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Richen Xiong, Samuel L. Brantly, Kaixiang Su, Jacob H. Nie, Zihan Zhang, Rounak Banerjee, Hayley Ruddick, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Seth Ariel Tongay, Cenke Xu, Chenhao Jin

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[Tunable exciton valley-pseudospin orders in moiré superlattices](https://mdr.nims.go.jp/datasets/51285ea0-3f9d-4394-9c49-85a3e62b4e9d)

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Tunable exciton valley-pseudospin orders in moiré superlatticesArticle https://doi.org/10.1038/s41467-024-48725-zTunable exciton valley-pseudospin orders inmoiré superlatticesRichen Xiong1, Samuel L. Brantly1, Kaixiang Su1, Jacob H. Nie1, Zihan Zhang1,Rounak Banerjee 2, Hayley Ruddick2, Kenji Watanabe 3, Takashi Taniguchi 4,Seth Ariel Tongay 2, Cenke Xu1 & Chenhao Jin 1Excitons in two-dimensional (2D) semiconductors have offered an attractiveplatform for optoelectronic and valleytronic devices. Further realizations ofcorrelated phases of excitons promise device concepts not possible in thesingle particle picture. Here we report tunable exciton “spin” orders in WSe2/WS2 moiré superlattices. We find evidence of an in-plane (xy) order of exciton“spin”—here, valley pseudospin—around exciton filling vex = 1, which stronglysuppresses the out-of-plane “spin” polarization. Upon increasing vex orapplying a small magnetic field of ~10 mT, it transitions into an out-of-planeferromagnetic (FM-z) spin order that spontaneously enhances the “spin”polarization, i.e., the circular helicity of emission light is higher than theexcitation. The phase diagram is qualitatively captured by a spin-1/2Bose–Hubbard model and is distinct from the fermion case. Our study pavesthe way for engineering exotic phases of matter from correlated spinorbosons, opening the door to a host of unconventional quantum devices.Properties and applications of correlated systems can gobeyond thoseattainable within the single particle framework. For example, magnetswith spontaneous electron spin ordering form the cornerstone ofspintronics1; superconductors2 and phase-change transistors3 offerpotential routes to next-generation electronics. Moiré superlatticesrecently emerged as a powerful playground to engineer correlatedphenomena4. Along with the strong light-matter interaction anduniqueoptical selection rules5,6, correlated excitons in semiconductingmoiré systems hold promises for novel applications in photonics andvalleytronics7–9. However, while various magnetic orderings and cor-related phases of electrons are reported, such as correlatedinsulator10–14, superconductivity15–17, intrinsic18,19 or exciton-mediatedferromagnetism20, and fractional quantum anomalous Hall states21–24;correlated phases of excitons remain unexplored until veryrecently25–28, and exciton “magnets” with “spin” orders have not beendemonstrated.Hereweobserve an intriguing phase diagramof interlayer-exciton“spin” orders in WSe2/WS2 moiré superlattices near one exciton perlattice site. Spin-up and spin-down exciton “spins” correspond to (andhereafter refer to) the K and K’ valleys—two degenerate but inequi-valent corners of the hexagonal Brillouin zone—and are related bytime-reversal symmetry5. Similar to magnetism of real spins, exchangeinteraction between excitons of different valley pseudospins can leadto spontaneous ordering of the valley degree of freedom. For example,an out-of-plane ferromagnetic (FM-z) spin order polarizes spins to thesame out-of-plane direction, which, in the exciton context, corre-sponds to a state where excitons are spontaneously polarized to thesame valley; On the other hand, an in-plane spin is a coherent super-position of spin-up and spin-down. Therefore, excitons in an in-plane(xy) order are each a superposition between the two valleys of equalamplitude29,30. To probe exciton “spin” order, we use a pump–probespectroscopy25 that isolates the low energy excitations of the system(Fig. 1a, see Methods: “Pump–probe spectroscopy”). Similar to elec-trical capacitance measurements31,32, the DC pump light controls thebackground exciton density andmaintains the quasi-equilibrium state,while the AC probe light injects a small perturbation of extra excitonsand isolates their responses through lock-in detection. Owing to theoptical selection rules in transition metal dichalcogenides, left- andReceived: 11 December 2023Accepted: 13 May 2024Check for updates1Department of Physics, University of California at Santa Barbara, Santa Barbara, CA, USA. 2School for Engineering of Matter, Transport, and Energy, ArizonaStateUniversity, Tempe, AZ, USA. 3ResearchCenter for FunctionalMaterials, National Institute forMaterials Science, Tsukuba, Japan. 