Nagaoka ferromagnetism observed in a quantum dot plaquette

J.P. Dehollain,1, ∗ U. Mukhopadhyay,1, ∗ V.P. Michal,1 Y. Wang,2 B. Wunsch,2 C.

Reichl,3 W. Wegscheider,3 M. Rudner,4 E. Demler,2 and L.M.K. Vandersypen1, †

1QuTech and Kavli Institute of Nanoscience, TU Delft, 2600 GA Delft, The Netherlands2Department of Physics, Harvard University, Cambridge 02138, USA

3Solid State Physics Laboratory, ETH Zurich, Zurich 8093, Switzerland4Center for Quantum Devices and Niels Bohr International Academy,

Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark

The analytical tractability of Nagaoka ferromagnetism makes it a convenient model to explore thecapabilities of quantum simulators of collective electron interactions. However, the small ground-to-excited state energy compared to electron interactions, as well as the difficulty of measuringmagnetization in few particle devices, have made the Nagaoka model experimentally unattainable.Here we present experimental signatures of the ferromagnetic ground state, predicted for 3 elec-trons in a 4 site square plaquette, engineered using electrostatically defined quantum dots. Wetest the robustness of the Nagaoka condition under different scenarios of lattice topology, devicehomogeneity and magnetic flux through the plaquette. This long-sought demonstration of Nagaokaferromagnetism establishes quantum dot systems as prime candidates for quantum simulators ofmagnetic phenomena driven by electron-electron interactions.

INTRODUCTION

The emergence of magnetism in itinerant electron sys-tems presents a fascinating and challenging problem atthe heart of quantum many-body physics [1, 2]. In con-trast to itinerant models, systems of localized spins–asdescribed for example by the Heisenberg or Ising models–present strong atomic potentials that quench their elec-tronic kinetic energy. The tendency toward magnetismin such systems can be captured through an effectivespin Hamiltonian where positive or negative values ofexchange constants directly favor parallel or antiparal-lel alignment of spins. The situation is more subtle initinerant systems, where freely mobile electrons interactvia spin-independent Coulomb interactions. Here, mag-netism must emerge from a delicate quantum mechani-cal interplay between the potential energy that can besaved through building appropriate symmetries and cor-relations into electronic wave functions, and the corre-sponding costs in kinetic energy. Due to the complexityof this quantum many-body problem, rigorous theoreti-cal results characterizing conditions in which ferromag-netism may emerge in itinerant electron systems are fewand far between. On the experimental side, the situationin natural materials is also often complicated by the pres-ence of a mixture of localized and itinerant electrons. Al-together, this makes the problem of itinerant magnetisma ripe target for quantum simulation in well-controlledmesoscopic quantum devices.

The Nagaoka model of ferromagnetism [3] is one of thebest known mathematically solvable models of ferromag-netism. It is a special instance of itinerant magnetism,relying on the simplicity of the Hubbard model [4], which

∗ These authors contributed equally to this work.† Corresponding author. Email: l.m.k.vandersypen@tudelft.nl

captures complex correlations between electrons in a lat-tice using only two Hamiltonian parameters. Using thissingle-band model, Nagaoka proved that for some latticeconfigurations and in the limit of very strong interactions,the presence of a single hole on top of a Mott insulatingstate with one electron per site renders the ground stateferromagnetic.

This elegant theoretical demonstration of ferromag-netism in the Hubbard model poses the question whetherthe ferromagnetic ground state will persist in an experi-mental setting, in the presence of long-range interactionsand disorder, as well as additional available orbitals. Thisquestion could be approached by either identifying a fer-romagnetic material where the Nagaoka conditions areapproximately realized naturally, or by using an engi-neered lattice with controllable parameters to perform aquantum simulation [5, 6]. The feasibility of the latterhas been explored for quantum dots [7–9] as well as opti-cal superlattices [10]. In spite of the maturity of quantumsimulations of the Hubbard model, led by the cold atomscommunity [11], there has been no experimental obser-vation of a high-spin ground state in an almost half-filledlattice or array of itinerant electrons–the smoking gun ofNagaoka ferromagnetism.

