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Beini Gao, Daniel G. Suárez-Forero, Supratik Sarkar, Tsung-Sheng Huang, Deric Session, Mahmoud Jalali Mehrabad, Ruihao Ni, Ming Xie, Pranshoo Upadhyay, Jonathan Vannucci, Sunil Mittal, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Atac Imamoglu, You Zhou, Mohammad Hafezi

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[Excitonic Mott insulator in a Bose-Fermi-Hubbard system of moiré WS2/WSe2 heterobilayer](https://mdr.nims.go.jp/datasets/d29fa79a-0454-4009-a5aa-7e8f74045ac2)

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Excitonic Mott insulator in a Bose-Fermi-Hubbard system of moiré WS2/WSe2 heterobilayerArticle https://doi.org/10.1038/s41467-024-46616-xExcitonic Mott insulator in a Bose-Fermi-Hubbard system of moiré WS2/WSe2heterobilayerBeini Gao 1,8, Daniel G. Suárez-Forero 1,8 , Supratik Sarkar 1,8,Tsung-ShengHuang1, Deric Session1,Mahmoud JalaliMehrabad 1, RuihaoNi 2,Ming Xie 3, Pranshoo Upadhyay1, Jonathan Vannucci 1, Sunil Mittal1,Kenji Watanabe 4, Takashi Taniguchi 4, Atac Imamoglu 5, You Zhou 2,6 &Mohammad Hafezi 1,7Understanding the Hubbard model is crucial for investigating various quan-tum many-body states and its fermionic and bosonic versions have been lar-gely realized separately. Recently, transition metal dichalcogenidesheterobilayers have emerged as a promising platform for simulating the richphysics of the Hubbard model. In this work, we explore the interplay betweenfermionic and bosonic populations, using a WS2/WSe2 heterobilayer devicethat hosts this hybrid particle density. We independently tune the fermionicand bosonic populations by electronic doping and optical injection ofelectron-hole pairs, respectively. This enables us to form strongly interactingexcitons that are manifested in a large energy gap in the photoluminescencespectrum. The incompressibility of excitons is further corroborated byobserving a suppression of exciton diffusion with increasing pump intensity,as opposed to the expected behavior of a weakly interacting gas of bosons,suggesting the formation of a bosonic Mott insulator. We explain our obser-vations using a two-band model including phase space filling. Our systemprovides a controllable approach to the exploration of quantum many-bodyeffects in the generalized Bose-Fermi-Hubbard model.The rich physics of the Hubbard model has brought fundamentalinsights to the study of many-body quantum physics1. Initially proposedfor electronson a lattice, different fermionic andbosonic versions of thismodel have been simulated in various platforms, ranging from ultracoldatoms2 to superconducting circuits3. Recently, bilayer transition metaldichalcogenides (TMDs) have become a versatile platform to study theHubbard model thanks to the coexistence of several intriguing featuressuch as the reduction of electron hoppingdue to the formation ofmoirélattice with large lattice constant, and the existence of both intra- andinterlayer excitons. These characteristics have enabled the realizationof numerous effects of many-body physics such as metal-to-Mott insu-lator transition4–9, generalized Wigner crystals10–14, exciton–polaritonswith moiré-induced nonlinearity15, stripe phases16, light-inducedferromagnetism17. Moreover, there have been recent exciting perspec-tives of exploring such effects in light–matter correlated systems3,18,19.While typically the fermionic and bosonic versions of the HubbardReceived: 30 May 2023Accepted: 4 March 2024Check for updates1Joint Quantum Institute (JQI), University of Maryland, College Park, MD, USA. 2Department of Materials Science and Engineering, University of Maryland,College Park, MD, USA. 3Condensed Matter Theory Center, University of Maryland, College Park, MD, USA. 4National Institute for Materials Science,Tsukuba, Japan. 5Institute for Quantum Electronics, ETH Zurich, Zurich, Switzerland. 6Maryland QuantumMaterials Center, College Park, MD, USA. 7Institutefor Theoretical Physics, ETH Zurich, Zurich, Switzerland. 