4International Center forMaterials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, Japan. e-mail: jinchenhao@ucsb.eduNature Communications |         (2024) 15:4254 11234567890():,;1234567890():,;http://orcid.org/0000-0002-9921-8493http://orcid.org/0000-0002-9921-8493http://orcid.org/0000-0002-9921-8493http://orcid.org/0000-0002-9921-8493http://orcid.org/0000-0002-9921-8493http://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/0000-0001-8294-984Xhttp://orcid.org/0000-0001-8294-984Xhttp://orcid.org/0000-0001-8294-984Xhttp://orcid.org/0000-0001-8294-984Xhttp://orcid.org/0000-0001-8294-984Xhttp://orcid.org/0000-0002-5965-7475http://orcid.org/0000-0002-5965-7475http://orcid.org/0000-0002-5965-7475http://orcid.org/0000-0002-5965-7475http://orcid.org/0000-0002-5965-7475http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-48725-z&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-48725-z&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-48725-z&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-48725-z&domain=pdfmailto:jinchenhao@ucsb.eduright-circularly polarized (LCP and RCP) light selectively couple tospin-up and spin-down excitons5,8,33. This allows us to independentlycontrol spin of the background excitons and the extra injected exci-tons through polarization of the pump and probe light, respectively;and obtain spin-resolved system response from polarization-resolvedphotoluminescence (PL) detection.We can thereby directly create andprobe low-energy spin excitations of the system.ResultsSpin-1/2 Bose–Hubbard modelFigure 1b shows the pump–probe PL spectrum of a 0-degree-alignedWSe2/WS2moiré deviceD1with unpolarized pump, probe light, and PLdetection (the electron density is kept at 0 throughout this study).Unpolarized light couples to the total population of excitons5,33,34. Themeasurement therefore directly obtains the energy to remove oneexciton from the system, i.e., its chemical potential. A sudden jump ofexciton chemical potential is observed at one exciton per moiré site(vex = 1, nex = 1.9 × 1012 cm−2, see Methods: “Calibration of backgroundexciton density”), corresponding to a bosonic-correlated insulatorstate25. The low and high energy peaks, labeled peak I and II, corre-spond to PL emissions from a singly occupied site and a doublon site(sitewith two excitons). Their energy shift of ~30meVprovides a directmeasurement of exciton-exciton on-site repulsion and indicates thestrong correlation between excitons25,35.We then switch experimental configurations to inspect spinexcitations of excitons. The minimum model to account for exciton“spins” is a two-component Bose–Hubbard model36, given byH =X<i,j>,α�tbyi,αbj,α +h:c:+Xi,αU ni,α � 1=2� �2 +XiVni,1ni,2 ð1ÞHere the α = 1, 2 label K and K’ valley pseudospin of excitons, t ishopping between nearest neighboring sites, and the interactionsbetween excitons consist of intra-species repulsion U and inter-speciesrepulsion V. To establish suchmodel and separately determineU and V,we use linearly polarized pump light to generate equal populationof two “spins” in the background, an LCPprobe light to selectively injectextra K valley (“spin”-up) excitons, and monitor “spin”-resolvedresponses by separately collecting RCP (K’ valley) and LCP (K valley) PLfrom the probe light only (Fig. 1c, d). The K’ and K responses are rathersimilar at vex < 1 (Fig. 1e), which can be understood in the single excitonpicture from a short valley lifetime that quickly relaxes valley polariza-tion. We directly capture such relaxation process by time-resolvedpump–probe PL measurements (Fig. 1g). The pulsed probe light selec-tively injects K excitons at time zero, and the K’ response remainsunchanged. Afterwards, the valley polarization quickly disappears overtime, resulting in similar overall responses from the two valleys (Fig. 1e).In contrast, the two valleys’ responses become dramatically dif-ferent above vex = 1 (Fig. 1f). Most strikingly, their responses haveopposite signs for peak I. The negative K’ response indicates thatadding extraK excitons will decrease the number of singly occupied K’sites. Such behavior is incompatible with the single exciton picturewhere adding K excitons always increase both K and K’ excitonpopulations34,37 and is instead a unique consequence of exciton cor-relation. Our observation can be naturally understood from theBose–Hubbard model: the K excitons injected by the probe