Electrostatically defined semiconductor quantumdots [12–14] have been gaining attention as excellentcandidates for quantum simulations of the Hubbardmodel [15, 16]. Recent results have demonstratedthe feasibility to extend these systems into 2D lat-tices [17–20]. The ability to reach interesting interactionregimes along with low temperatures, as well as theability to perform spin correlation measurements, makequantum dot arrays particularly appealing for overcom-ing the challenges of observing evidence of Nagaokaferromagnetism.

In this article, we present experimental signatures ofNagaoka ferromagnetism, using a quantum dot device de-signed to host a 2× 2 lattice of electrons [20]. Using the

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high degree of parameter tunability, we study how exter-nal magnetic fields and disorder in local potentials affectthe ferromagnetic ground state. Furthermore, by effec-tively tuning the geometry of the system from periodicto open boundary conditions, we experimentally demon-strate the suppression of magnetism expected from theLieb-Mattis theorem [21].

NAGAOKA MODEL IN THE QUANTUM DOTPLAQUETTE

The single-band Hubbard model provides a simple de-scription of interacting electrons in a lattice, through aHamiltonian that contains competing kinetic energy andelectron-electron interaction terms:

HH = −∑〈i,j〉σ

ti,jc†iσcjσ +

∑i

Uini↑ni↓ −∑i

µini, (1)

where ti,j is the matrix element accounting for electrontunneling between sites i and j, Ui is the on-site Coulombrepulsion energy on site i and µi is a local energy offsetat dot i, which can be electrostatically controlled. The

operators ciσ, c†iσ and niσ represent the second quantiza-tion annihilation, creation and number operators for anelectron on site i with spin projection σ = {↑, ↓}.

To study the conditions under which Nagaoka ferro-magnetism can manifest itself on a square plaquette, werestrict the system to 3 electrons (i.e. one less than halffilling), with nearest-neighbor only coupling and periodicboundary conditions (see schematic in inset of Fig. 1a).This case is analytically solvable [7] for homogeneous in-teractions (Ui = U , ti,j = t, µi = 0) and in the limitU � t, where the eigenstates have energies:

E3/2 = −2t and E1/2 = −√

3t− 5t2

U, (2)

where E3/2 is the energy of the ferromagnetic quadru-plets (with total spin s = 3/2 and spin projectionsm = {±1/2,±3/2}) and E1/2 is the energy of the 2 setsof low-spin s = 1/2 doublets, which are degenerate inthis model [22].

The simple Hamiltonian in Eq. (1) does not account forsome of the essential features of the experimental device.For comparison with experimental results, we employ amore general model Hamiltonian, in which we accountfor interdot Coulomb repulsion (in Fig. 2a), spin-orbitand hyperfine interactions (in Fig. 3b), as well as theeffects of external magnetic fields (in Fig. 5a-b). Theimplementation of these terms is described in detail in thesupplementary text. In addition to this model, we havealso performed a ab initio calculation based on multipleorbitals solved from quantum wells, showing very similarresults to those obtained with Eq. (1) [22].

Tunnel coupling t [μeV]20 60 100 140

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t12U1 U2

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FIG. 1. Device schematic and Nagaoka model. (a) Falsecolored SEM image of a device from the same batch as the oneused in the experiments. The gate structure used to define thequantum dots is colored in dark gold. A slab of silicon nitride(colored in green) is laid over gates C3 and P3, to electricallyisolate those gates from the D0 gate (colored in bright gold)which runs over them and contacts the substrate at the centerof the structure. A sketch of the expected 2DEG density inblue shows the 4 dots forming a plaquette in the center of thedevice, along with nearby charge sensors and electron reser-voirs. (b) Energy spectrum as a function of tunnel couplingusing the solution expressed in Eq. (2), with U = 2.9 meV.Shaded area shows the experimentally accessible range of t inthis system.

