Abstract
In quantum simulation, manybody phenomena are probed in controllable quantum systems. Recently, simulation of Bose–Hubbard Hamiltonians using cold atoms revealed previously hidden local correlations. However, fermionic manybody Hubbard phenomena such as unconventional superconductivity and spin liquids are more difficult to simulate using cold atoms. To date the required singlesite measurements and cooling remain problematic, while only ensemble measurements have been achieved. Here we simulate a twosite Hubbard Hamiltonian at low effective temperatures with singlesite resolution using subsurface dopants in silicon. We measure quasiparticle tunnelling maps of spinresolved states with atomic resolution, finding interference processes from which the entanglement entropy and Hubbard interactions are quantified. Entanglement, determined by spin and orbital degrees of freedom, increases with increasing valence bond length. We find separationtunable Hubbard interaction strengths that are suitable for simulating strongly correlated phenomena in larger arrays of dopants, establishing dopants as a platform for quantum simulation of the Hubbard model.
Introduction
Quantum simulation offers a means to probe manybody physics that cannot be simulated efficiently by classical computers, using controllable quantum systems to physically realize a desired manybody Hamiltonian^{1,2,3}. In the analogue approach to quantum simulation exemplified by cold atoms in optical lattices^{4,5}, the simulator’s Hamiltonian maps to the desired Hamiltonian. Compared to digital quantum simulation, realized via complex sequences of gate operations^{6,7}, analogue quantum simulation is usually carried out with simpler building blocks. For example, the Heisenberg and Hubbard Hamiltonians of great interest in manybody physics are directly synthesized by cold atoms in optical lattices^{2,3}. Although of immense interest and proposed long ago^{8}, analogue simulation of fermionic Hubbard systems has proven to be very challenging^{2,3}. The anticipated regime of the intensely debated spin liquid, unconventional superconductivity and pseudogap^{9,10,11} has yet to be accessed even for cold atoms. Here, the required low temperature T<t/30 is problematic due to the weak tunnel coupling t of cold atoms^{5,12}. Moreover, experimentally resolving individual lattice sites, crucial elsewhere in Bose–Hubbard simulation^{4}, remains very challenging in quantum simulation of the Hubbard model^{5}.
Here, we perform atomic resolution measurements resolving spin–spin interactions of individual dopants, realizing an analogue quantum simulation of a twosite Hubbard system. We demonstrate the much desired combination of low effective temperatures, singlesite spatial resolution, and nonperturbative interaction strengths of great importance in condensed matter^{9,10,11}. The dopants’ physical Hamiltonian , determined at the time of fabrication^{3}, maps to an effective Hubbard Hamiltonian , where is the onsite Coulomb repulsion, (c_{iσ}) creates (destroys) a fermion at lattice site i with spin σ, is the number operator, and h.c. is the Hermitian conjugate. Here, it is desirable to achieve nonperturbative (intermediate) interaction strengths associated with quantum fluctuations and emergent phenomena^{9,10,11}, that is, beyond perturbative Heisenberg interactions (large ) realized in photonbased^{13} and ionbased^{14} simulations, and magnetic ions on metal surfaces^{15}. We focus on the system ground state, prepared by relaxation on cooling^{3}, rather than system dynamics.
Because the states of our artificial Hubbard system are coupled and interacting, tunnelling spectroscopy locally probes the spectral function. The spectral function is of key interest in manybody physics because it provides rich information on interactions^{16,17}, and is highly sought after in future ‘coldatom tunnelling microscope’ experiments^{18}. For our fewbody system, the local spectral function describes the quasiparticle wavefunction (QPWF)^{19,20,21,22} and the discrete coupledspin spectrum of the dopants. We find that interference of atomic orbitals directly contained in the QPWF allows us to quantify the electron–electron correlations and the entanglement entropy. The entanglement entropy is a fundamental concept for correlated manybody phases^{23,24,25,26} that has thus far evaded measurement for fermions. In the counterintuitive regime of our experiments, entanglement entropy increases as the valence bond is stretched, as Coulomb interactions overcome quantum tunnelling. In our system, the entanglement entropy is directly related to the Hubbard interactions , and we find that is tunable with dopant separation, increasing from 4→14 for d/a_{B}=2.2→3.7, where a_{B}=1.3 nm is the effective Bohr radius. This range, of interest to simulate unconventional superconductivity and spin liquids^{9,10,11}, is realized here due to the large Bohr radii of the hydrogenic states. The semiconductor host allows for electrostatic control of the chemical potential^{27,28}, desirable to dynamically control filling factor^{9,11} but not possible for ions on metal surfaces^{15}.
