Abstract
The Hubbard model constitutes one of the most celebrated theoretical frameworks of condensed-matter physics. It describes strongly correlated phases of interacting quantum particles confined in lattice potentials1,2. For bosons, the Hubbard Hamiltonian has been deeply scrutinized for short-range on-site interactions3,4,5,6. However, accessing longer-range couplings has remained elusive experimentally7. This marks the frontier towards the extended Bose–Hubbard Hamiltonian, which enables insulating ordered phases at fractional lattice fillings8,9,10,11,12. Here we implement this Hamiltonian by confining semiconductor dipolar excitons in an artificial two-dimensional square lattice. Strong dipolar repulsions between nearest-neighbour lattice sites then stabilize an insulating state at half filling. This characteristic feature of the extended Bose–Hubbard model exhibits the signatures theoretically expected for a chequerboard spatial order. Our work thus highlights that dipolar excitons enable controlled implementations of boson-like arrays with strong off-site interactions, in lattices with programmable geometries and more than 100 sites.
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Acknowledgements
C.L. and F.D. thank S. Suffit for support during sample fabrication, together with A. Reserbat-Plantey and B. Urbaszek for a critical reading of the manuscript. Work at CNRS was funded by IXTASE from the French Agency for Research (ANR-20-CE30-0032-01). The work at Princeton University (L.P. and K.B.) was funded by the Gordon and Betty Moore Foundation through the EPiQS initiative grant GBMF 9545, and by the National Science Foundation MRSEC grant DMR 1420541. Research at ICFO acknowledges support from ERC AdG NOQIA; Agencia Estatal de Investigación (R&D project CEX2019-000910-S, funded by MCIN/ AEI/10.13039/501100011033, Plan National FIDEUA PID2019-106901GB-I00, FPI, QUANTERA MAQS PCI2019-111828-2, QUANTERA DYNAMITE PCI2022-132919, Proyectos de I+D+I "Retos Colaboración” QUSPIN RTC2019-007196-7); MCIN via European Union NextGenerationEU (PRTR); Fundació Cellex; Fundació Mir-Puig; Generalitat de Catalunya through the European Social Fund FEDER and CERCA programme (AGAUR grant no. 2017 SGR 134, QuantumCAT U16-011424, co-funded by ERDF Operational Programme of Catalonia 2014-2020); the computer resources and technical support at Barcelona Supercomputing Center MareNostrum (FI-2022-1-0042); EU Horizon 2020 FET-OPEN OPTOlogic (grant no 899794); National Science Centre, Poland (Symfonia grant no. 2016/20/W/ST4/00314); European Union’s Horizon 2020 research and innovation programme under the Marie-Skłodowska-Curie grant agreement no. 101029393 (STREDCH) and no. 847648 (‘La Caixa’ Junior Leaders fellowships ID100010434: LCF/BQ/PI19/11690013, LCF/BQ/PI20/11760031, LCF/BQ/PR20/11770012, LCF/BQ/PR21/11840013). R.W.C. acknowledges support from the Polish National Science Centre (NCN) under Maestro grant no. DEC2019/34/A/ST2/00081.
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K.B. and L.P. realized the GaAs bilayer and C.L. and F.D. designed and fabricated the gate electrodes to realize the 250 nm period electrostatic lattice. C.L. and F.D. performed all experiments and data analysis. C.L., U.B., T.G, R.W.C., T.S., M.L., M.H. and F.D. contributed to the theoretical developments. All authors contributed to writing the manuscript. F.D. directed the project.
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Extended data figures and tables
Extended Data Fig. 1 Theoretical hallmarks of CB order.
a, Lowest band structure factor S1(k) at T=100 mK obtained by exact diagonalisation of a 8 site square lattice (Betts cluster) with periodic boundary conditions. It exhibits a dominant peak at quasi-momentum k = (π/a, π/a), which is a characteristic signature of CB order. A second strongly suppressed quasi-peak lies at k = (0, 0) (due to finite size effects), corresponding to a homogenous liquid without any density order. b, ∣S1(π/a, π/a)∣ (black) and ∣S1(0, 0)∣ (blue) are plotted versus temperature T. Up to T ≲ Tc = 420 mK, the structure factor signalling CB order remains at least twice as large as the structure factor for a homogeneous liquid. c, CB order parameter deduced from mean-field calculations as a function of the chemical potential μ and temperature (T = 4, 125, 247, 389, 450 mK in blue, violet, black, red and green respectively). The order parameter is given by the population difference ∣nA − nB∣ between two sub-lattices, A and B, of the square lattice. Below around 410 mK ∣nA − nB∣ is significant manifesting CB order.
Extended Data Fig. 2 Spatially resolved PL intensity and intensity fluctuations.
a, Spatial variations of the PL intensity \(\overline{{A}_{\max }}\) (black line) and \(\sigma ({A}_{\max })/\overline{{A}_{\max }}\) (violet bars) measured at T = 330 mK and P = 17 nW, that is for the MI phase. Both \(\overline{{A}_{\max }}\) and \(\sigma ({A}_{\max })/\overline{{A}_{\max }}\) vary weakly in the 3 μm central region of the laser excited region, evidencing that the MI phase is homogeneous across more than 100 lattice sites. Outside this region we note that \(\sigma ({A}_{\max })/\overline{{A}_{\max }}\) increases steeply while \(\overline{{A}_{\max }}\) drops, which signals that excitons realise a normal fluid. b, Same measurements obtained for P = 8.2 nW, that is for the CB phase. Results are extracted from the experiments reported in Fig. 2.
Extended Data Fig. 3 Exciton compressibility versus average lattice filling.
Fluctuations of the maximum of the PL intensity (\(\sigma ({A}_{\max })/\overline{{A}_{\max }}\)) as a function of the power P of the loading laser, in a different region of our two-dimensional square lattice. As for Fig. 2, experimental results are obtained by statistically analysing a series of 10 measurements for every value of P. The laser excitation profile was set close to the one for the experiments shown in Fig. 2. Remarkably we recover that two insulating phases emerge for P = 7 and 14.4 nW, in good agreement with the findings discussed in the main text. Experiments were realised at T = 330 mK; error bars display statistical confidence while the level of Poissonian fluctuations is given by the grey shaded region.
Extended Data Fig. 4 Residuals at \(\bar{n}=\) 1/2 and 1.
a, PL spectrum measured at \(\bar{n}\)=1/2 (top) together with the modelled profile (black line). The bottom panel displays the residuals between modelled and measured profiles (black line), compared to the amplitude of poissonian fluctuations (grey area). b, Same measurements for \(\bar{n}\)= 1. Experimental results are taken from the data reported in Fig. 2b–d.
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Lagoin, C., Bhattacharya, U., Grass, T. et al. Extended Bose–Hubbard model with dipolar excitons. Nature 609, 485–489 (2022). https://doi.org/10.1038/s41586-022-05123-z
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DOI: https://doi.org/10.1038/s41586-022-05123-z
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