formdoublons at vex > 1, which will decrease the number of singly occupiedsites by converting them into doublon sites. The decrease in K’ sitestherefore indicates that K excitons selectively form doublon sites withK’ excitons (Supplementary Fig. 2a). This is further confirmed by theperfect valley balance in doublon emission (peak II) regardless ofexperimental configuration (Supplementary Fig. 2b), which requires Kand K’ excitons to be symmetric within any doublons.We note that while peak I is expected to disappear at vex ≥ 1 inexciton chemical potential measurement, as is observed in Fig. 1b, itshould persist in spin-resolved measurements until vex = 2 (Fig. 1c, d)due to the spin-selective formation of doublons discussed above. Thiscan also be understood by the cancellation of peak I signals whenFig. 1 | Spin-1/2 Bose–Hubbard model. a Schematics of our pump–probe spec-troscopy on a type-II hetero-bilayer WSe2/WS2. The background interlayer-excitondensity (red) is controlled by the pump light and the charge density is kept at zero.Theprobe light injects anextra interlayer-exciton (blue), whose response is isolatedthrough lock-in detection. b Exciton-filling dependence of probe-induced PLspectrum using unpolarized pump and probe light. A sudden jump of excitonchemical potential at νex = 1 is observed (white arrow), indicating an incompressiblestate of excitons. Low (high) energy emission peak is denoted as peak I (II).c, d Polarization-resolved probe-induced PL spectra as a function of exciton filling.A linear pump light is used to generate equal numbers of K and K’ valley excitons inthe background, while an LCPprobe light selectively excites extraK valley excitons.K’ valley (c) or K valley (d) PL response induced by the probe light is collectedseparately. UKK (UKK’) refers to pump injecting unpolarized excitons, probeinjecting K valley excitons and PL detecting K (K’) valley excitons. e, f Linecuts of(c, d) at low (e) and high (f) exciton density. g, h Probe-induced TRPL signals frompeak I at low (g) and high (h) exciton density. A constant background signal fromthe CW linear pump light is subtracted. Pink boxes in (e, f) denote the spectral filterused to isolate peak I response. The negative signals in UKK’ configuration (h)indicate that K excitons selectively form doublons with K’ excitons.Article https://doi.org/10.1038/s41467-024-48725-zNature Communications |         (2024) 15:4254 2adding up the two spin channels at vex ≥ 1 (Fig. 1c, d), which confirmsthat the probe light cannot add more single-occupied sites and canonly create doublons.Meanwhile, peak II should ideally only emerge atvex≥ 1. The weak peak II features observed at vex < 1 in Fig. 1c, d aremainly due to the large intensity of the pulsedprobe light usedhere fortime-resolved measurements, as well as the spatial inhomogeneity inthe exciton density (seeMethods: “Discussions on doublon emission”).These effects could be reduced by using a weaker probe and a morehomogeneous sample, such as in ref. 25. A systematic study on probeintensity and sample homogeneity would help further quantify theseeffects.Our results thus unambiguously establish a “spin”-dependent on-site repulsion between excitons. The ~30meV jump of exciton che-mical potential at vex = 1 (Fig. 1b) corresponds to the opposite-“spin”repulsion V, while the same-“spin” repulsion U is much greater than V.Consequently, doublons only formby twoexcitons of opposite “spins”like electrons in a Fermi–Hubbard model, which offers a rare realiza-tion of spin-½ Bose–Hubbard model.Inter-site spin-dependent interactions between excitonsNext, we investigate inter-site spin interactions that may lead to spinorders. We vary the pump light polarization and keep the probe LCP.Different pump polarization maintains background excitons of dif-ferent spins, while the LCP probe light always injects spin-up excitons.Any difference in the measured probe response can therefore directlyreflect spin-dependent interaction between excitons. Figure 2a–cshows the probe-induced spin imbalance (the difference between Kand K’ emission induced by the probe light) as a function of excitonfillings for LCP, RCP, and linear pump, respectively. See SupplementaryFig. 4 for results on another device D2. Peak II always shows zero spin-imbalance signal, as expected for doublon emission (SupplementaryFig. 