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Quantum simulation of a Fermi–Hubbard model using a semiconductor quantum dot array

Abstract

Interacting fermions on a lattice can develop strong quantum correlations, which are the cause of the classical intractability of many exotic phases of matter1,2,3. Current efforts are directed towards the control of artificial quantum systems that can be made to emulate the underlying Fermi–Hubbard models4,5,6. Electrostatically confined conduction-band electrons define interacting quantum coherent spin and charge degrees of freedom that allow all-electrical initialization of low-entropy states and readily adhere to the Fermi–Hubbard Hamiltonian7,8,9,10,11,12,13,14,15,16,17. Until now, however, the substantial electrostatic disorder of the solid state has meant that only a few attempts at emulating Fermi–Hubbard physics on solid-state platforms have been made18,19. Here we show that for gate-defined quantum dots this disorder can be suppressed in a controlled manner. Using a semi-automated and scalable set of experimental tools, we homogeneously and independently set up the electron filling and nearest-neighbour tunnel coupling in a semiconductor quantum dot array so as to simulate a Fermi–Hubbard system. With this set-up, we realize a detailed characterization of the collective Coulomb blockade transition20, which is the finite-size analogue of the interaction-driven Mott metal-to-insulator transition1. As automation and device fabrication of semiconductor quantum dots continue to improve, the ideas presented here will enable the investigation of the physics of ever more complex many-body states using quantum dots.

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Figure 1: Gate-defined quantum dot array as a platform for quantum simulations of the Fermi–Hubbard model.
Figure 2: Hamiltonian engineering using a scalable toolbox of local control and measurements.
Figure 3: Collective Coulomb blockade physics in the Fermi–Hubbard phase space.

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Acknowledgements

We acknowledge discussions with M. Veldhorst, A. F. Otte, R. Sensarma and the members of the Vandersypen group, sample fabrication by F. R. Braakman, set-up preparation by T. A. Baart and experimental assistance from M. Ammerlaan, J. Haanstra, R. Roeleveld, R. Schouten and R. Vermeulen. This work was supported by the Netherlands Organization of Scientific Research (NWO) VICI programme, the European Commission via the integrated project SIQS, the Japan Society for the Promotion of Science (JSPS) Postdoctoral Fellowship for Research Abroad, LPS-MPO-CMTC and the Swiss National Science Foundation.

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Authors

Contributions

T.H., T.F., C.J.v.D. and L.J. performed the experiments and analysed the data, C.R. and W.W. grew the heterostructure, X.L. and S.D.S. performed the theoretical analyses with X.L. carrying out the numerical simulations, T.H., T.F., X.L., L.J., S.D.S. and L.M.K.V. contributed to the interpretation of the data, and T.H. wrote the manuscript (X.L. wrote part of the Methods), with comments from T.F., X.L., S.D.S. and L.M.K.V.

Corresponding author

Correspondence to L. M. K. Vandersypen.

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The authors declare no competing financial interests.

Additional information

Reviewer Information thanks S. Bose, S. Rogge, J. Salfi and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Figure 1 Gate-to-dot cross-talk.

a, Cross-talk measurement of gates P1 and B12 on the left dot detuning. (See Fig. 1 for details of the triple-quantum-dot array and associated gates.) The slope of the charge transition (fit in white) yields the relative effect (δB12P1 = −α11/α14) of the two gates on the single-particle energy offset ε1 of the leftmost dot. Note also the non-zero background in charge sensor response we find in experiments, which is due to a direct coupling between the swept gate voltages and the sensing dot conductivity. Text in brackets denotes electron filling. b, Charge stability diagram showing the anti-crossing (white) and polarization line (red) between the left and middle dot, yielding the relative effect α11 = α21 + [(δP2P1)(α22α12)] of the two plunger gates (P1 and P2) on their respective dots. Automated edge finding and fitting procedures are outlined in Methods. c, Measured matrix elements αij/α22 (top left of each panel) as a function of tunnel coupling, tavg. No visual distinction is made between the measured matrix elements at different electron filling. No error bars are shown, as the small uncertainty in the slope fits yields errors smaller than marker size. Note that α22/α22 = 1 is shown as a line and αii/α22 values as red-filled circles, indicating that they derive from the slopes of polarization lines, such as the one shown in red in b. Source data

Extended Data Figure 2 Simulated classical charge addition spectra.

ac, Simulated charge addition spectra (see Methods for details and nomenclature) for a triple quantum dot at zero tunnel coupling, U2 = 1.05U1 = 0.95U3 and up to two particles per dot, connected to a reservoir at μ = 0 and kBT = 0.02U (>10 times larger than for the experiments described in the main text), with Vij = 0 and δi = 0 (a) or with V12 = V12 = 2V13 = 0.2U and δi = 0 (b) or δ1 = δ3 = 0 and δ2 = U/15 (c). denotes total electron filling, the sum of local interactions. States are denoted by charge occupation (n1 n2 n3) and specific degeneracy points A–F are referred to in Methods. The relation between εi and μ′ specified in the boxes at lower left applies to the vertical line at zero (horizontal) detuning. Source data