8These authors contributed equally: Beini Gao, Daniel G. Suárez-Forero, Supratik Sarkar.e-mail: dsuarezf@umd.edu; hafezi@umd.eduNature Communications |         (2024) 15:2305 11234567890():,;1234567890():,;http://orcid.org/0000-0002-4544-4962http://orcid.org/0000-0002-4544-4962http://orcid.org/0000-0002-4544-4962http://orcid.org/0000-0002-4544-4962http://orcid.org/0000-0002-4544-4962http://orcid.org/0000-0002-2757-6320http://orcid.org/0000-0002-2757-6320http://orcid.org/0000-0002-2757-6320http://orcid.org/0000-0002-2757-6320http://orcid.org/0000-0002-2757-6320http://orcid.org/0000-0003-2645-2307http://orcid.org/0000-0003-2645-2307http://orcid.org/0000-0003-2645-2307http://orcid.org/0000-0003-2645-2307http://orcid.org/0000-0003-2645-2307http://orcid.org/0000-0002-9809-9998http://orcid.org/0000-0002-9809-9998http://orcid.org/0000-0002-9809-9998http://orcid.org/0000-0002-9809-9998http://orcid.org/0000-0002-9809-9998http://orcid.org/0000-0001-9923-8809http://orcid.org/0000-0001-9923-8809http://orcid.org/0000-0001-9923-8809http://orcid.org/0000-0001-9923-8809http://orcid.org/0000-0001-9923-8809http://orcid.org/0000-0002-7836-7967http://orcid.org/0000-0002-7836-7967http://orcid.org/0000-0002-7836-7967http://orcid.org/0000-0002-7836-7967http://orcid.org/0000-0002-7836-7967http://orcid.org/0000-0002-7177-5431http://orcid.org/0000-0002-7177-5431http://orcid.org/0000-0002-7177-5431http://orcid.org/0000-0002-7177-5431http://orcid.org/0000-0002-7177-5431http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-0641-1631http://orcid.org/0000-0002-0641-1631http://orcid.org/0000-0002-0641-1631http://orcid.org/0000-0002-0641-1631http://orcid.org/0000-0002-0641-1631http://orcid.org/0000-0002-9854-545Xhttp://orcid.org/0000-0002-9854-545Xhttp://orcid.org/0000-0002-9854-545Xhttp://orcid.org/0000-0002-9854-545Xhttp://orcid.org/0000-0002-9854-545Xhttp://orcid.org/0000-0003-1679-4880http://orcid.org/0000-0003-1679-4880http://orcid.org/0000-0003-1679-4880http://orcid.org/0000-0003-1679-4880http://orcid.org/0000-0003-1679-4880http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-46616-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-46616-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-46616-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-46616-x&domain=pdfmailto:dsuarezf@umd.edumailto:hafezi@umd.edumodel are explored independently, combining these two models in asingle system holds intriguing possibilities for studying mixed bosonic-fermionic correlated states20,21.In this work, we demonstrate Bose–Fermi–Hubbard physics in aTMD heterobilayer. We independently control the population of fer-mionic (electronic) particles by doping with a gate voltage (Vg), and thepopulation of bosonic (excitonic) states by pumping with a pulsedoptical drive of intensity I. Harnessing these two control methods, werealize strongly interacting excitons. In particular, we show the incom-pressibility of excitonic states near integer filling by observing an energygap in photoluminescence, accompanied by an intensity saturation.Remarkably,weobserve the suppressionofdiffusion, a strong indicationof the formation of a bosonic Mott insulator of excitons.ResultsPhysical system and experimental designTo demonstrate these effects, we use amoiré lattice created by stackingtwomonolayersofWS2andWSe2,with symmetric topandbottomgates.Figure 1a shows a schematic illustration of the heterobilayer device (seeSupplementary Note 1 for details). Due to the type-II band alignment ofthe heterostructure (Fig. 1b), negative doping results in a population ofelectrons in the WS2 subject to the moiré potential of the bilayer. Theratio between the density of this population and the density of moirésites in the structure determines the so-called electronic filling factor(νe). The optical pump results in the formation of an energeticallyfavorable interlayer exciton (X)22, by pairing between an electron inWS2and a hole in WSe2 (represented in Fig. 1b). In order to explore differentregimes of Bose–Fermi–Hubbard model, we control the bosonic andfermionic populations by changing I and Vg, respectively. This can becompared to the ultracold atom implementation of Bose–Fermimixturewhere the respective populations are fixed in each experiment23. Beforediscussingour experimental observation,wediscuss three limiting casesthat determine the phase space of our system, as indicated in Fig. 1c. Thecorresponding physical scenarios are represented in panels d–f. First, inthe weak excitation limit and low electronic filling factor (νe ~0) regime,the system’s photoluminescence (PL) emission originates exclusivelyfrom the few X states in the quasi-empty lattice (panel d). This emissioncomes from excitons in lattice sites where they are the only occupantparticles, namely, “single occupancy states” (X1). Upon