EXPERIMENTAL ACCESS TO THE NAGAOKAREGIME

The quantum dot plaquette (Fig. 1a) is formed by bias-ing metallic gates patterned on top of an AlGaAs/GaAsheterostructure, to control the local density of a 2-dimensional electron gas (2DEG) located 90 nm below

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FIG. 2. Experimental protocol. (a) Simulated charge stability diagram showing the approximate gate space used in theexperiment. In the experiment we pulse in a straight line in gate space from point M to point N and back. Top-right insetshows a schematic of the local energies at points N and M , highlighting in the latter how the measurement of 2 spins in thesinglet-triplet basis is performed through spin-to-charge conversion. Lower-left inset shows a measured charge stability diagramof the dotted region, with the same gate voltage ratios as the simulation, which we use in the experiment to calibrate thegate voltages at point N . (b) Calculated energy spectrum as a function of detuning proportion, using the theoretical model(Eq. (1) and supplementary text) without spin-coupling effects. Parameter values were set to Ui = [2.9, 2.6, 2.9, 3.0] meVand ti,j = 16 µeV, as extracted from the experiment. Inset shows a zoomed-in spectrum of the region where the 3 spins aredelocalized on all 4 dots, where there are a total of 8 states: the s = 3/2 quadruplets (red) and the 2 sets of s = 1/2 doublets(blue), of which one set connects with the |T 〉 branch and the other with the |S〉 branch at point M . Line colors representthe spin state of the system in each region, denoted by the labels in the figure. The energies extracted from the numericalsolutions are offset by the energy of |s,m〉 = |3/2,+3/2〉. (c) Pulse sequence used in the experiment (see main text for detaileddescription).

the surface of the substrate. We use two additionalnearby dots as charge sensors, to measure charge sta-bility diagrams where we can observe charge tunnelingevents either between an electron reservoir and a dot, orbetween two dots in the plaquette. These diagrams (suchas the one in Fig. 2a) allow us to map out the charge oc-cupation of the system, as a function of voltage in thegates. The device is tuned to a regime where the systemis loaded with 3 electrons, and the charge configurationenergies of the electrons are resonant. We set the localenergy reference at this regime as µi(N) = 0 eV for alldots, and refer to this condition as point N (see insetof Fig. 2a). Different features of the charge stability dia-grams are also used to estimate the effective Hamiltonianparameters in our experimental system. The effectiveon-site interaction Ui is measured by extracting the localenergy offset in dot i required to change the occupationfrom 1 electron to 2 electrons. The effective tunnel cou-pling term ti,j is measured by analyzing the width of thestep in the charge sensing signal as the detuning betweendots i and j is swept to transfer a single electron betweenthem. Virtual gates provide knobs to effectively controlthe µi and ti,j parameters in the experimental system,

by canceling the effects of cross-talk between gates. Amore detailed description of the fabrication, operation,measurement protocols and implementation of the vir-tual gates can be found in the supplementary materialand in Ref. [20].

The simple model described by Eqs. (1) and (2) alreadyprovides some useful insight into the parameter regimesthat are relevant to the experiment. The ferromagneticstate is the ground state at large U/t, with a transi-tion to a low-spin ground state occurring at U/t = 18.7.The quantum dot array used in this work has an averageU ≈ 2.9 meV across the four sites, with tunable nearest-neighbor tunnel couplings in the range of 0 < t <∼ 20 µeV.Unless otherwise stated, the couplings in these measure-ments are tuned to ti,i+1 ≈ 16 µeV. This means that weare probing the regime where the ground state is expectedto be ferromagnetic and the transition to the low-spinstate is out of experimental reach (see Fig. 1b). More-over, the expected energy gap between the ferromagneticand low-spin states in the system is E1/2−E3/2 ≈ 4 µeV,which is comparable to the measured electron tempera-ture kBTe ≈ 6 µeV (70 mK) [20]. This complicates themeasurement, because we cannot distinguish the ground

4

state of the system in its equilibrium state. Instead, weneed to drive the system out of equilibrium in order to tryto amplify the probability in the ground state. To thisend, we have developed techniques to probe the differentenergy levels and probe the spin state of the system ontimescales faster than the thermal relaxation.