2). Peak I is insensitive to pumppolarization at low exciton density,suggesting negligible spin interaction effects. At increasing excitondensity, in contrast, the signals vary dramatically with pump polar-ization. Under both LCP and RCP pump (Fig. 2a, b), we observe a sharpsignal enhancement at vex ~ 1.1 followed by a quick drop at vex > 1.2(black arrows). Both features are absent in the linear pump case(Fig. 2c), indicating their origin from spin interactions. While thestrongest spin interaction in the system is the on-site AFM exchange, itcannot account for the symmetric behaviors between the LCP and RCPpump or the sensitive filling dependence (see Methods: “Spin-1/2Bose–Hubbard model”). These features therefore indicate rapidlychanging inter-site spin interactions when doping slightly away from abosonic-correlated insulator.We first investigate the feature at vex = 1.1 and monitor its evolu-tion under an out-of-plane magnetic field Bz. Figure 2d–f shows themagnetic field-dependent spin imbalance spectra for fixed vex = 1.1 anddifferent pump polarization. Surprisingly, the probe-induced spinimbalance under CP pumps or linear pump are either suppressed orenhanced by anorder ofmagnitude, respectively, upon applying a tinymagnetic field of 5 mT. Such sensitive magnetic field dependence andlow saturation field (~20 mT) of exciton spin polarization have notbeen reported before38–41, and generally indicates adjacent phasetransitions with strong spin fluctuations42.To further confirm the phase transition, we also performpump–only PL measurement. Such measurement collects PL from allbackground excitons in the system and is therefore less sensitive tospin interactions than the pump–probemeasurement. However, a spinorder will affect not only low energy excitations but all excitons in thesystem and should therefore be observable in such measurements.Figure 3a shows example spectra of K and K’ PL at vex = 1.39 with RCPpump, fromwhichweobtain the PL rawhelicityηPL =IK,PL�IK0 ,PLIK ,PL + IK0 ,PL. IK ,PL andIK 0 ,PL are the PL emission intensity from K and K’ excitons (peak I inFig. 3a), which areproportional to the number of singly occupiedK andK’ sites, respectively. Figure 3b summarizes ηPL under different pumppolarization and exciton fillings. To reveal spin orders, we introduceFig. 2 | Inter-site spin-dependent exciton interaction. a–c Probe-induced spinimbalance (arbitrary units) as a function of background exciton fillings for LCP (a),RCP (b), and linear (c) pump, respectively. For both LCP and RCP pump, a sharpenhancement of signals at νex ~ 1.1 followed by a quick drop at νex ~ 1.2 are observed(black arrows). In contrast, these features aremissing under linearpump, indicatingrapidly changing exciton spin interaction when doping slightly away from thecorrelated insulator state.d–f Evolutionof probe-induced spin imbalance (arbitraryunits) at νex ~ 1.1 under out-of-planemagnetic fieldBz for LCP (d), RCP (e), and linear(f) pump, respectively. The signals show sensitive and symmetric change under asmall |Bz| ~ 5 mT. All the measurements are performed at base temperature 3 Kunless specified.Article https://doi.org/10.1038/s41467-024-48725-zNature Communications |         (2024) 15:4254 3generalized helicity GH = ηPL=ηpump, where ηpump is the helicity ofpump light (seeMethods: “Data analysis” and Supplementary Fig. 9). Inthe case of no spin order, the LCP and RCP components of pump lightshould independently contribute to PL emission, and thereforeGHwillbe a constant over pumppolarization. A spin order, on the other hand,generates a mean field that depends on all exciton spins in the systemand thus on the pumppolarization. Consequently, GHwill changewithpump helicity ηpump.Figure 3c, d showsGHandnormalizedGHover pumppolarization(helicity) at different exciton fillings. Since GH is not well-defined atηpump = 0 (linear pump), only data at |ηpump| >0.02 are obtainedexperimentally (symbols); and GH at ηpump = 0 can be extrapolatedfrom the limit of ηpump→0 (see Methods: “Data analysis”). At lowexciton density GH is indeed a constant over pump polarization. Incontrast, GH becomes a “Λ” shape at vex = 1.1 and quickly transitionsinto a “V” shape at vex > 1.25, echoing the two features in thepump–probe PL spectra (Fig. 2a, b). When we further apply an out-of-plane magnetic field, the constant GH at low exciton filling remainsintact (Fig. 3e). The “Λ” shape GH at vex = 1.1, on the other hand,changes dramatically and becomes a “V” shape under both ±30 mTfield (see Supplementary Fig. 7a for data at 30mT). Such sensitive andsymmetric magnetic field dependence is consistent with thepump–probe results and again signifies an adjacent phase transition.We have alsomeasured normalized GH at 60K as a reference (Fig. 3f),which is flat over the whole exciton density