Extended Data Figure 3 Miniband width and electron temperature.

a, Measured charge stability diagrams of the (222)–(333) miniband as a function of homogeneous filling (only P1 values are shown) and offset in the outer two dot energies by changing P1 and P3 in opposite directions, akin to the simulations of Extended Data Fig. 2c. For crosses and dashed line, see b. b, Similar measurement as a function of the offset in the middle dot energy, controlled by P2. The P1 values are somewhat different from those in a because these measurements were taken at slightly different tunnel coupling tunings. The white diamond and asterisk indicate (roughly) the position of the same degeneracy points as shown in Extended Data Fig. 2. c, Broadening of a charge addition line (blue) due to the finite temperature of the (rightmost) Fermi reservoir. A Fermi–Dirac fit of the transition is shown in red, which yields an effective reservoir temperature of 72(1) mK. Source data

Extended Data Figure 4 Determining lever arm and tunnel coupling.

a, Example of a photon assisted tunnelling (PAT) measurement, which at low tunnel couplings is the measurement method of choice for both lever arm and tunnel coupling. Plotted is the difference in charge sensor response between applying a microwave excitation (y axis) or not as a function of detuning (x axis). Dashed red line is a fit to the hybridized charge state spectrum of the double dot24. The energy difference between bonding and antibonding states yields the minimum in frequency (2tij, here for dots 1 and 2) and the slope away from the transition gives the lever arm between detuning voltages applied to the gates and single-particle energy difference change between the two dots. The need to generate AC excitations and transmit them without severe losses through coaxial cables in the dilution refrigerator set-up, however, limits the maximum tunnel frequency we can accurately determine with this method to roughly 20 GHz (83 μeV). b, Example of a polarization line width measurement (blue circles), with fit in red. As an alternative to PAT, one can determine the tunnel coupling by assessing the width of the polarization line37. The excess charge (say on the left dot) transition is broadened both by an effective electron temperature and by the tunnel coupling. Charge sensor response is however not a direct measurement of excess charge. Not only does there exist a finite cross-talk between the gate voltages and the charge sensor response that leads to a finite slope away from the transition, we also typically find a back-effect of the excess charge on the sensing dot, leading to a different slope on either side of the transition. We fit the data with the following equation, taking this back-effect into account to first order in excess charge: , where V(ε) is the charge sensor response as a function of the detuning ε = δ(εiεj) away from to the transition and V0, δV and are the background signal, sensitivity and gate–sensor coupling, respectively. Note that ε is a linear combination of the swept gate voltages, taking the relevant cross-capacitances and the lever arm into account. Excess charge on the left dot is described by , with and effective temperature kBTe ≈ 6.5 μeV (1.6 GHz). c, Excess charge as function of detuning for three different tunnel couplings, showing that this characterization method works up to tunnel couplings that are several times higher than those measurable by PAT. d, Comparison of PAT and polarization line width measurements. The data are well explained by assuming a constant lever arm α22 = 83(1) μeV per mV between gate P2 and the middle dot. Text in brackets denote relevant charge states; error bars are 1σ fit uncertainties. Source data

Extended Data Figure 5 Simulations of collective Coulomb blockade for the simplified Hubbard model.

a, Cartoon diagram of a triple-dot system, which is a simplified version of the model used to describe the experiments in the main text. Specifically, we have set a uniform tunnel coupling t and Hubbard U (not shown), while ignoring the inter-site Coulomb interaction term Vij. We describe two levels per dot with a level splitting Δ that separates the single-particle energies of the first and second orbital. Each energy level is doubly degenerate owing to the spin degrees of freedom. b, Peaks in the electron addition spectrum for the triple-dot system in a. It is known that the classical Coulomb blockade effect arises purely from the charging effects of the quantum dots. When electron tunnelling between quantum dots is allowed, however, quantum fluctuations compete with the classical charging effects and give rise to a rich phase diagram, which is known as collective Coulomb blockade20. The metal–insulator transition in such a system is best captured by the charge addition spectrum, which is precisely what we measure in the experiment (Fig. 3b). The numbers in b (at top and bottom of the panel) indicate the average electron numbers in the system when the chemical potential μ resides at the respective gap. Here we use Δ/U = 0.2, and kBT/U = 0.04 (>20 times larger than for the experiments described in the main text). cf, Line cuts for the addition spectrum in b at different values of t/U (shown at top left of each panel). As discussed in the main text, there will be three different regimes in this phase diagram: at weak tunnel couplings the quantum dot states split into minibands but the isolated Coulomb blockade of each individual dot is preserved; at intermediate tunnel couplings the Coulomb blockade of individual dots is lost, but the gap between minibands remains open; finally, in the large tunnel coupling limit the gap between minibands can become comparable to temperature, and the system will be in a metallic state. The same can be seen in these line cuts. At t = 0 we can see that there are four critical chemical potentials μ at which electrons can be added to the triple dot. For the present model, these four peaks occur at μ = 0, U, 2U + Δ, and 3U + Δ, respectively. Each peak is triply degenerate, as the energy cost of adding electrons to any of the three dots is identical. For non-zero but small tunnel couplings (d, e) each triply degenerate peak at t = 0 starts to split into a miniband, indicating the breakdown of Coulomb blockade in each dot. However, different minibands are still separated by gaps that arise from a collective origin, reminiscent of the energy gap in a Mott insulator. Finally, at sufficiently high tunnel couplings we find non-zero at the middle gap (f), indicating that Coulomb blockade is completely overwhelmed by temperature. Source data

