# Nagaoka ferromagnetism observed in a quantum dot plaquette

## Abstract

Engineered, highly controllable quantum systems are promising simulators of emergent physics beyond the simulation capabilities of classical computers1. An important problem in many-body physics is itinerant magnetism, which originates purely from long-range interactions of free electrons and whose existence in real systems has been debated for decades2,3. Here we use a quantum simulator consisting of a four-electron-site square plaquette of quantum dots4 to demonstrate Nagaoka ferromagnetism5. This form of itinerant magnetism has been rigorously studied theoretically6,7,8,9 but has remained unattainable in experiments. We load the plaquette with three electrons and demonstrate the predicted emergence of spontaneous ferromagnetic correlations through pairwise measurements of spin. We find that the ferromagnetic ground state is remarkably robust to engineered disorder in the on-site potentials and we can induce a transition to the low-spin state by changing the plaquette topology to an open chain. This demonstration of Nagaoka ferromagnetism highlights that quantum simulators can be used to study physical phenomena that have not yet been observed in any experimental system. The work also constitutes an important step towards large-scale quantum dot simulators of correlated electron systems.

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## Data availability

The datasets obtained from the measurements described in this work are available in the repository Zenodo with the identifier https://doi.org/10.5281/zenodo.3258940.

## Code availability

The code used to plot the datasets and implement the models used to reproduce all the figures in the main manuscript is available in the repository Zenodo with the identifier https://doi.org/10.5281/zenodo.3258940.

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## Acknowledgements

We acknowledge input and discussions with M. Chan, S. Philips, Y. Nazarov, F. Liu, L. Janssen, T. Hensgens, T. Fujita and all of the Vandersypen team, as well as experimental support by L. Blom, C. van Diepen, P. Eendebak, F. van Riggelen, R. Schouten, R. Vermeulen, R. van Ooijik, H. van der Does, M. Ammerlaan, J. Haanstra, S. Visser and R. Roeleveld. L.M.K.V. thanks the NSF-funded MIT-Harvard Center for Ultracold Atoms for its hospitality. This work was supported by grants from the Netherlands Organisation for Scientific Research (FOM projectruimte and NWO Vici) (J.P.D., U.M., L.M.K.V.), the European Research Council (ERC-Synergy) (V.P.M., L.M.K.V.), the Postdoctoral Fellowship in Quantum Science of the Harvard-MPQ Center for Quantum Optics and AFOSR-MURI Quantum Phases of Matter (grant number FA9550-14-1-0035) (Y.W.), the Swiss National Science Foundation (C.R., W.W.) and The Villum Foundation (M.S.R.).

## Author information

B.W., M.S.R., E.D. and L.M.K.V. had equal contribution in conceptualization. Experimental investigation and methodology was performed with equal contribution from J.P.D. and U.M. Theoretical investigation was led by J.P.D., V.P.M. and B.W. (extended Hubbard models), and Y.W. (ab initio model). J.P.D. led the data curation and software, with support from U.M. and V.P.M. J.P.D. and U.M. had equal contribution in the formal analysis, with support from V.P.M. L.M.K.V. led the funding acquisition and supervision. U.M. led the resources (device fabrication) with support from J.P.D. C.R. and W.W. led the resources (heterostructure growth). J.P.D. led the writing—original draft, with review and editing support from U.M., V.P.M., Y.W., M.S.R., E.D. and L.M.K.V. Please refer to the Casrai Credit Taxonomy for definitions of each of these roles.

Correspondence to L. M. K. Vandersypen.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks Joe Salfi, Igor Zutic 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 Fig. 1 Charge stability diagram of the relevant voltage regions.

a, Measured charge stability diagram showing both point N and point M, as highlighted in Fig. 2a. b, Measured charge stability diagram focusing on the 2001:1101 charge transition, where spin measurements are performed (point M).

### Extended Data Fig. 2 Tuning the gate voltages to the Nagaoka condition using charge stability diagrams.

a, Sample charge stability diagram where we have highlighted the visible interdot transitions, where the electrochemical potentials of two dots become resonant (that is, an electron is allowed to tunnel between the two dots). b, Charge stability diagram similar to a, where we have modified gates P1 and P3 such that the interdot transitions appear at different locations in the diagram. Dashed black lines delimit the regions with a fixed total electron occupation in the system. c, In this diagram, gates P1 and P3 have been tuned to observe the Nagaoka condition, where the three visible interdot transitions are aligned in the three-electron configuration. The intersite interaction in the system provides an effective isolation from the reservoirs for a narrow range of gate voltages, such that the system can remain stable with three electrons in the resonant configuration.