increasing νe, thenumberof singlyoccupied sites decreases, and in the limitingcaseof νe≥1, as represented in panel e, the optically generated excitons can onlyform in lattice sites already occupied by charged particles. In this case,the required energy to form the exciton increases due to the on-siteCoulomb repulsion, and hence the PL emission has new branch withhigher energy than the previous regime. Consequently, the PLoriginatesfrom lattice sites with an electron-exciton double occupancy (X2).Finally, we consider the case where the electronic doping is below thethreshold required to reach a fermionicMott insulator (0 < νe < 1) but I isstrong enough to optically saturate the single-occupancy states. Theextra excitons create a number of sites with electron-exciton orexciton–exciton double occupancies (panel f). In this case, the PLemission corresponds tomixed contributions fromexciton–exciton andexciton–electron interaction (Uex-ex and Uex-e); the individual peakscannot be distinguished in a single spectrum due to the broadness oflinewidths. Therefore, in this regime, the emitted light is only a combi-nation of the X1 and X2 PL emission. This interplay between exciton andelectron occupancy can lead to situations in which the moiré lattice iscompletely filled with a mixed population of fermions and bosons,forming a hybrid incompressible state. Specifically, in the limit of weakelectronic tunneling, excitons can form a Mott insulating state, in theremainder of sites that are not filled by electronic doping. Note the linein Fig. 1c denoting panel f is an asymptote since optical pumping can notfully saturate an exciton line. At νe=0, this intensity is denoted as I* (seeSupplementary Note 8 for details).System’s properties for varying electronic (fermionic)occupationTo experimentally investigate these regimes, we perform PL mea-surements, with varying pumppower and backgate voltage. A detaileddescription of the optical setup canbe found in SupplementaryNote 2.---WS2 WSe2Interlayer excitonb)d)X100 1IntensityI*c)feX2f)X1-+-- -- - -++ + +d-e)X2- - - -+a)WS2WSe2SiO2SiBGTGhBNGraphiteFig. 1 | WS2/WSe2 bilayer as aplatform for correlatedphysics. a Schematic of theWS2/WSe2 dual-gate device. The TMD heterobilayer is embedded between twosymmetric gates: top gate (TG) and bottom gate (BG). b Depiction of the type-IIband alignment of the bilayer. The blue and red curves denote bands fromWS2 andWSe2, respectively. The shaded ellipse indicates the formation of interlayer exci-tons composed of an electron from the WS2 conduction band and a hole from theWSe2 valence band. c Phase diagram of the system. The population of the moirélattice can be controlled via two independent parameters: the gate voltage changesthe electronic filling factor (νe), and the optical pump creates a population ofexcitons, proportional to the input intensity. In the gray area, the systembehaves asa mixed gas of bosonic and fermionic particles. As one approaches the upper limit(black line), the system becomes incompressible due to the saturation of single-occupancy states. d–f Interlayer exciton formation under optical excitation forthree different regimes governed by the pump intensity (I) and νe: c low I and νe ~ 0,d low I and νe ~ 1, e high I and 0 < νe < 1. X1 (X2) denotes PL emission from singly(doubly) occupiedmoiré lattice sites. X2 can originate from either electron-exciton(Uex-e) or exciton–exciton (Uex-ex) double occupancies.Article https://doi.org/10.1038/s41467-024-46616-xNature Communications |         (2024) 15:2305 2We use pulsed excitation to achieve high exciton density while redu-cing thermal effects by keeping low average power. Experiments withCW excitation are consistent with the presented data, as shown inSupplementary Note 6. Figure 2a–c shows the PL dependence at threedifferent intensities as schematically shown in panel d. Figure 2a showsthe normalized doping-dependent PL spectrum for low/(0.08μW/μm2), which corresponds to low bosonic occupation. The fermionicoccupation νe is varied between 0 and 1.1. For low νe, PL emission isdetectedonly fromX1. However, at Vg ≈ 2.98 V,wedetect a transition inthe PL emission to X2. This transition corresponds to the formation ofX’s in the presence of an incompressible fermionic Mott insulator24,25.From the reflectivity measurement and calculations from a capacitormodel, we attribute Vg = 2.98 V to νe = 1 (see