (e 2 /h) 3 2 G 2 (e 2 /h) E PZ c ev SD (1,2) (µev) e ev SD (1,2) (µev) E PZ 2 1 A 1 2 A (µev) (µev) Figure S11: (a) Inelastic transitions between pseudo-zeeman-split states of the double dot at an N e = 3 LBTP, in zero magnetic field. An electron of either spin can occupy dot 1. (b) Experimental conductance G 1 for dot 1. The four traces correspond to different values of E PZ, with E PZ > 0 meaning dot 1 is favored to hold the unpaired electron. (c) Calculated spectral function A 1 for dot 1. (d) Experimental conductance G 2 for dot 2. (e) Calculated spectral function A 2 for dot 2. For all panels, Γ 1, Γ 2 4 mev. Γ 1S and Γ 2S were both tuned to be 2 3% of Γ 1D and Γ 2D, respectively, such that the biased leads probe the equilibrium local density of states on their respective dot. The bias is applied to both dots simultaneously. The parameters used for the calculations were T = 40 mk, B = 1 T, U 1 = mev, U 2 =1.5 mev, U =0.1 mev, 1 =17 mev, 2 =148 mev. Note that α 1 = α 2 = 1 serve only as normalization factors in the calculation. The ɛ 1,ɛ 2 used are in Table S4. 24 NATURE PHYSICS

25 SUPPLEMENTARY INFORMATION E PZ (mev) ɛ 1 (mev) ɛ 2 (mev) Table S4: Parameters ɛ 1 and ɛ 2 used for each value of experimental E PZ in Fig. S11. S7.2 N e =1LBTP, T field Fig. S12 shows the orbital state-resolved bias spectroscopy and calculated spectral functions at an N e = 1 LBTP, in a T Zeeman field. By considering the cartoon of Fig. S12a, and identifying each electron-like process with a corresponding hole-like process in Fig. 4a, the relationship between the N e = 1 LBTP and N e = 3 LBTP becomes clearer. We again consider ω and ev SD as equivalent. In dot 2, all of the expected features are observed (Fig. S12d): a weak peak at ω = E Z, a peak (threshold) that tracks with E PZ for E PZ <E Z, and a purely orbital Kondo peak at ω = 0 for E PZ = 0. The overall shapes of the curves are in rough qualitative agreement with the spectral functions in Fig. S12e, although the relative peak heights may differ. However, in dot 1 (Fig. S12b), the purely orbital Kondo peak at ω = 0 for E PZ =0 is obscured by poorly understood background conductance at positive ω. Additionally, an unexpected feature is observed at ω = 30 µv that does not track with E PZ. It is tempting to suggest that the LBTP being measured is actually a (1,1)/(2,0) LBTP. In this interpretation, both dots could hold an unpaired electron, and both dots should exhibit a peak at ω = ±E Z. In other words, the spectral functions for both dots should look similar to Fig. S12e, with ω ω for dot 1. However, the increasing conductance at positive ω in Fig. S12b is in qualitative agreement with Fig. S12c, and would not be expected in this alternate explanation. Additionally, our ability to maintain electron occupation number assignments is supported by Fig. S1. Therefore, the unexpected feature is instead likely associated with a low-lying excited state. NATURE PHYSICS 25

26 b E PZ d E Z -E PZ E Z a 1 G 1 (e 2 /h) G 2 (e 2 /h) E PZ E Z E PZ 1 2 E Z c 1 A ev SD (1,2) (µev) E PZ = 0 µev 12 µev 18 µev 26 µev 36 µev E PZ e 2 A ev SD (1,2) (µev) -E Z -E PZ E Z E Z (µev) (µev) Figure S12: (a) Inelastic transitions between Zeeman-split states of dot 1 and dot 2 at an N e = 1 LBTP. (b) Experimental conductance G 1 for dot 1 in a T Zeeman field. The five traces correspond to different values of E PZ, with E PZ > 0 meaning dot 2 is favored to hold the unpaired electron. (c) Calculated spectral function A 1 for dot 1. (d) Experimental conductance G 2 for dot 2. (e) Calculated spectral function A 2 for dot 2. For all panels, Γ 1, Γ 2 4 mev. Γ 1S and Γ 2S were both tuned to be 2 3% of Γ 1D and Γ 2D, respectively, such that the biased leads probe the equilibrium local density of states on their respective dot. The bias is applied to both dots simultaneously. The parameters used for the calculations were T = 40 mk, B = 1 T, U 1 = mev, U 2 = 1.5 mev, U =0.1 mev, 1 =17 mev, 2 =148 mev. Note that α 1 = α 2 =1 serve only as normalization factors in the calculation. The ɛ 1,ɛ 2 used are in Table S5. 26 NATURE PHYSICS