Results
Spectroscopy of coupledspin system
Subsurface boron acceptors in silicon were identified at 4.2 K as individual protrusions^{29,30} (density ∼10^{11} cm^{−2}) in constant current images due to resonant tunnelling at a sample bias U=+1.6 V, and due to the acceptor ion’s influence on the valence density of states at U=−1.5 V. The sample was prepared by ultrahigh vacuum flash annealing at 1,200 °C and hydrogen termination. The observed subsurface acceptors had typical depths^{29,30} <3 nm, and correspondingly, a volume density >25 times less than the bulk doping 8 × 10^{18} cm^{−3}. Pairs of nearby acceptors with d≲5 nm were also found, with a smaller density ∼10^{9} cm^{−2}.
The spectrum and spatial tunnelling probability of the coupled acceptors were investigated at T=4.2 K via singlehole tunnelling from a reservoir in the substrate to the dopant pair, to the tip^{29,30} (Fig. 1a). For the dopant pair in Fig. 1b (top), dI/dU measured along the interdopant axis (Fig. 1b, bottom) contains a peaks for each state entering the bias window, at U≈0.2, 0.45, 0.55 and 0.8 V. Consistent with our singleacceptor^{29} and singledonor^{31} measurements near flatband bias conditions, the bias for each peak in the spectrum (Fig. 1b, bottom) is independent of tip position. This rules out distortion of our quantum state images by inhomogenous tipinduced potentials^{32} observed in other multidopant systems^{33}. These results can be attributed to weak electrostatic control by the tip (Fig. 1c) and the states’ proximity to flatband^{29,30,31}, though a large tip radius may also play a role.
The spectral and spatially resolved measurements (Fig. 1b) directly demonstrate that the holes are interacting, as follows. First, two peaks centred on dopant ions A or B are resolved in real space (Fig. 1b). Second, energy differences between the peaks resolved in real space are smaller than the ∼350 μeV thermal resolution. However, for orbitals at the same energy to not interact, their overlap must vanish. Since the measured orbitals have a strong overlap, the sites are tunnel coupled, irrespective of the details of the tunnelling current profile. The number of states observed, their energy differences, and their energies relative to the Fermi energy confirm that the observed states are twohole states (Supplementary Figs 1 and 2).
Correlations and entanglement from Hubbard interactions
The ground state of a Hubbard model with nonperturbative interactions is governed by in Fig. 2a in the subspace of , , and , where creates a localized electron on site i∈{A, B} with spin , and is the vacuum state. The ground state is a superposition , where γ_{c} (γ_{i}) is the probability amplitude for a covalent (ionic) configuration (Fig. 2b). Rewriting the state in a basis of even and odd orbitals, , where γ_{ee} (γ_{oo}) is the probability amplitude of the ‘even/even’ (‘odd/odd’) configuration.
In limit of small tunnel couplings (large , Fig. 2b) the Hubbard system may be described by perturbative Heisenberg spin interactions. For vanishing , the ground state is a Heitler–London singlet of localized spins, , with no contributions from and . Due to vanishing wavefunction overlap the electrons can be associated with sites A and B (they are distinguishable^{23,34,35}), and the spin at site A depends on the spin at site B as for a maximally entangled Bell state. In the limit of vanishing interactions (, Fig. 2b) corresponding to a tightbinding approximation, the spins delocalize and . In a molecular orbital (MO) basis, the ground state is , which is a single Slater determinant. Although this state is a singlet (one spin up, one spin down) due to fundamental indistinguishability, the electrons can be ascribed independent properties because they occupy the same orbital, and the state is uncorrelated^{23,34,35}.
For intermediate , where tunnelling and Coulomb interactions compete nonperturbatively^{2,3,9,11}, tunnelling hybridizes the doublyoccupied configurations and into the ground state, such that the particles lose their individual identities. Here, the von Neumann entanglement entropy quantifies genuine entanglement (interdependency of properties), distinguishing it from exchangecorrelations due to indistinguishability^{23,26,35}. Employing the convention^{36} (1) for zero (maximal) entanglement, increases as increases and coherent localization occurs (Fig. 2c), saturating at value of 1.