Supplementary Note 3).The energy gap between X1 and X2 is ΔE ≈ 29meV, which correspondsto the on-site Coulomb repulsion energy between an electron and anexciton (Uex-e).We elaborate on this energy gap later in the sub-section“Energy map along the phase space”. The dim mid-gap featuresbetween X1 and X2 at νe ~1 are strongly position-dependent and dis-appear at higher power. This indicates that such emission is fromlocalized excitons. Figure 2b shows the PL spectrum under pumpintensity equal to 12.1μW/μm2. It is worth noticing that the Vg at whichthe PL signal fromX2 is detected is lower than in panel a. The system istherefore in the regime depicted in Fig. 1f. Upon further increasing thepump intensity X2 can bedetected even at νe =0, as observed in Fig. 2c.In this case, the PL emission originates solely from double occupancyof excitons in a moiré lattice site, suggesting that, for high I, purelybosonic states of strongly interacting excitons are created. ComparingFig. 2a, c, one can observe that in the former case, the emergence ofthe X2 peak corresponds to a sharp suppression of X1, while in thelatter case, both peaks coexist. This indicates the nature of the doubleoccupancy: in the first scenario, the exciton is forming in the presenceof an electron, and after its recombination, there are no other opticalexcitations in the system. In contrast, the coexistence of both peaks inpanel c shows that upondouble excitonoccupancy, the recombinationof X2 precedes the recombination of X1.From the observation described in the previous paragraph, weconclude that the detection of PL emission with X1 and X2 energiesbenchmarks the formation of exciton states in singly and doublyoccupied lattice sites, respectively. At νe = 0, the X1 peak in Fig. 2c isblueshifted with respect to Fig. 2a. We associate this feature with amean-field effect due to exciton–exciton interaction. As we increasethe electronic doping, fewer sites are available to create X1 excitonsand on those occupied sites, only X2 is created. Consequently, theeffective population of X1 excitons is decreased. Therefore themean-field shift is suppressed to the point that at high filling (νe ~1)the X1 energy is the same as in the case of low pump intensity. Next,in order to understand the interplay between fermionic and bosoniclattice occupancies in each regime, we perform a quantitative ana-lysis of their respective integrated intensity. We extract these valuesfrom the collected PL spectra using a computational fitting method(see Supplementary Note 7 for further details). Figure 2e–g displaysthis intensity dependence on νe for the same I range of panels a–c.We notice that as electrons fill the system’s phase space (uponincreasing Vg), the number of accessible single-occupancy statesdecreases. As a consequence, the integrated intensity of X1 reduceswith increasing νe. Remarkably, for each intensity, there is a criticalνe after which the PL emission of X2 exceeds that of X1. The gatevoltage at which the crossing takes place (V crg ) is highlighted on eachpanel by a vertical dashed line. This line indicates a constant ratiobetween the X1 and X2 populations. The crossing takes place at lowerνe upon increasing I, as expected. In Fig. 2h, we track V crg as a functionof the total collected PL emission, which gives an indication of thetotal number of excitons in both X1 and X2 branches. We observe aclear trend: a higher total population of excitons results in a fastersaturation of the single-occupancy states and hence an increasingnumber of double occupancy states.e)X1X2)s/stnuoC(rewoPLPGate Voltage (V)Normalized spectrumh)Vgcr(V)Gate Voltage (V)f))s/stnuoC(rewoPLPX1X2g)Gate Voltage (V))s/stnuoC(rewoPLPX1X2d)(a)(b)(c)00 1IntensityI*Photon Energy (eV)Gate Voltage (V)Vg=2.98Va)X1X2X1b)Vg=2.6VX2Photon Energy (eV)Gate Voltage (V)Vg=1.95VX1X2c)Photon Energy (eV)Gate Voltage (V)0.5 0.5 0.50.5 0.5 0.5Total PL Power (Counts/s)0.60.40.2604020Fig. 2 | System’s properties for increasing electronic (fermionic) occupation.a–c Normalized PL spectrum as a function of gate voltage (νe) for three differentpump intensities: I =0.08μWμm2 (a), I = 12.1μW/μm2 (b) and I = 1229μW/μm2 (c).The peaks associated with single (X1) and double (X2) occupancy are indicated oneach panel. The dashed lines indicate the gate voltages at which the PL intensity X2exceedsX1. The dashed black lines of (d) indicate themeasurement ranges of (a–c).e–g