Measurement protocol

Since the sensing dots are only sensitive to chargetunneling events, a spin-to-charge conversion protocol isneeded in order to perform measurements of the spinstate of the system. We do this at point M , where µMi ≈[−2.5, 0.0, 1.0,−0.5] meV (see inset of Fig. 2a). There,the ground charge state is [2, 0, 0, 1] (where [n1, n2, n3, n4]corresponds to the number of electrons with dot numberin the subscript), while the first excited charge state is[1, 1, 0, 1]. These states have an uncoupled spin in dot 4,with the remaining 2 spins in a singlet |S〉 (triplet |T 〉)configuration for the ground (first excited) state. Thecharge stability diagram in Fig. 2a is simulated and mea-sured (inset) using a gate combination that allows to seeboth point N and M in the same diagram.

Fig. 2b shows the lowest three multiplets of the energyspectrum of the 3-electron system, along the line thatconnects M to N . Close to M we see a typical doublequantum dot spectrum corresponding to the [2, 0, 0, 1]↔[1, 1, 0, 1] charge transition with the |S〉 and |T 〉 branches,while in the region around N the spins delocalize andwe see branches corresponding to the quadruplets anddoublets of the 3-electron system.

With this device, we can probe the spin state of the3-electron system using the following protocol: 1 - re-peatedly (10000 times) pulse rapidly from point N toM , 2 - for each repetition, perform single-shot |S〉/|T 〉measurements using dots 1 and 2 and taking 2 out of the3 electrons, and 3 - extract the triplet probability PT .Under ideal conditions, this constitutes a 2-spin projec-tive measurement of the 3-electron system, resulting in

P(3/2)T = 1 when the 3-electron system is in a ferromag-

netic state (any of the s = 3/2 quadruplets). In the low-spin sector (s = 1/2), there are two sets of doublet statesavailable, one of which projects 2 spins to |S〉, while theother projects to |T 〉 [22]. In this system the doublets areeffectively degenerate (see Fig. 2b), and their hybridiza-

tion will result in P(1/2)T = 0.5.

Due to the low ratio of energy level splitting to tem-perature at point N , we cannot probe the ground state ofthe system by way of relaxation. Instead, we apply a gatepulse sequence that follows the detuning range shown inthe energy spectrum plotted in Fig. 2b. Using the pulsesequence drawn in Fig. 2c, a 2-spin singlet state with athird, free spin sitting on dot 4, is initialized by waiting atpoint M for 500 µs. Next we apply simultaneous pulsesto the Pi gates of different amplitudes, such that we ef-fectively pulse along the ‘detuning proportion’ pε axis inFig. 2b (see also the line along the charge stability dia-

0 4 8 12Wait time at N τwait [μs]Ramp time τramp [μs]

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0.4 0.8

FIG. 3. Characterization of the Nagaoka condition. (a)Measured PT vs pε using the protocol described in the maintext. Different curves correspond to different values of τramp.The main figure focuses on the region close to point N , whilethe inset is zoomed out to the entire detuning range, for the2 extreme values of τramp. τwait is fixed to 50 ns (500 ns) forthe main figure (inset). (b) Average PT in the detuning re-gion 1.00 < pε < 1.01 for a set of 40 measurements within thesame τramp range shown in (a). Black line is a fit made viatime evolution simulations in which we initialize a statisticalmixture of the two lowest energy eigenstates at pε = 0.8 andsweep the potentials to pε = 1 at different rates. The fit hasthe hyperfine coupling parameter δN as a free fitting parame-ter, and the extracted PT curve from the model is scaled andoffset to match the data at the minimum and maximum τramp,to account for measurement imperfections [22]. (c) Thermalrelaxation measurements. PT is measured for increasing waittimes at point N , for diabatic (dark) and adiabatic (light)passages. Solid lines are exponential fits as guide to the eye.