range. This further con-firms the origin of nontrivial shapes in GH from exciton spin orders.Tunable transient exciton spin ordersTo unravel the nature of the exciton spin orders, we performeddetailed magnetic field dependence at vex = 1.1 and 1.3. Figure 4a, bshow the PL raw helicity and GH at vex = 1.1 from 0 to 30 mT. Intrigu-ingly, the GH at Bz > 20 mT exceeds unity in a wide pump polarizationrange. AGH> 1means that the spin polarization of the system is higherthan the pump. This cannot be explained by field-induced symmetrybreaking between the two spins, which would favor one spin over theother and lead to asymmetric GH between RCP and LCP pump. Incontrast, the observed GH is symmetric against ηpump = 0 and exceedsunity on both sides (Fig. 4b). If excitons in the system were notdecaying over time—equivalently, all PL emissions are re-absorbed bythe system—the systemwould keep amplifying the spin polarization. Atiny initial spin-up/down injection would then eventually develop intoa close to fully spin-up/down state. Such spontaneous spinpolarizationis the hallmark of an FM-z order. On the other hand, because hereexcitons are in a quasi-equilibrium between decaying and pumping,any system memory is lost over the exciton lifetime and there shouldnot be hysteresis. Hence all orders identified experimentally are oftransient nature at the timescale of exciton decay.Our observation indicates a transient FM-z order of excitons atvex = 1.1 andmagnetic field Bz > 20mT. The zero-field state at vex = 1.1 ismore exotic. Phenomenologically, it shows opposite behaviors fromthe high field FM-z order in both pump–probe and GHmeasurements(Figs. 2a–c and4b), indicatingdistinct exciton spin states. Its extremelysensitive magnetic field dependence and low saturation field (~20mT)is particularly surprising. The most well-known effect from an out-of-plane magnetic field is the Zeeman splitting that lifts the degeneracybetween the two exciton spins. Similarly, the dominant effect ofmagnetic field on excitons is an energy splitting between the twovalleys, termed “valley Zeeman effect” 38,39, with a g-factor of 4 inmonolayer TMD. However, the splitting should be <~0.1meV under10 mT34, which is too small compared to the expected energy scale ofspin interaction (~1meV, see Methods: “Discussions on magnetic fielddependence”). In addition, applying positive and negative Bz shouldlead to opposite Zeeman splitting. Instead, in our experiment theyhavemostly symmetric effects and eventually result in a transition intothe same FM-z order (Figs. 2d–f, 3e and Supplementary Fig. 7a). Bothobservations exclude a simple linear coupling between the ZeemanFig. 3 | Evidence of exciton spin orders from generalized helicity (GH).a Pump–only PL fromK (black) and K’ (red) valley with RCPpumpat vex = 1.39. PeakI shows a large PL helicity while peak II has no PL helicity.b–d PL raw helicity ηPL (b),generalized helicity (GH, defined as ηPL/ηpump) (c) and normalized GH (d) of peak Iunder different pumphelicity ηpump and exciton fillings νex. GHdoes not dependonηpump at low exciton density, consistentwith the single-particle picturewith no spinorder. In contrast, GH becomes “Λ” shape at νex ~ 1.1 and quickly transitions into “V”shape at νex > 1.25, indicating existence of mean-field from exciton spin order.eNormalized GH at Bz = −30mT (see Bz = 30mT in Supplementary Fig. 7a). The “Λ”shape at νex ~ 1.1 becomes “V” shape under ±30 mT field. The sensitive and sym-metric Bz dependence echoes with the pump–probe measurement results.f Normalized GH at 60K. GH remains flat over the whole exciton filling range. GHand normalized GH at νex = 0.03, 1.12, 1.39 are highlighted in (c–f). Data are shownas lines and symbols for νex = 1.12 and 1.39; and only lines are shown at other fillingsfor visual clarity. Error bars represent standard deviation in PL helicity, GH andnormalized GH (b–f, see Methods: “Data analysis”).Article https://doi.org/10.1038/s41467-024-48725-zNature Communications |         (2024) 15:4254 4field and the order parameter and suggest a finite in-plane componentin the zero-field order, i.e., an xy order (For more discussions, seeMethods: “xy order”).Indeed, such phase transition can fully explain our experimentalobservations. Under linear pump, the xy spin order creates an in-planemean field, which will efficiently mix up and down spins and suppressspin polarization. On the other hand, spins under CP pumps are initi-alized to be along the z direction and the in-planemean field is weaker.We therefore expect a stronger suppression of spin polarization