Evolution of the PL intensity for X1 (red) and X2 (blue) as a function of gatevoltage for the same values of pump intensities displayed in (a–c). The electronfilling factor at which X2 exceeds X1 decreases as pump intensity increases. h showsthe gate voltage at which the intensity of X2 exceeds that of X1, as a function of thetotal PL intensity. The error bars represent the standard errors for the parameterestimates in the fitting routine.Article https://doi.org/10.1038/s41467-024-46616-xNature Communications |         (2024) 15:2305 3System’s properties for varying excitonic (bosonic) occupationNext, in order to trace the role of the optical pump and the opticalsaturation that leads to the formation of incompressible bosonic states,we investigate the PL for varying I for different νe. In Fig. 3a–c, we focuson three different values of νe, as indicated in panel d, and study the PLspectrum for increasing emitted PL power. For zero fermionic occu-pancy (panel a), X2 contributes to the emission only at very high total PLemission intensity. In panels b and c, we increase the electronic dopingto νe=0.7 and νe=0.95, respectively, and as expected, the total PL atwhich we detect X2 decreases. In the low-power region, panel c showsthe PL emission frommid-gap states also observed in Fig. 2a. Apart fromthe energy gap in the emission, we observe a blueshift of the X1 line withincreasing PL power. Assuming the weak tunneling regime, this shiftshould be equal to Uex-exhx̂yx̂i, where x̂y is the creation operator of anexciton. For example, in Fig. 3b for total PL power at 2 counts/s, thebosonic occupation is hx̂yx̂i ’ 0:2. This corroborates with the energygap that occurs at 10 counts/s for an estimated unity filling (hx̂yx̂i ’ 1).We present a fully quantum theoretical analysis of this observation inSupplementary Note 10. Panels e–g show the intensities of X1 and X2 forthe values of νe in panels a–c. As expected, in panel e, one can observethat the intensity of the X1 PL emission increases monotonically, and itstarts to saturateonly at very high total PLemission regimes.Uponfillingthe moiré lattice with one exciton or one electron per site, the X1 PLintensity saturates. With higher νe, the saturation occurs at lower I, asshown in panels f and g. Since this saturation corresponds to filling thesingle-occupancy states, we associate it with the establishment of anincompressible bosonic Mott insulator. Note that this bosonic Mottinsulator is in a drive-dissipative regime, similar to the demonstration insuperconducting qubit systems26.To quantitatively analyze this saturation effect, we fit the X1 PLpower (P1) to the function P1 = Pmax1PP +Psat, where P is the total PLpower. From the fitting, we extract Pmax1 which is the asymptotic valueof the X1 emitted PL power, and Psat which determines the total PL ofsaturation. This functional form corresponds to the expected systembehavior when the charge gap U is sufficiently large to permit theutilization of a phase-space filling model to treat both single anddouble occupancy states (details in Supplementary Note 8). Figure 3fincludes an example of the fitting function (dashed black line).According to our model, the value of Psat should decrease withincreasing νe because a lower excitonic population is required toachieve the incompressible states. The compiled data for the full rangeof νe, shown in panel h with brown marks, is in good agreement withthe expected trend. Fromthemodel, we canalso infer that thequantityPsat=Pmax1 should be independent of the electronic doping levelbecause both quantities depend linearly on 1 − νe; higher electronicoccupancy implies less single-occupancy states available to host anexciton. The greenmarks in Fig. 3h represent this behavior, which is ingood agreement with the model. We conclude that the saturation ofsingle-occupancy states is directly reflected in the intensity of X1,enabling the extraction of the conditions under which the incom-pressible states occur. Importantly, this enables a direct calibration ofthe bosonic and fermionic fractions in the system.Exciton diffusion measurementsIn order to further validate the incompressible nature of excitonicstates,weperformdiffusionmeasurements of the interlayer excitons27.For a steady population of excitons createdby a continuous-wave laserpump, the diffusion length carries information about the nature of thestate: an incompressible