Single-molecule quantum dot as a Kondo simulator

Structural flexibility of molecule-based systems is key to realizing the novel functionalities. Tuning the structure in the atomic scale enables us to manipulate the quantum state in the molecule-based system. Here we present the reversible Hamiltonian manipulation in a single-molecule quantum dot consisting of an iron phthalocyanine molecule attached to an Au electrode and a scanning tunnelling microscope tip. We precisely controlled the position of Fe2+ ion in the molecular cage by using the tip, and tuned the Kondo coupling between the molecular spins and the Au electrode. Then, we realized the crossover between the strong-coupling Kondo regime and the weak-coupling regime governed by spin–orbit interaction in the molecule. The results open an avenue to simulate low-energy quantum many-body physics and quantum phase transition through the molecular flexibility.

Supplementary Figure 3 shows the differential charge distribution calculated for the P2 configuration. The charges distribute not only in the interface region between FePc and Au(111) but also in the region between the Fe 2+ ion and the apex atom of tip. The differential charge distribution ∆ ( ) is defined as ∆ = N − b%c − de *** − =f? , (3) where N is the charge distribution of a molecular quantum dot consisting of FePc on Au(111) and the STM tip, b%c represents that of FePc, and de *** and =f? represent those of the Au(111) substrate and the STM tip, respectively. First, the total geometric structure of the molecular quantum dot is optimized. Then, N , b%c , de *** and =f? are calculated by using the optimized structure parameters. Finally, the differential charge distribution is calculated by using Supplementary Eq.

Supplementary Note 4 | Two-orbital and two-channel Kondo model
In this section, we describe the two-orbital and two-channel Kondo model discussed in the main text. The present DFT calculations show two localized spins, one in the W 0 orbital and the other in the degenerate g orbitals, in the electronic ground state of FePc on Au(111). These localized spins interact with the surface electrons and form the Kondo resonance states, respectively, when the tip is far away from the molecule. As the tip approaches to the molecule, the spectrum shows the crossover from the FK resonances to the inelastic step structure. This situation is well described by the following Hamiltonian, . The second term describes the s-d Kondo exchange couplings (denoted as * and 3 ) of the two localized spins (denoted as * and 3 ) with the surface electrons. * and 3 are spin in the W 0 and g orbitals, respectively. The degeneracy in the g orbitals is neglected for simplicity. = 1 and = 2 specify the W 0 (screening channel 1) and g (screening channel 2) orbitals, respectively. The first term of * represents the Hund's coupling between the two spins. We set w at -0.8 eV. The second term describes the SOI-splitting. We assume a uniaxial MA term of Therefore, we calculated the spectral evolution by treating * and 3 as model parameters with fixing w and , and qualitatively compared the calculated and the measured spectra.
The decrease of * and 3 during tip approaching discussed above can be confirmed by the DFT calculation qualitatively. Supplementary Figure 4 shows the PDOS spectra of W 0 orbital calculated for FePc on Au(111) as a function of the molecular configuration (P0, P2 and P3 in Fig. 3a). In order to extract the variation in the hybridization of the d orbital with the surface and remove the hybridization with the tip, we calculated the PDOS spectra by using the model structure in which the structure of FePc on Au(111) is fixed in the same as Fig. 3a without the tip. In the majority-spin PDOS of P0, multiple peaks extend from -5 to -1 eV as a result of the hybridization with the substrate electronic states ( Supplementary Fig. 4a). These peaks merge to a main peak around -2 eV from P0 to P2 and P3. Overall spectral width obviously becomes narrower from P0 to P3. Similar spectral variation is observed in the minority-spin PDOS as shown in Supplementary Fig. 4b.