### Extended Data Fig. 3 Effects of spin-coupling mechanisms.

Calculated spectra of the system in the region of pε close to the level crossing of the s = 1/2 and s = 3/2 energies, comparing the effects of different mechanisms for spin coupling. a, Spectrum without any spin coupling effects. b, Spectrum including only spin–orbit coupling (SOC) effects. c, Sample spectrum with both spin–orbit and hyperfine induced Overhauser field gradients, using a single combination of hNa fields (as defined in the Supplementary Methods) selected from a normal distribution with standard deviation δN = 73 neV. The Supplementary Methods describe the implementations of these spin-coupling terms in the theoretical model.

### Extended Data Fig. 4 Characterization of the Nagaoka condition.

a, Average PT in the detuning region 1.00 < pε < 1.01 for 40 values of τramp within the same range shown in Fig. 3. Solid lines are fits using the time evolution simulations described in the Supplementary Methods, for different values of distance $$\ell$$ between neighbouring dots. Inset shows the unscaled results of the time-evolution simulations, where the probability of s = 3/2 is the sum of the lowest four eigenstate probabilities from the final evolved state. b, Thermal relaxation measurements. PT is measured for increasing wait times at point N, for diabatic (blue) and adiabatic (red) passages. Solid lines are exponential fits as guide to the eye.

### Extended Data Fig. 5 Ab initio simulations from 2D to 1D.

a, Schematic of the methodology used in the ab initio simulations to reproduce the effect of the four-dot system transition from a 2D plaquette to a 1D chain. We gradually vary the angle θ, which effectively varies the distance between two of the dots. b, c, The ground-state energy and spin configuration (b) and the ferromagnetic to low-spin energy gap ΔE as a function of θ (c). The ground state soon becomes a low-spin state for the rotating angle at 0.3°.

### Extended Data Fig. 6 Ab initio simulations for local energy offsets.

a, Schematic of the methodology used in the ab initio simulations to reproduce the effect of a local energy offset. The amplitude of the potential V of one of the quantum wells is changed by an amount dV. The variation of the single-well potential by positive or negative dV gives unbalanced site energies. Besides, with the change of eigenstate basis, the hybridization and interaction parameters are also affected in the ab initio calculation. b, c, The ground-state energy and spin configuration (b) and the ferromagnetic to low-spin energy gap ΔE as a function of dV (c). When the potential detuning is dV = 0.11 meV or dV = −0.07 meV, the system undergoes a transition to a low-spin ground state. The transitions at these two directions have a different nature, as drawn in the insets. For dV > 0, the particular quantum dot is deeper and tends to trap more electrons. However, a negative dV raises the energy cost on the particular quantum well and leads to a lower probability of occupation in a three-electron system. Without the ‘mobile’ hole in the ‘half-filled’ system, the ground state becomes a low-spin state instead of a Nagaoka ferromagnetic state.

### Extended Data Fig. 7 Local energy offsets on all dots.

Same measurement as in Fig. 6, applying the ±50 μeV offset on each of the four dots. Panels correspond to offsets in dots 1 to 4, clockwise from the top left. Note that the asymmetry in the plots is related to the fact that the local energies at point M are in an asymmetric detuning configuration and we pulse linearly from this configuration to point N. As expected, the simulated energies of the different spin states at point N (pε = 1), are the same in all four plots.

### Extended Data Fig. 8 Large local offsets.

Each pair of panels show experimental measurements (left) and simulated spectra (right), where point N has been redefined such that the chemical potential of dot 1 is offset by the amount shown on the top right of each panel. Green crosses highlight the detuning points used to obtain the values in Fig. 6b. For experimental plots, these points where obtained using a peak-finding algorithm (local maxima by simple comparison with neighbouring values); for simulated plots, the points correspond to the energy-level crossings.

## Supplementary information

### Supplementary Information

This file contains Extended Fermi-Hubbard models used to simulate different experiments in the main text, Ab initio exact diagonalisation simulations of the 2×2 plaquette and Mapping 3-spin states onto 2-spin measurements.

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Dehollain, J.P., Mukhopadhyay, U., Michal, V.P. et al. Nagaoka ferromagnetism observed in a quantum dot plaquette. Nature 579, 528–533 (2020). https://doi.org/10.1038/s41586-020-2051-0

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