bosonic state is expected to have a lowerdiffusion length than a weakly interacting one. We spatially image thediffusion pattern with spectral resolution and extract the diffusionh)Normalized spectrumPsat /P1maxPsat (Counts/s x100)PL Power (Counts/s)Total PL Power (Counts/s)e)X1X2PL Power (Counts/s)g)X1X2Vg(V)(a) (b) (c)00 1IntensityI*d)PL Power (Counts/s)Fitting functionf)X1X2Photon Energy (eV)1.41.451.51.55X1X2b)0.5 1 5 10X1X2Photon Energy (eV)a)1.41.451.51.550.5 1 5 10 50X1X2c)0.1 0.5 1 5 10Photon Energy (eV)1.41.451.51.552Total PL Power (Counts/s)Total PL Power (Counts/s) Total PL Power (Counts/s)Total PL Power (Counts/s) Total PL Power (Counts/s)604020Fig. 3 | System’s properties for increasing excitonic (bosonic) occupation.a–cNormalized PL spectrum as a function of the total collected PL power for threedifferent electronic filling factors. The peaks associated with single (X1) and double(X2) occupancy are indicated on each panel.d indicates the ranges of I and νe for themeasurements shown in (a–c). e–g evolution of the PL power for X1 (red) and X2(blue) as a function of the total collected PL power for the same values of νedisplayed in (a–c). f displays the fitting function (dashed black line) employed toextract Psat and Pmax1 (described in the text). h Evolution of Psat (brown) as afunctionof the gate voltage (νe). As expected fromour phase-spacefillingmodel, itsvalue reduces with increasing filling factor. The quantity Psat=Pmax1 (green) showsgood agreement with the theoretical model. The error bars represent the standarderrors for the parameter estimates in the fitting routine.Article https://doi.org/10.1038/s41467-024-46616-xNature Communications |         (2024) 15:2305 4length (LX1) of the single-occupancy excitons. The choice of LX1 as anappropriate quantity to benchmark the incompressibility of bosonicMott insulating states, assumes a constant exciton lifetime with vary-ing population. This is supported by previous reports in the literaturethat show the independence of this quantity over three orders ofmagnitude of pumping power28. The downward diffusion image haspatterns that originate in the inhomogeneous surface of the bilayer.Although the inhomogeneities on that side hinder the extraction of LX1,the optically induced suppression of the diffusion length for constantνe can be clearly observed in this region (Fig. 4a, b). The populationinjected at y =0 (dotted line) propagates, and its emission pattern ismonitored along a range of 5μm (dashed rectangle). The color scale isthe same for both panels. Panel b shows a reduction of the diffused X1population in comparison to panel a. For the quantitative analysis ofthis observation, it is necessary to use a fitting routine, for which thesmooth pattern on top of the injection point (y <0) is more reliable.Figure 4c shows the extracted LX1 as a functionofVg for different pumpintensities from the exponentially decaying spatial diffusion pattern inthis region.We providemore details about the analysis of the diffusiondata in Supplementary Note 9. For low electronic density, the excitondiffusion length increases as the power is augmented. This trend,highlighted by the upward arrow, is in agreement with the expectedbehavior for weakly interacting bosons28,29. Remarkably, as the elec-tronic filling factor increases, the trend completely inverts (inset). Thisis a direct signature of the bosonic Mott insulator formation since thebulk is incompressible and the melting only occurs at the edge.Energy map along the phase spaceThe implemented fitting algorithm allows us to track the changes inthe energy of both species of excitons and the energy gap betweenthem. These results are presented in Fig. 5. Panels a and b show thecentral energies of the peaks X1 and X2 in the space of parameters forwhich each peak is detectable. In the range where both of them canbe detected, their energy difference ΔE (panel c) provides importantinformation about the nature of the interactions taking place in thesystem. In the case of low electronic occupancy and high excitondensity (top left corner of the panel), ΔE corresponds to theexciton–exciton interaction gap (Uex-ex ~32meV). Conversely, athigh νe and low exciton density (bottom right corner), this gapdepends on the exciton–electron interaction (Uex-e ~27meV). Thegradual change in the nature of the interactions taking place in thesystem along the parameters