gram in Fig. 2a), defined such that µi(pε) = (1− pε)µMi .We then wait a time τwait at µi(pε), before finally puls-ing back to point M to perform the measurement. Im-portantly, the level crossings seen in Fig. 2b are in factavoided level crossings with spin-orbit and nuclear hyper-fine mediated coupling between the spin states [22]. Thisavoided level crossing allows to probe the different statesin the region around pε = 1, by varying the ramp ratein the pulse sequence: a slow (fast) ramp rate resultsin an adiabatic (diabatic) passage through the avoidedlevel crossings, so the ground (excited) state is reached.

5

In practice, in order to avoid leakage to excited statesalong the way, 80% of the pulse is performed adiabat-ically, with the variable ramp time τramp only for theremaining 20%. Varying τwait allows to study the relax-ation dynamics in the system. As long as τwait is shorterthan the thermal relaxation time-scale, the measurementof PT will be able to distinguish between ferromagneticand low-spin states at point N .

MEASUREMENT RESULTS

Fig. 3a shows plots of PT (pε) when we apply the ex-perimental protocol described above. The inset of thefigure shows the entire range of pε, highlighting that PTremains at a low value for most of the range, with a sharpincrease as pε approaches 1 (point N). This is consistentwith expectation based on the energy spectrum plottedin Fig. 2b, where the initialized singlet state is not sub-ject to any energy-level crossing until the region closeto point N , where the levels cross and the electrons be-come delocalized in the array, leading to a sharp increasein the observed PT . The non-zero triplet fraction at lowvalues of pε is attributed partly to infrequent thermal ex-citations during the initialization stage–as a consequenceof the finite electron temperature–and partly to a smallprobability of leakage to excited states during the pulse.

The main figure shows the measurement aroundpoint N , for a range of τramp. In the region where0.99 < pε < 1.03, a clear increase of PT is observedas τramp is increased, consistent with a gradual transi-tion from diabatically pulsing into the low-spin state, toadiabatically pulsing into the ferromagnetic state, wherePT is maximum. For the faster pulses, we see ‘peaks’of PT at pε = 0.99 and 1.03, where the pulse reachesthe energy-level crossings, as all the spin states can beexpected to quickly (i.e., much faster than the experi-mental timescales) mix by the nuclear hyperfine fieldsand spin-orbit coupling [23, 24].

From the τramp timescale for the diabatic to adiabatictransition (see Fig. 3b) we can extract information aboutthe spin-coupling mechanisms at the avoided crossings.To this end we have expanded the model in Eq. (1), toinclude the effects of spin-orbit interaction and the hy-perfine induced magnetic field gradients [22]. The modelsuggests that the random hyperfine field gradients domi-nate the spin coupling present at the avoided level cross-ing (i.e., around pε ≈ 0.97). We can use the model tofit the data in Fig. 3b, through time-evolution simula-tions [22], from which we estimate a hyperfine couplingparameter δN = 73 ± 3 neV, defined as the standarddeviation of the Gaussian probability distribution of thehyperfine field in each dot [22]. The extracted δN is inagreement with previous observations and calculations,which have estimated 70 neV to 120 neV in similar GaAsquantum dot systems [24–26]. We note that the observedbehavior can be qualitatively captured by a simple two-level Landau-Zener model [22].

(a)

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= 1000 nsτramp = 5 nsτramp

s = 1/2

s = 3/2s = 1/2

s = 3/2

FIG. 4. From ring to chain. (a) Comparison of 3measurements with the following values of tunnel couplings[t12, t23, t34, t41] (in µeV): [18.6, 15.3, 17.4, 18.6](orange);[15.7, 7.9, 20.3, 19.0](green); [18.2, 0.0, 21.1, 20.7](purple).The offsets between the curves are not attributed to thetopology, but are due to small measurement-to-measurementvariations in the thermal excitation rate during the initializa-tion stage of the protocol. (b) Calculated energy spectrum asa function of detuning proportion, using the tunnel couplingvalues corresponding to the green (left) and purple (right)plots from (a).