Supplementary Note 5 | Evaluation of and from the DFT calculations
We evaluate the variation of * and 3 with the movement of Fe atom. The evaluation was carried out by fitting the PDOS spectra of Fe 3 W 0 orbital in the P0-P3 configurations shown in Supplementary Fig. 4a with an Anderson-type model Hamiltonian 3,4 . The results of spectral fitting are shown in Supplementary   Supplementary Table 2. The fitting procedure is described below.
In FePc on Au(111), the Fe 3 W 0 orbital couples the conduction electrons of both wide sp-band and narrow d-band of Au(111). Therefore, we used the following Anderson-type model Hamiltonian: Here, ƒ" l n ( † l n ) is the creation operator of conduction electron with spin of the sp-(d-) band, ‡l n is that of 3 W 0 orbital with its energy ‡ , ‡l is the number operator of the 3 W 0 orbital, and is the onsite Coulomb interaction in the 3 W 0 orbital. Within the Hartree-Fock approximation, the Green's function of the Fe 3 W 0 orbital can be written as where ‡ = ‡ + < ‡l > and Σ ƒ" and Σ † ( ) are the self-energies caused by the hybridization with sp-and d-bands, respectively. The self-energy is described as The subscript specifies sp-or d-bands. The local density of states at the Fe atom can be obtained by ρ #… = − * g ‡l ( ) . In the spectral fitting, we approximated that the hybridization ‡ˆ is independent of the electron wavevector . We also adopted the approximations in the estimation of self-energies as follows: For the sp-band, we took the wide-band-limit and approximated as Here is the hopping energy between the nearest-neighboring orbitals and defines the band center.
We focused on the spectral feature around -2 eV in each PDOS spectrum calculated for the configurations of P0, P2 and P3, and fitted the spectrum by using Supplementary Eqs.  should vary with the movement of Fe 2+ ion. As the ion moves away from the Au(111) surface, Δ ƒ" is reduced by 43% in the P2 configuration, and 55 % in the P3 configuration compared to that in the P0 configuration. In the NRG calculations of spectral evolution shown in Fig. 4, we used 40%-and 60%-reduced values for * and 3 in medium-coupling regime 2 (P2) and weak coupling regime (P3), respectively. The values for * and 3 in the strong-coupling regime (P0) were taken to reproduce the widths of the Kondo peaks in the high-and low-U channels.

Supplementary Note 6 | NRG calculations of tunneling spectra
In order to explain the evolution of the tunneling spectrum with approaching the STM tip to the Fe 2+ ion, we calculated the differential conductance (d d ) spectrum. We take into account both elastic and inelastic electron tunneling processes. The former is the normal tunneling and the latter the tunneling with inelastic spin excitation. The elastic process consists mainly of the tunneling through the Kondo resonance appearing as a sharp peak in the DOS spectrum. Thus, the elastic tunneling process provides the DOS spectrum modified with the Fano effect (hereafter denoted as FK resonance spectrum). In the inelastic process, a tunneling electron excites the spin-flip transition between the substates separated by the SOI-splitting renormalized with the Kondo effect. Thus, the d d spectrum is described as a linear combination of the FK resonance spectra arising from the channels 1 and 2 and the inelastic spin excitation spectrum. Previous theoretical works reveal that the ratio of inelastic ( f$>c ) to the elastic ( >c ) current is determined by the exchange coupling of the tunneling electron with the localized spin and the tunneling barrier [5][6][7][8] . Actually, the exchange coupling cannot be evaluated quantitatively, and thus the ratio is practically treated as a parameter. We calculate the elastic ( >c ) and inelastic ( f$>c ) currents independently in the following way and describe the total tunneling current as their linear combination.
The elastic contribution to the d d spectrum can be written as  Fig. 6b). Initially, a sharp peak emerges in the regime (i). The peak structure is the Kondo resonance arising from the localized electron in the g orbitals. In the regime (ii), the Kondo peak is drastically suppressed and a small gap opens at the Fermi level. This gap also appears as a result of the competition. The Kondo signature is furthermore suppressed from the regime (ii) to regimes (iii) and (iv).
The spectra in Fig. 4e are drastically different from those in Supplementary Fig. 6b due to the Fano effect.
The peak and gap structures in Supplementary Fig. 6b appear as dip and peak (hump) structures in Fig. 4e.
Comparing the decays of Kondo signatures in the channels 1 and 2, the decay in the channel 2 is much faster than that in the channel 1. Thus, we conclude that the spectral evolution observed in the STM experiments mainly comes from the variation of the FK resonance spectrum in the channel 1.
We set * = 3.0 and 3 = −0.3 for calculating the FK resonance spectra shown in Fig. 4. These values should change with approaching the STM tip to the Fe 2+ ion. However, we used the constant values because the variations of the FK resonance spectra (the DOS spectra) and the contribution from the inelastic excitation process are dominant.
Supplementary Figures 6c and 6d show the inelastic spin excitation spectra calculated in the four regimes by the NRG technique. Moving from the regime (i) to the regime (iv), the excitation energy increases, which correlates with the widening of the gap observed in the DOS spectra. As the Kondo couplings decrease with the approach of the tip to the ion, the contribution from the inelastic excitation spectrum enhances, and then the tunneling spectra especially in the regimes (iii) and (iv) look identical to the inelastic spectra. The response of the inelastic spin excitation spectrum to an external magnetic field was also calculated by using Supplementary Eqs. (10)- (12). Supplementary Figure 7 shows the inelastic spin excitation spectra calculated at 0 and 10 T perpendicular to the molecular plane. The spectrum at 10 T consists of the two steps because the step at 0 T splits due to the spin Zeeman effect. These spectra reasonably reproduce the experimental results shown in Fig. 2e, providing a strong support that the