space is reflected in the change of ΔE.Interestingly, the largest energy gap takes place for states with highoccupation of bosons and fermions (top right corner), which isconsistent with a blueshift of the X2 PL peak due to the high popu-lation of excitons with large Bohr radius repelling through dipolarinteraction.DiscussionIn summary, we demonstrated a Mott insulating state of excitons in ahybrid Bose–Fermi Hubbard system formed in a TMD heterobilayer.While our incompressibility observation was based on spatiallyresolved diffusion in the steady-state limit, one can explore interestingnon-equilibriumphysics due to the relatively long lifetime of interlayerexcitons. More generally, spatiotemporally resolved measurements,combined with independent tunability of fermionic and bosonicpopulations, make it possible to investigate the equilibrium and non-equilibrium physics of Bose–Fermi mixtures. Moreover, a quantummicroscopic model capable of fully describing such a driven-dissipative Bose–Fermi mixture remains an open area of research.The novel experimental diffusion method used to benchmark theexcitonic incompressibility opens exciting perspectives for the simu-lation of complexdynamics inmany-bodyquantumsystems that rangefrom a single bosonic particle in a Fermi sea to a strongly interactinggas of bosons. Particularly intriguing examples are the optical inves-tigation of charge and spin physics in integer and fractional fillings,e.g., Mott excitons30,31 or spin liquids32–35, and fractional Cherninsulators36,37.MethodsDevice fabricationThe WSe2/WS2 heterostructure was fabricated using a dry-transfermethodwith a stampmade of a poly(bisphenol A carbonate) (PC) layeron polydimethylsiloxane (PDMS). All flakes were exfoliated from bulkcrystals onto Si/SiO2 (285 nm) and identified by their optical contrast.The top/bottom gates and TMD contact are made of few-layer gra-phene. The PC stamp and samples were heated to 60 °C during thepick-up steps and released from the stamp to the substrate at 180 °C.The PC residue on the device was removed in chloroform, followed bya rinse in isopropyl alcohol and ozone clean. Sample transfer wasperformed in an argon-filled glovebox for improved interface quality.The electrodes consist of 3.5 nmof chromium and 70 nmof gold. Theywere fabricated using standard electron-beam lithography techniquesand thermal evaporation.LX1X2X1a)X1 diffusion052.5-2.5ExcitationSpotExcitationSpotb)c)X1 diffusionX2X1052.5-2.5Increasing ILonger diffusionNeutral regionIncreasing IShorter diffusionCounts/s (x100)5101520Fig. 4 | Exciton diffusion and incompressibility. Spectrally and spatially resolveddiffusion pattern at νe =0:73ðVg = 2:34V) for low (a) and high (b) I. The dashedrectangle highlights the regionwhere the suppression of diffusion canbe observed.c Exciton diffusion length as a function of the gate voltage for a range of νe and fordifferent input intensities. For low νe, the diffusion length increases with I due toexciton repulsive interaction. Upon further filling the moiré lattice, the trendinverts, indicating the optical realization of incompressible states. The inset is azoom-in of the reddotted rectangle to highlight the reductionof LX1 with increasingI. The error bars represent the standard errors for the diffusion length estimatedfrom the exponential fitting.Article https://doi.org/10.1038/s41467-024-46616-xNature Communications |         (2024) 15:2305 5Optical measurementsThe sample is kept in a dilution refrigerator at a temperature of 3.5 K.For PL measurements, we use a confocal microscopy setup. Ourpumping source is a pulsed Ti:Sapphire laser tuned at 720 nm(1.722 eV), with a pulse duration of 100 fs and a repetition rate of~80MHz. In addition, an optical chopper system at 800Hz is used toprevent sample heating while having a high pump intensity. The resi-dual pump is removed with a spectral filter before collecting the PLemission in a spectrometer-CCD camera device. A complete descrip-tion of the setup is presented in the Supplementary Note 2.For the diffusionmeasurements,weused a continuous-wave (CW)laser. The rest of the optical measurement setup was similar.Data availabilityThe PL and diffusion data generated in this study have been depositedin the Figshare database under accession links: https://doi.org/10.6084/m9.figshare.25246012.v1; https://doi.org/10.6084/m9.figshare.25246006.v1; https://doi.org/10.6084/m9.figshare.25246009.v1;https://doi.org/10.6084/m9.figshare.25246015.v1 .References1. Hubbard, J. Electron correlations in narrow energy bands. Proc.Royal Soc. Lond. Ser. A. Math. Phys. Sci. 276, 238–257 (1963).2. Greiner, M., Mandel, O., Esslinger, T., Hänsch, T. W. & Bloch, I.Quantum phase transition from a superfluid to a Mott insulator in agas of ultracold atoms. Nature 415, 39–44 (2002).3. Carusotto, I. et al. Photonic materials in circuit quantum electro-dynamics. Nat. Phys. 16, 268–279 (2020).4. Ghiotto, A. et al. Quantum criticality in twisted transition metaldichalcogenides. Nature 597, 345–349 (2021).5. Li, T. et al. 