We now discuss the spatial tunnelling maps of the twohole ground states for different interacceptor distances. Obtained by integrating the lowest voltage dI/dU peak, the maps are shown in Fig. 3a–c for distances d/a_{B}=2.2, 2.7 and 3.5 (a_{B}=1.3 nm) having orientations ±2° from 〈110〉, 8±2° from 〈100〉 and 3±2° from 〈110〉, respectively. The multinm spatial extent of the states reflects the extended wavelike nature of the acceptorbound holes, owing to their shallow energy levels, which contrasts Mn ions on GaAs surfaces^{37}, magnetic ions on metals^{15}, and Si(001):H dangling bonds^{38}. Consequently, their envelopes are amenable to effectivemass analysis with lattice frequencies filtered out^{19,20,28,39}. Consistent with measurements of single acceptors at similar depths on resonance at flatband^{29,30}, the states have predominantly slike envelopes with slight extension along [110] directions, as expected when symmetry is not strongly perturbed by the surface. Depths of the d/a_{B}=2.7 and d/a_{B}=3.5 pairs were estimated to be ∼0.9 nm, and for d/a_{B}=2.2, ∼0.6 nm (see Supplementary Fig. 3).
We employed fullconfiguration interaction calculations of the singlet ground state to confirm that Coulomb correlations of coupled acceptors influence the ground state in a way that mimics the S=1/2 Hubbard model. In particular, for d/a_{B}∼2, is predominantly composed of , a singlet of two even ±‘3/2’ spin MOs. With increasing d, interactions enhance the probability amplitude of the singlet with two odd orbitals, analogous to the Hubbard Hamiltonian (Fig. 2b). The spins ±‘3/2’ are predominantly composed of valence band (VB) Bloch states. In particular, the lowlying ±‘1/2’ spin excitations of each acceptor^{30}, which are predominantly composed of Bloch states, do not qualitatively change the description. We also note that for d/a_{B}≳2, the MOs are essentially linear combinations atomic orbitals having the effective Bohr radii of single acceptors.
Singlehole tunnelling transport through our coupleddopant system locally probes the spectral QPWF^{19,20,21}. When (Fig. 1a), the singlehole tunnelling rate is essentially governed by Γ_{out}, the tunnelout rate^{31}. In the present case, singlehole tunnelling from the twohole system to a singlehole final state (Fig. 1) contributes , where is the QPWF, is the field operator, creates a singlehole MO eigenstate ϕ_{j}(r) of the system^{19}, and the total tunnel rate is .
From our QPWF description of coupled dopants, we obtain a spatial tunnelling probability for the ground state. Here, γ_{ee}^{2} and (γ_{oo}^{2}) contain constructive (destructive) interference corresponding to even (odd) linear combinations of atomic orbitals ϕ_{e}(r_{1}) (ϕ_{o}(r_{1})) (note: γ_{ee}^{2}+γ_{oo}^{2}=1). To obtain γ_{oo}^{2}, data were fit to Γ(r, γ_{ee},γ_{oo}), assuming linear combinations of parametrized slike atomic orbitals for ϕ_{e}(r) and ϕ_{o}(r) appropriate for subsurface acceptors. The QPWF and atomic orbitals are described in Supplementary Figs 4–6.
The leastsquares fits in Fig. 3d–f (coloured lines) of Γ(r, γ_{ee}, γ_{oo}) are in good agreement with data (squares), for d/a_{B}=2.2, 2.7 and 3.5. For comparison with the data, grey curves are shown for both the uncorrelated (maximally correlated) state with γ_{oo}=0 (γ_{oo}/γ_{ee}=1) in Fig. 3d–f. We note that all three separations exhibit interaction effects at the midpoint of the ions, where the atomic orbital quantum interference is strongest. We obtain γ_{oo}^{2}=0.12±0.06, 0.23±0.07 and 0.39±0.08 for d/a_{B}=2.2, 2.7 and 3.5. Data taken at higher tip heights gave identical results to within experimental errors (see Supplementary Figs 7 and 8), independently verifying that the tip does not influence our results.
The Coulomb correlations, embodied both in (Fig. 4a) and the entanglement entropy (Fig. 4b), could be evaluated directly from the fit, and both increase with increasing d. The onetoone mapping from to (Fig. 2c) was used to determine the effective Hubbard interactions from the entanglement entropy in Fig. 4b. We obtain , 6.4 and 14, for d/a_{B}=2.2, 2.7 and 3.5, respectively (Fig. 4c), which increase as the tunnel coupling decreases.