Supplementary Note 7 | Comparison of Kondo and mixed valence models
The majority-and minority-spin PDOS spectra show that the occupation in the W 0 orbital exceeds 1 and deviates from the half filling configuration suitable to the Kondo Hamiltonian used in the previous. From this feature, one might think that the mixed valence model of the d 6 +d 7 configuration 11 better describes the electronic ground state of FePc on Au(111) rather than the Kondo model and that the asymmetric peak in the spectrum A in Fig. 2a is the d-resonance. In this section, we discuss the variation of the excitation spectrum caused by the deviation from the half filling, and then we demonstrate that the Kondo model is more reasonable than the mixed valence model.
In order to discuss the influence of the occupation deviation, we consider the following Hamiltonian: Supplementary Figure 9 shows the calculated excitation spectra. The spectra in Supplementary Figs. 9a, 9c and 9e correspond to the Kondo (or mixed valence) regime while the spectra in Supplementary Figs. 9b, 9d and 9f to the SOI-dominant regime. One sees a sharp peak at the Fermi level in the spectrum of channel 1 ( W 0 orbital) in Supplementary Fig. 9a. A similar peak appears in Supplementary Fig. 9c. These peaks originated from the Kondo effect. In contrast, the spectrum is drastically changed at †* = − as shown in Supplementary Fig. 9e; a broad asymmetric peak appears below the Fermi level. This broad peak originates from the charge fluctuation in the mixed valence configuration. The spectra of channel 2 ( g orbital) are hardly changed as shown in Supplementary Figs. 9a, 9c and 9e, and each sharp peak is the Kondo resonance state.
The spectra in the SOI-dominant regime show a dip structure around the Fermi level in the channel 1 as shown in Supplementary Figs. 9b, 9d and 9f. The dip arises from the energy gap between the W = 0 − 17 − and W = ±1 states formed by the SOI. Each spectral shape is the convolution of the Kondo peak (or charge fluctuation peak) and the dip. Comparing the spectral intensity outside the gap in Supplementary   Fig. 9a with the counterpart in Supplementary Fig. 9c, the intensity is reduced drastically. The same is true for Supplementary Figs. 9b and 9d. The reduction stems from the suppression of the Kondo effect by the SOI. The reduction of the Kondo peak is also observed more clearly in the channel 2 where the half filling is satisfied. Thus, the drastic reduction is characteristic response of the Kondo resonance state to the SOI in this model. In contrast, when comparing the spectra in Supplementary Figs. 9c and 9f, the peak associated with the charge fluctuation is not reduced in intensity in the SOI-dominant regime but instead it is enhanced. This is because the charge fluctuation does not compete with the SOI and the decrease of Δ * makes the peak sharper.
The spectral variation derived from the SOI discussed above enables us to solve the problem which model better describes the electronic ground state of FePc on Au(111), Kondo or mixed valence model.
We conclude that the Kondo model is more reasonable for the electronic ground state of FePc on Au(111).