If we keep pε = 1 fixed and vary the wait time τwaitspent at point N, we observe relaxation of the s = 1/2and s = 3/2 states, reflected in the decay of PT to anintermediate level between the PT observed for slow andrapid sweeps, at the shortest τwait (see Fig. 3c). Thisis consistent with thermal equilibration in the system,in which the electron temperature is comparable to theenergy gap between the s = 1/2 and s = 3/2 states atpoint N . The thermal equilibration occurs on a timescaleτrelax ∼ 2 µs. We note that we cannot directly assign thevalues of PT to s = 1/2 and s = 3/2 populations, becausethe observed PT is subject to measurement imperfectionscaused by mechanisms that are difficult to disentangle,such as the finite measurement bandwidth, the signal tonoise ratio and |T 〉 to |S〉 relaxation, as well as unwantedleakage to other states during the pulsed passages.

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Boundary conditions and the Lieb-Mattis theorem

Considering that the square plaquette can be thoughtof as a 1D ring, the observation of a ferromagnetic groundstate may appear to be in contradiction with the Lieb-Mattis theorem [21] which states that the ground state ofa 1D array of electrons has the lowest possible spin. How-ever, as later pointed out by Mattis himself [7] the Lieb-Mattis theorem was only proven for 1D chains with openboundary conditions, explicitly excluding arrays with pe-riodic boundary conditions such as the case of the pla-quette. We can intuitively understand the difference be-tween these two configurations when we consider howthe hole tunnels to its next-nearest neighbor [27]. In a2D plaquette, the hole has 2 possible paths to the next-nearest neighbor. If the system is initialized in any of thes = 1/2 configurations, the 2 paths will leave the systemin 2 different configurations. On the other hand, for ans = 3/2 system the 2 paths are identical, and interfereconstructively to lower the kinetic energy of the system.In contrast, in an open boundary 1D array, the kineticenergy of the hole is independent of the spin configura-tions of the neighboring electrons (i.e., there is only onepath for the hole to follow through the array), thereforethe total energy of the system will obey the Lieb-Mattistheorem.

One powerful feature of the quantum dot system isthat the tunnel barriers can be tuned independently, al-lowing us to test different array topologies. In Fig. 4 wecompare diabatic and adiabatic sweeps, as we raise thetunnel barrier that controls t23, effectively transformingthe plaquette into a open chain of quantum dots. Inthe latter regime, we see that PT becomes insensitive tosweep rate. Additionally, we no longer observe the peaksof PT for the fast sweep rate, which we had associatedwith mixing at the avoided level crossings. From theseobservations we infer that for the open chain, the instan-taneous ground state does not exhibit an avoided crossingbetween an s = 1/2 state and an s = 3/2 state as thesystem is taken to point N. This interpretation is alsoconsistent with the numerical simulations of the energyspectrum shown in Fig. 4b.

Destroying ferromagnetism with magnetic fields

Given that Nagaoka ferromagnetism can be explainedthrough interference effects due to the trajectories ofthe hole around the ring, it then follows that a mag-netic flux through the plaquette will add an Aharonov-Bohm phase [28] that disturbs the interference ef-fects. We capture this effect in the theoretical modelby modifying the second term in Eq. (1) as [22]:

−∑〈j,k〉σ tj,k exp (−iϕjk) c†jσckσ. We use the gauge in

which ϕ41 = 2πΦΦ0

, where Φ = B`2 is the flux generated bya magnetic field B through the plaquette with estimateddistance between nearest-neighbor dots `, and Φ0 = h/e