27 SUPPLEMENTARY INFORMATION E PZ (mev) ɛ 1 (mev) ɛ 2 (mev) Table S5: Parameters ɛ 1 and ɛ 2 used for each value of experimental E PZ in Fig. S12. S8 Technical details S8.1 Electronics For the data taken in Fig. 1b and 4 of the paper, custom current amplifiers designed by Y. Chung of Pusan National University (early version of that which is presented in [10]) were used in place of commercial Ithaco / DL Instruments 1211 current amplifiers, which have been previously employed in our measurement setup [11]. The custom amplifiers are crucial to this experiment in that the input offset voltage of the current amplifiers must remain stable over a period of days to avoid applying an uncontrolled source-drain bias across the dot. Over a continuous interval of 2.8 days, the standard deviation of the input offset voltage was measured to be µv for the amplifier attached to dot 1, and µv for the amplifier attached to dot 2. The amplifiers were characterized in the same locations where they were used for measurement, as no active temperature control of the amplifiers was performed during measurement or characterization. S8.2 Electron temperature calibration Our electron temperature was calibrated by Coulomb blockade peak thermometry, using the same device and during the cool-down when the data of Figs. 2 and 3 were measured. Only a single dot was formed and measured during the electron temperature calibration. In Fig. S13, we show a temperature-limited Coulomb blockade peak. Experimental data (black circles) are compared with a theoretical lineshape (solid blue line) describing the limit E k B T Γ, where E is the level spacing [12]. The NATURE PHYSICS 27

28 Measured Temperature limited Including finite Γ V SD (µv) V V Figure S13: Left: Temperature-limited Coulomb blockade peak (black circles). The solid blue line is a fit to (8), and the dashed red line is a fit to a convolution of that and a narrow Lorentzian lineshape. Right: Bias spectroscopy on this peak gives a clean Coulomb blockade diamond, where the slopes may be confidently extracted to determine α g. lineshape is given by: G(V g )=y 0 + G 0 4k B T 1 cosh 2 ( αgev g 2k B T ) (8) where α g determines the conversion between gate voltage and energy, e is the electron charge, V g is the gate voltage away from resonance, G 0 is a temperatureindependent prefactor, and y 0 allows for a small offset due to instrumentation. We extract and fix α g =572 from the slopes of lines seen in bias spectroscopy. The temperature extracted from the fit is insensitive to y 0. Additionally, we fit to a convolution of the inverse-cosh-squared lineshape and a Lorentzian, to describe both finite T and Γ (dashed red line). Agreement is seen with both lineshapes. Neglecting Γ we find T e = 23 mk, or T e = 20 mk with Γ = 1.1 µev. The uncertainty in temperature is small, and becomes smaller at higher temperatures. A temperature dependence was performed on this peak by heating the mixing cham- 28 NATURE PHYSICS

29 SUPPLEMENTARY INFORMATION ber of our dilution refrigerator, and the electron temperatures extracted from the fits as a function of temperature were recorded. By comparing with a ruthenium-oxide resistance thermometer on our measurement probe, we establish a correspondence between the resistance reported on the resistance thermometer and the electron temperature. S8.3 Magnetic field calibration 0.33 B z (T) R (Ω) B y (T) Figure S14: Four-wire resistance as a function of the y-axis (in-plane) and z-axis (perpendicular) magnetic fields. The slopes of the solid white and dashed white lines are m = 206 and m = 203, respectively. This corresponds to a misalignment between the y-axis field and the plane of the sample. Because of small but uncontrolled sample tilt with respect to axes defined by the two-axis magnet in our experimental dewar, energizing only the in-plane coil will give rise to a perpendicular component as seen by the sample, and vice versa. To apply a magnetic field precisely in the plane of the sample, as is done in Fig. 4, we calibrate in situ using a four-wire current-biased measurement of Shubnikov-de Haas oscillations in resistance, as a function of both the nominally perpendicular field B z and nominally in-plane magnetic field B y. Fig. S14 shows the Shubnikov-de Haas oscillations observed near a perpendicular magnetic field of 0.3 T, and how they track with an added in-plane field. The geometry of the 2DEG mesa is not well defined, so both even and odd components of magnetoresistance contribute to the measured resistance. The observed stripes correspond to a NATURE PHYSICS 29