We conclude the analysis of the QPWFs with some critical remarks on correlations extracted from our fitting model, recalling that the large spatial overlap of the spectrally overlapping acceptorbound holes directly shows their states are tunnel coupled. First, the Coulomb correlations have a systematic effect on interference in the QPWF such that the leastsquares error is significantly worse if γ_{oo}^{2} is forced to zero in the fitting model (Supplementary Table 1). Second, if applied to very far apart dopants where the ground state can still be resolved, our fitting model would not give a spurious result that the two dopants are highly correlated. This follows because the difference between ϕ_{e}(r)^{2} and ϕ_{o}(r)^{2}, which reflects the interference of atomic orbitals and is used to detect correlations, tends to zero as d/a_{B} increases. Data (Fig. 3a–c) presented here are for coupled dopants that we found to be (i) well isolated from other dopants or dangling bonds, and (ii) at identical depths, as evidenced by the spatial extent and brightness of the atomic orbitals. When the latter is not satisfied, the atomic levels can be detuned, introducing more parameters to the fit.
Comparison with theory
These experimental results obey the trends predicted by our theory calculations for the spinorbit coupled VB. Predictions in Fig. 4a,b for displacements along 〈100〉 (blue solid line) and 〈110〉 (red solid line) both show increasing correlations and entanglement with increasing dopant separation. Moreover, we find that the observed and predicted entanglement entropy qualitatively reproduce a singleband model (Fig. 4a,b, dashed lines). This result implies that interhole Hubbard interactions follow an essentially hydrogenic trend with atomic separation, even for nonperturbative interactions .
The hydrogenic nature of and persists in spite of the ± ‘1/2’ spin excited states of a single acceptors. Such ±‘1/2’ singleacceptor excited states states are found nominally Δ∼1–2 meV above the ±‘3/2’ spin ground state due to inversion symmetry breaking at the interface^{30}. Although t>Δ, and remain hydrogenic in our calculations because the ‘1/2’ spin excited state has an slike envelope whose spatial extent is similar to (1) the slike ± ‘3/2’ ground state and (2) the scaled hydrogenic ground state. Otherwise, single particle ±‘1/2’ states would hybridize stronger than single particle ±‘3/2’ states, form the 2hole singlet at smaller separations, and localize more slowly relative to molecular hydrogen with increasing d. Furthermore, the polarization of the ±‘3/2’ and ±‘1/2’ states into and components, respectively, limits the mixing of ±‘1/2’ states into the ground state.
Spinexcited states and effective temperature
Finally, we discuss the observed excited states, which confirm that the interacceptor tunnel coupling dominates thermal and tunnelcoupling effects of the reservoir. The energies of the states were determined by fitting the singlehole transport lineshapes^{40} of the coupled acceptors (Supplementary Figs 1 and 2). For the first excited state we found 5.2±0.6 and 1.2±0.2 meV for d/a_{B}=2.2 and 3.5, respectively ( orientation), and 1.6±0.7 meV for d/a_{B}=2.7 ( orientation). Shown in Fig. 5a, these energies are too small to add another hole, which would require ≈50 meV for an acceptor in bulk silicon. However, the energies agree well with our predictions for twohole excited states of coupled hole spins ±‘3/2’ and ±‘1/2’, that is, 8.5 and 1.5 meV for d=2.2a_{B} and d=3.5a_{B} (〈110〉 orientation), and 2.0 meV (〈100〉 orientation). Here we note that some of the predicted coupledspin excited states (Fig. 5b) are unconventional: a singlet and triplet of two ‘3/2’ holes (orange lines) and two ‘1/2’ holes (black lines) are obtained, where is the ground state for all separations. More subtly, two manifolds , , i=1 … 4, containing four states are predicted (green lines), where one ±‘3/2’ spin level and one ±‘1/2’ spin level is occupied. For d/a_{B}=2.2 and 2.7 (d/a_{B}=3.5), the measured energies are in better agreement with predictions for excitations.
The interacceptor tunnel couplings t (ratios t/T) were estimated to be 12 meV (30), 7 meV (20) and 3.5 meV (10) for d/a_{B}=2.2, 2.7 and 3.5, respectively, at T=4.2 K. Such couplings t exceed the reservoir coupling Γ_{in} (Supplementary Table 2) to the substrate by more than 50 × . Combined with bias U∼0.2–0.3 V needed to bring the level into resonance, this rules out coherent interactions with substrate and tip reservoirs^{41}. Note that the measured energy splittings imply small thermal excitedstate populations of ≲10^{−5}, ≲10^{−2} and ≲10^{−1} for d/a_{B}=2.2, 2.7 and 3.5, respectively.