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s = 1/2

s = 3/2

s = 1/2

s = 3/2

s = 1/2

s = 3/2−38

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FIG. 5. Applying an external magnetic field. (a) Lowesteigenenergies of the s = 1/2 (blue) and s = 3/2 (red) statesat point N as a function of magnetic field, obtained from thenumerical model after including the effect of an Aharonov-Bohm phase (details in main text). (b) Same as (a) but withthe addition of the Zeeman effect, and the lowest 4 eigenen-ergies of each s states are shown. (c) Diabatic passage mea-surements for different fields in the range of 0 to 16 mT. In-set shows a numerically calculated spectrum at 12 mT, withthe Aharonov-Bohm phase and Zeeman effect included in themodel.

is the flux quantum. Using this gauge, the phases forthe other links vanish. In addition, the application of anexternal field subjects the system to the Zeeman effect,causing a spin-dependent energy offset EZ = gµBBm toeach eigenstate, where g = −0.4 is the electron g-factorin GaAs and µB is the Bohr magneton.

Fig. 5a shows the effect of the magnetic flux on thespectrum, ignoring the Zeeman effect. The lowest s =1/2 and s = 3/2 levels at point N are shown as a func-tion of the applied field, where periodic crossings can beobserved. In the range 30 < B < 160 mT, the systemground state transitions to the low-spin state, with the–perhaps counterintuitive–implication that we can destroythe ferromagnetic state by applying a magnetic field. Ad-ditionally, this effect highlights that the ferromagneticstate in this system is dominated by the Nagaoka effectand not by long-range interactions. Indeed, the ab initiocalculations suggest that long-range interactions only ac-count for ∼ 20% of the ferromagnetic polarization. Whenwe include the Zeeman effect (see Fig. 5b) the picture be-

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0.2

0.3

0.4

0.5

0.6

0.7

Trip

let p

ropo

rtio

n PT

0.96 0.98 1.00 1.02 1.04 1.06Detuning proportion pε

0.96 0.98 1.00 1.02 1.04 1.06Detuning proportion pε

0.98 1.00 1.02 1.04 1.06Detuning proportion pε

FIG. 6. Local energy offsets. Adiabatic and diabatic passage measurements with point N purposefully redefined to havea ±50 µeV offset on each of the 4 dots. Panels correspond to offsets in dots 1 to 4, clockwise from the top-left. Insets shownumerically calculated spectra for the same experimental condition.

comes more complicated, because both Zeeman and or-bital effects cause perturbations of similar energy scales.

From this initial numerical analysis it is clear thatthe experimental characterization of the applied externalfield will be challenging, due to the increased complexityof the spectral structure of the spin states as a functionof field. The small energy splittings that appear bothat point N , as well as at lower pε values (see inset ofFig. 5c) are expected to cause mixing of the spin statesduring the adiabatic pulses. To minimize this mixing, weadjusted the pulsing protocol such that we pulse adia-batically (1 µs ramp) to pε = 0.2, then pulse diabatically(5 ns ramp) the rest of the way. The results in Fig. 5cshow that from 4 to 8 mT PT increases at point N, andwe stop observing the characteristic dip. Note that therange of field that we were able to probe is still belowthe estimated ground state transition point (∼ 30 mT).Therefore, we infer that the observed increase in PT isthe effect of hybridization of the s = 1/2 and s = 3/2states as their energy gap reduces. We cannot claim thatthe observed hybridization of states is occurring solely

at point N , as it is evident from the increase in PT atpε < 0.97 (i.e. prior to the energy-level crossings) thatsome of the mixing is occurring during the pulse. How-ever, we do see that PT in all plots converge at the energy-level crossings (pε ≈ 0.97 and pε ≈ 1.03) suggesting thatthe Aharonov-Bohm orbital effects are partly responsiblefor the additional mixing in the region around point N .