30 Magnetic field (T) Splitting (µev) g Table S6: Approximate spin state splittings and corresponding g-factors as a function of magnetic field. constant perpendicular field. The slope of the stripes gives a compensation factor such that any perpendicular component introduced by the in-plane magnetic field may be cancelled out by application of an added perpendicular field to within a few percent. Even an applied field in the plane of the sample will subtly modify orbital states because of the finite extent of the electronic wavefunctions normal to the plane, an effect we neglect in our analysis. S8.4 g-factor calibration The Zeeman energy E Z is related to the magnetic field B by E Z g µ B B, where µ B is the Bohr magneton and g is the g-factor. Among GaAs/AlGaAs heterostructures, the g-factor can vary considerably, and so we calibrate in situ for our device by looking for a Zeeman splitting in the bias spectroscopy as we vary an in-plane magnetic field. Fig. S15 displays conductance through dot 2, demonstrating the Zeeman splitting. A splitting is seen to emerge by B =T, though the exact splitting is not resolved owing to the width of the level. As the field is increased, we can extract the splitting by reading off the value of V SD(2) above which the source-drain voltage drop is large enough to allow for inelastic spin flip scattering processes. From this value, any offset for true zero bias is then subtracted (usually a few µv or less). Table S6 summarizes the extracted splittings and corresponding g-factors. We find g consistent with that of bare GaAs, g =4, and take this value in calculating E Z for given B. 30 NATURE PHYSICS

31 SUPPLEMENTARY INFORMATION 300 B = T 300 B = 2.0 T V SD (2) (µv) G 2 (e 2 /h) V SD (2) (µv) V V 300 B = 3.0 T 300 B = 4.0 T V SD (2) (µv) V SD (2) (µv) V V Figure S15: Conductance G 2 as a function of source-drain bias V SD (2) across dot 2 and gate voltage V P 2, at in-plane magnetic fields of B = T (top-left), B = 2.0 T (top-right), B = 3.0 T (bottom-left), and B = 4.0 T (bottom-right). The color scale is fixed for all four values of magnetic field, which are labeled in the upper-left of each plot. Blue solid lines correspond to the alignment of the source lead Fermi energy with the ground state, and blue dotted lines correspond to alignment of the drain lead Fermi energy with the ground state. White arrows denote where V SD (2) is read off to extract the splitting. NATURE PHYSICS 31

32 S8.5 Bias spectroscopy To apply and maintain a particular E PZ while changing the applied source-drain biases V SD1(2) across dot 1 (2) requires some care. Gates P1 and P2 as well as leads S1 and S2 all have capacitances to both dot 1 and dot 2. These capacitances must all be characterized every time the W gates or magnetic field are changed. Once the capacitances are known, electrostatic gating of the dots by the biased source leads may be compensated by changes in V P 1 and V P 2. Further details have been published previously [13]. 32 NATURE PHYSICS

33 SUPPLEMENTARY INFORMATION References [1] Wilson, K. G. The renormalization group: Critical phenomena and the Kondo problem. Rev. Mod. Phys. 47, (1975). [2] Bulla, R., Costi, T. A. & Pruschke, T. Numerical renormalization group method for quantum impurity systems. Rev. Mod. Phys. 80, (2008). [3] Weichselbaum, A. & von Delft, J. Sum-Rule Conserving Spectral Functions from the Numerical Renormalization Group. Phys. Rev. Lett. 99, (2007). [4] Tóth, A. I., Moca, C. P., Legeza, Ö. & Zaránd, G. Density matrix numerical renormalization group for non-abelian symmetries. Phys. Rev. B 78, (2008). [5] We used an open-access Budapest NRG code, Legeza, O., Moca, C. P., Tóth, A. I., Weymann, I. & Zaránd, G. arxiv: (2008) (unpublished). [6] Oliveira, W. C. & Oliveira, L. N. Generalized numerical renormalization-group method to calculate the thermodynamical properties of impurities in metals. Phys. Rev. B 49, (1994). [7] Goldhaber-Gordon, D. et al. From the Kondo Regime to the Mixed-Valence Regime in a Single-Electron Transistor. Phys. Rev. Lett. 81, (1998). [8] Costi, T. A., Hewson, A.C. & Zlatic, V. Transport coefficients of the Anderson model via the numerical renormalization group. J. Phys: Cond. Matt. 6, (1994). [9] Le Hur, K., Simon, P. & Loss, D. Transport through a quantum dot with SU(4) Kondo entanglement. Phys. Rev. B 75, (2007). [10] Kretinin, A. V. & Chung, Y. Wide-band current preamplifier for conductance measurements with large input capacitance. Rev. Sci. Instrum. 83, (2012). [11] Potok, Ron M. Probing many body effects in semiconductor nanostructures. Ph. D. dissertation. Dept. of Physics, Harvard University (2006). NATURE PHYSICS 33

34 [12] Beenakker, C. W. J. Theory of Coulomb-blockade oscillations in the conductance of a quantum dot. Phys. Rev. B 44 (4), (1991). [13] Amasha, S. et al. Pseudospin-Resolved Transport Spectroscopy of the Kondo Effect in a Double Quantum Dot. Phys. Rev. Lett. 110, (2013). 34 NATURE PHYSICS