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Rev.B 106, 094417 (2022).E1 (eV)10010210-110-210-3101a)E2 (eV)810010210-110-210-3101b)E (eV)Uex-exUex-e10010210-110-210-3101c)1.461.471.451.441.43Fig. 5 | X1 andX2 exciton energiesalong thephasediagram. Energy of theX1 (a) andX2 (b) PL emission as a functionof gate voltage andpump intensities. Thewhite areascorrespond to the range of parameters where the corresponding peak completely vanishes. When X1 and X2 coexist, we extract the energy difference, as shown in (c).Article https://doi.org/10.1038/s41467-024-46616-xNature Communications |         (2024) 15:2305 6https://doi.org/10.6084/m9.figshare.25246012.v1https://doi.org/10.6084/m9.figshare.25246012.v1https://doi.org/10.6084/m9.figshare.25246006.v1https://doi.org/10.6084/m9.figshare.25246006.v1https://doi.org/10.6084/m9.figshare.25246009.v1https://doi.org/10.6084/m9.figshare.25246015.v1https://arxiv.org/abs/2111.09440https://arxiv.org/abs/2111.0944034. Rademaker, L. Spin-orbit coupling in transition metal dichalco-genide heterobilayer flat bands. Phys. Rev. B 105, 195428 (2022).35. Kiese, D., He, Y., Hickey, C., Rubio, A. & Kennes, D. M. Tmds as aplatform for spin liquid physics: a strong coupling study of twistedbilayer WSe2. APL Mater. 10, 031113 (2022).36. Li, H., Kumar, U., Sun, K. & Lin, S.-Z. Spontaneous fractional Cherninsulators in transition metal dichalcogenide moiré superlattices.Phys. Rev. Res. 3, L032070 (2021).37. Crépel, V. & Fu, L. Anomalous hall metal and fractional cherninsulator in twisted transition metal dichalcogenides. Phys. Rev. B107, L201109 (2023).AcknowledgementsThe authors acknowledge fruitful discussions with N. Schine and A.Kollar. This work was supported by AFOSR FA95502010223, MURIFA9550-19-1-0399, FA9550-22-1-0339, NSF IMOD DMR-2019444, ARLW911NF1920181, and Simons and Minta Martin foundations. Ming Xie issupported by the Laboratory for Physical Sciences. R. Ni and Y. Zhou aresupportedby theU.S. Department of Energy,Officeof Science,OfficeofBasic Energy Sciences Early Career Research Program under Award No.DE-SC-0022885.Author contributionsB.G., D.G.S.F., S.S. and M.H. conceived and designed the experiments.K.W. and T.T. supplied the necessary material for the fabrication of thesample. B.G., D.S. and R.N. designed and fabricated the sample. J.V. andS.M. collaborated with the preparation of the setup at its initial stage.B.G., D.G.S.F. and S.S. performed the experiments. B.G., D.G.S.F., S.S.,T.S.H., M.J.M., M.X., A.I., Y.Z. andM.H. analyzed the data and interpretedthe results. T.S.H. and M.H. elaborated on the theoretical models pre-sented in the manuscript. B.G., D.G.S.F., S.S., M.J.M. and M.H. wrote themanuscript, with input from all authors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-46616-x.Correspondence and requests for materials should be addressed toDaniel G. Suárez-Forero or Mohammad Hafezi.Peer review information Nature Communications thanks the anon-ymous reviewers for their contribution to the peer review of this work. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2024Article https://doi.org/10.1038/s41467-024-46616-xNature Communications |         (2024) 15:2305 7https://doi.org/10.1038/s41467-024-46616-xhttp://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Excitonic Mott insulator in a Bose-Fermi-Hubbard system of moiré WS2/WSe2 heterobilayer Results Physical system and experimental�design System’s properties for varying electronic (fermionic) occupation System’s properties for varying excitonic (bosonic) occupation Exciton diffusion measurements Energy map along the phase�space Discussion Methods Device fabrication Optical measurements Data availability References Acknowledgements Author contributions Competing interests Additional information