Discussion
We performed atomic resolution measurements resolving spin–spin interactions of interacting dopants, realizing quantum simulation of a twosite Hubbard system. Analyzing these local measurements of the spectral function^{17}, we find increasing Coulomb correlations and entanglement entropy as the system is ‘stretched’^{23,35,42} in the regime of nonperturbative interaction strengths . Our experiment is the first to combine low effective temperatures t/T∼30 at 4.2 K and singlesite measurement resolution, considered essential^{3,5,12} to simulate emergent Hubbard phenomena^{9,11}. Lower effective temperatures t/T∼420 are possible at T=0.3 K. For example, 4 × 4 Hubbard lattices with and t/T∼40 have recently been associated with both the pairing state and pseudogap in systems exhibiting unconventional superconductivity^{11}.
The approach generalizes to donors, which can be placed in silicon with atomicscale precision^{27} and spatially measured in situ after epitaxial encapsulation^{43,44}. In contrast to disordered systems^{45}, atomically engineered dopant lattices will require weak coupling to a reservoir, displaced either vertically as demonstrated herein, or a laterally^{27}. Strain could be used to further enhance the splitting between light and heavy holes, or suppress valley interference processes of electrons^{31,46}. Interestingly, open Hubbard systems which may exhibit unusual Kondo behaviour^{47,48} could also be studied by this method. The demonstrated measurement of spectral functions could be used to directly determine excitation spectra, evaluate correlation functions^{45} or obtain quasiparticle interference spectra^{17}, all of which contain rich information about manybody states, including chargeordering effects. We envision insitu control of filling factor^{9,11}, using a backgate or patterned sidegate^{27}. These capabilities will allow for quantum simulation of chains, ladders or lattices^{9,11,49} at low effective temperatures, having interactions that are engineered atombyatom.
Methods
Sample preparation
Samples were prepared by flash annealing a boron doped (p≈10^{19} cm^{−3}) silicon wafer at ∼1,200 °C in ultrahigh vacuum (UHV) followed by slow cooling at a rate 1 °C_{min}^{−1} to 340 °C. Then, hydrogen passivation was carried out ∼340 °C for 10 min by thermally cracking H_{2} gas at a pressure =5 × 10^{−7} mbar.
Measurements
Atomic resolution singlehole tunnelling spectroscopy was performed at 4.2 K using an UHV Omicron low temperature scanning tunnelling microscope. Current I was measured as a function of sample bias U and dI/dU was obtained by numerical differentiation. Details for the analysis of the data are provided in Supplementary Figs 1–3 and 5–8 and Supplementary Notes 1, 2, 4 and 5.
Theory
Theory calculations of interacting states were carried out using the configuration interaction approach, in the Luttinger–Kohn representation including a realistic description of the heavyhole (J=3/2, m_{J}=3/2), lighthole (J=3/2, m_{J}=1/2) and splitoff hole (J=1/2, m_{J}=1/2) degrees of freedom. Details for the theory are provided in Supplementary Fig. 4 and Supplementary Notes 3, 6 and 7.
Additional information
How to cite this article: Salfi, J. et al. Quantum simulation of the Hubbard model with dopant atoms in silicon. Nat. Commun. 7:11342 doi: 10.1038/ncomms11342 (2016).
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Acknowledgements
We thank H. Wiseman, M.A. Eriksson, M.S. Fuhrer, O. Sushkov, D. Culcer, J.S. Caux, B. Reulet, G. Sawatzky, J. Folk, F. Remacle, M. Klymenko and B. Voisin for helpful discussions. This work was supported by the European Commission Future and Emerging Technologies Proactive Project MULTI (317707), the ARC Centre of Excellence for Quantum Computation and Communication Technology (CE110001027) and in part by the US Army Research Office (W911NF0810527) and ARC Discovery Project (DP120101825). S.R. acknowledges a Future Fellowship (FT100100589). M.Y.S. acknowledges a Laureate Fellowship. The authors declare no competing financial interests.
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Experiments were conceived by J.S., J.A.M. and S.R. J.S. carried out the experiments and analysis, with input from J.A.M., R.R., L.C.L.H. and S.R. Theory modelling was carried out by J.S., J.A.M., R.R. and L.C.L.H. and S.R., with input from all authors. J.S. and S.R. wrote the manuscript with input from all authors.
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Salfi, J., Mol, J., Rahman, R. et al. Quantum simulation of the Hubbard model with dopant atoms in silicon. Nat Commun 7, 11342 (2016). https://doi.org/10.1038/ncomms11342
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DOI: https://doi.org/10.1038/ncomms11342
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