Extended Data Figure 6 Characterizing model parameters.

a, Simulated charge stability diagram for a triple-dot system with parameters t = 0.006, U1 = 3.98, U2 = 3.48, U3 = 2.70, V12 = 0.41, V23 = 0.35, V13 = 0.11 (all energies in meV). As described in Methods, the eigenstates can be obtained exactly in the t = 0 limit, as the eigenstates of the triple-dot system can be represented simply by the charge states (n1 n2 n3). In this regime, one can show that on the ε2ε3 plane the border between the (111)/(112) region and the border between the (111)/(110) region are exactly separated by an energy of U3. Similarly, the border between the (111)/(121) region and the border between the (111)/(101) region are separated by an energy of U2. In the presence of a non-zero but small tunnel coupling as is the case here, we expect that such an estimate is still reasonable. Now that the tunnel coupling is non-zero, the ground state of the system is no longer an exact charge state (n1 n2 n3), but generally a superposition of different charge states. To retain a connection to the t = 0 limit, we keep labelling sections of the charge stability as (n1 n2 n3), but with the distinction in mind that (n1 n2 n3) no longer denotes the exact ground state, but instead the charge state with the largest weight in the actual ground state. As a check, we can determine the values of U2 and U3 from the simulated charge stability diagram using the method described above and find that U2 = 3.44 meV and U3 = 2.71 meV, respectively, which is reasonably close to the corresponding model parameters. Because the data in Fig. 2c are taken at t/U = 0.002, we can thus trust the extracted U. b, Charge stability diagram for a triple-dot system with parameters t = 0.17, U1 = 2.92, U2 = 2.39, U3 = 2.53, V12 = 0.55, V23 = 0.47, V13 = 0.27 (all energies in meV). We find that the structure of the charge stability diagram remains qualitatively the same as that in a, and if we again extract the values of U2 and U3 using the same method, we find that U2 = 2.48 meV and U3 = 2.56 meV, which still agrees reasonably well with the original model parameters. Granted, at sufficiently large t/U the structure of the charge stability diagram will change drastically, and the present method to extract model parameters is bound to fail. However, as we never enter those regimes, our fitting method serves the purpose of this experiment. c–e, Calibrated tunnel couplings (c) and measured inter-site Coulomb (d) and on-site Coulomb (e) terms at calibrated values of the average tunnel coupling, corresponding to the experimental parameter space plot shown in Fig. 3b. Plotting symbols are defined in each key; blue shows data from the first subband from 0 to 6 electrons, red shows data from the second subband from 6 to 12 electrons. Error bars are 1σ fit uncertainties. Source data

Extended Data Figure 7 Isolated versus collective Coulomb blockade in charge and transport.

a–d, Charge stability diagrams (a and c) and transport through the array (b and d), at low (a and b) and high (c and d) tunnel coupling. Panels a and c show charge stability diagram around the (333) regime in the low (a) and high (c) tunnel coupling regimes, using a combination of all seven gates (only P1 values are shown) that change the local fillings equally. To further investigate the distinct phases, we focus on the regime with around nine electrons in total, corresponding to half-filling of the second band, and look at both charge sensing and transport. In the localized phase (t/U < 0.02 in a), the charge stability diagram shows transition lines following three distinct, well-defined directions, corresponding to the filling of the separate lithographically defined dots. In the delocalized phase (t/U > 0.15 in c), this distinct nature is all but lost, highlighting the incipient formation of a large single dot. The same effect can also be seen in transport measurements, as we observe Coulomb diamond sizes as a function of filling. b, Transport through the array following the zero-detuning line of Fig. 2b as a function of applied bias (60% on leftmost and 40% on bottom right reservoir). In the (333) state, this applied bias has to overcome the local (strong) Coulomb repulsion in order for current to flow, similar to a Mott insulator whose Fermi energy resides inside the gap. Adjacent Coulomb diamonds correspond to a Fermi-level inside the miniband and are significantly smaller, allowing current to flow at much smaller bias voltages. d, Similar data in the high tunnel coupling regime. Whereas the individual nature of the dots is all but gone, global (weaker) Coulomb repulsion still prohibits transport at small bias, as expected for the collective Coulomb blockade phase. The notion of a large gap at half-filling is gone, and it is only the charging energy of the entire system that prohibits transport occurring, regardless of filling. The dots are in collective Coulomb blockade, and its transport characteristics are similar to that of a small, metallic island. Source data

Extended Data Table 1 Example of simulated transition points
Extended Data Table 2 Experimental and theoretical miniband widths

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Hensgens, T., Fujita, T., Janssen, L. et al. Quantum simulation of a Fermi–Hubbard model using a semiconductor quantum dot array. Nature 548, 70–73 (2017). https://doi.org/10.1038/nature23022

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