Sensitivity to local energy offsets

We also use the tunability available in quantum dotsystems to study the effects of disorder of the local poten-tial present in each dot. For the plots in Fig. 6, we mod-ified the experimental protocol used to probe the statesat point N , pulsing instead to a point N ′, where the localenergy of one of the dots is offset by ±50 µeV. We cando this by employing the virtual gates technique [16, 20],which gives access to control knobs that map a linearcombination of Pi gates onto local dot energy offsets.The insets of the panels in Fig. 6 show the expected

8

energy spectra when we simulate the experimental con-ditions using the model in Eq. (1). The spectra showthat for all cases there remains a region in the detuningtrajectory where the ferromagnetic state is the groundstate, but both the width and the position of this regionaround point N ′ varies with the different local offsets ap-plied. The experimental results in the main panels showexcellent qualitative agreement with the variations ob-served in the calculated spectra, further confirming thevalidity of the experimental protocol. Remarkably, wehave also pushed the offset of dot 1 to the range −100 to+800 µeV and the system still shows signs of the ferro-magnetic ground state.

DISCUSSION

In this work we have presented the first measurementsshowing experimental evidence of Nagaoka’s 50-year oldtheory in a small scale system. The large degree of tun-ability, high ratio of interaction strength to temperature,and fast measurement techniques available to quantumdot systems, allowed observing both the ferromagneticground state and the low-spin excited state of an almost-half-filled lattice of electrons. Even though the problemof 3 electrons in a 4-site plaquette can be solved analyt-ically using the Hubbard picture, a complete descriptionof this experimental system that includes all its availableorbitals is not easily tractable, analytically or numeri-cally. Indeed, the computational cost of the ab initiocalculation [22], with all interaction terms being consid-ered, is on the order of 10000 CPU hours. In addition,this small scale quantum simulation provides value be-yond proof-of-principle in two important ways. First,by performing a quantum simulation involving chargeand spin states, it builds on previous demonstrations [16]that quantum dot systems can be useful simulators of theHubbard model, despite their inhomogeneities in the po-tential shape and local energies. Additionally, small scalesimulations on tractable models can be used to systemat-ically benchmark the performance of devices as the scale-up technology develops towards devices that can perform

classically intractable simulations. Finally, in this workwe showed a flavor of the capabilities for studying thesensitivity to disorder, and these experiments already re-vealed some surprising effects, when we found that theNagaoka condition can still be observed after offsettinga local energy by amounts much larger than the tunnelcoupling. This can readily be studied in further detail,along with other possibilities for exploring the effects ofdisorder, which could bring insights into e.g., the stabilityof the ferromagnetic state.

This experiment is an important step forward in an-swering the question of whether itinerant magnetism canoccur in real systems. Larger realizations of similar quan-tum dot systems (or any other experimentally control-lable system), such as 2 ×N or M ×N arrays can shedmore light on the discussion. As the system becomeslarger the exchange interaction grows proportionally tothe system size, creating a competition against the hop-ping energy that is characteristic of Nagaoka ferromag-netism, and leaving the fate of the Nagaoka mechanismin larger systems in the realm of the unknown.

ACKNOWLEDGMENTS

We acknowledge useful input and discussions with M.Chan, S. Philips, Y. Nazarov, F. Liu, L. Janssen, T.Hensgens, T. Fujita and all of the Vandersypen team, aswell as experimental support by L. Blom, C. van Diepen,P. Eendebak, R. Schouten, R. Vermeulen, R. van Ooijik,H. van der Does, M. Ammerlaan, J. Haanstra, S. Visserand R. Roeleveld. L.M.K.V. thanks the NSF-fundedMIT-Harvard Center for Ultracold Atoms for its hospi-tality. Funding: J.P.D., U.M., L.M.K.V. acknowledgesupport from the Netherlands Organization for ScientificResearch (FOM projectruimte and NWO Vici) and theSwiss National Science Foundation. Y.W. acknowledgesthe Postdoctoral Fellowship in Quantum Science of theHarvard-MPQ Center for Quantum Optics and AFOSR-MURI Quantum Phases of Matter (Grant No. FA9550-14-1-0035). M.S.R. acknowledges support from The Vil-lum Foundation.

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