nearlinear pump and a weaker suppression near CP pumps, i.e., a “Λ” shapeGH (Fig. 4b). The high field FM-z order, on the contrary, amplifies spinpolarization. This leads to a sharp rise of ηPL with ηpump and a largeGH> 1 when |ηpump| is small. At large |ηpump|, ηPL saturates since itcannot exceed 1 (Fig. 4a); and GH is always smaller than 1 when |ηpump| = 1 (CP pumps).We therefore expect a “V” shapeGH as observedexperimentally (Fig. 4b). The transition regionbetween the xy and FM-zorders at intermediate field is more complicated, where GH becomesasymmetric between positive and negative ηpump or Bz (Fig. 4b). Thisindicates extrinsic symmetry breaking between the two spins by themagnetic field and thus no well-defined order (see Methods: “Discus-sions on magnetic field dependence”).We now turn to the feature at vex > 1.25. Its zero field behaviors arequalitatively similar to the high field behaviors at vex = 1.1 in all mea-surements: pump–probe measurement (Fig. 2) shows a stronger spinimbalance signal under linear pump compared to CP pump; GH showsa “V” shape with GH> 1 over a wide range of pump helicity. Uponapplying an out-of-plane magnetic field, these behaviors are qualita-tively unchanged and quantitatively enhanced. For example, GH isenhanced to a giant value of 2.3 near linear pump (Fig. 4d), corre-sponding to a rapid increase and saturation of spin polarization asηpump increases that can be clearly seen in the PL raw helicity (Fig. 4c).These results provide strong evidence that at vex > 1.25 the system isalready in an FM-z order without magnetic field, i.e., suggesting afilling-controlled transition between xy to FM-z order at vex ~ 1.25.The pump–probe measurement results (Fig. 2) are also naturallyexplained by the competition between the in-plane and out-of-planespin interactions of excitons. At vex = 1.1, the dominant in-plane spininteractions under a linear pump rapidly quench out-of-plane spinimbalances, while a CP pump reduces such quench by forcing excitonspins tobeout-of-plane.We therefore observe stronger probe-inducedspin imbalance signals under both LCP and RCP pumps compared tothe linear pump case. Upon increasing exciton filling and/or magneticfield Bz, the FM-z spin interactions dominate and enhance spin imbal-ances under linear pump as the system is not fully polarized. Once allspins are polarized under CP pump, the system is not susceptible tospin excitations anymore and the probe-induced spin imbalancebecomes vanishingly small.We next measure the temperature dependence of these orders.Figure 5a, b shows normalized GH at vex = 1.1 and 1.3, respectively, fortemperatures from 3 to 60K. The “Λ” shape GH at vex = 1.1 and “V”shape GH at vex = 1.3 disappear at around 35 and 50K, respectively,indicating melting of the associated spin orders. To quantify thetemperature dependence, we define ΔGH = GH CP pumpð Þ�GH linear pumpð ÞGH CP pumpð Þ . Apositive and negative ΔGH correspond to a “Λ” shape and “V” shape GHand indicates xy- and z- spin order, respectively. Figure 5c, d shows thephase diagram of ΔGH at 0 and −30 mT (see Supplementary Fig. 7b for30 mT data). We also mark regions where GH (linear pump) >1 withdotted texture,which is thehallmark of anFM-zorder.At zerofield, thesystem first enters an xy order upon increasing exciton filling to vex ~ 1and then transitions into anFM-zorder at vex ~ 1.25. Atmagnetic field of−30mT the xy order is suppressed, and the FM-z order is favored overthe filling range of vex > 1.1. Its melting temperature keeps increasingwith the exciton filling.DiscussionOur observations provide strong evidence of phase transitions from a(transient) xy order to FM-z order driven by both exciton filling andmagnetic field. This can be naturally understood in a spin-½Fig. 4 | Tunable exciton spin orders with magnetic field and exciton filling.a, bMagnetic field dependence of PL raw helicity ηPL (a) andGH (b) at νex = 1.1 from0 to 30mT. The GH transforms from a “Λ” shape to a “V” shape and saturates at ~20mT, suggesting a phase transition. At high field, GH exceeds 1 in a wide range ofpump helicity, indicating spontaneous increase of spin polarization, which is thehallmark of an FM-z spin order. c, d Same as (a, b) for νex = 1.3. The zero field GH issimilar to the high field GH at νex = 1.1, signifying an FM-z spin order at zero field.The “V” shapeGH is further enhancedbyBz. Data are shown as lines and symbols forBz = 0 and 30mT; andonly lines are shownat othermagneticfields for visual clarity.Error bars represent standard deviation in PL helicity andGH. Error bars in (a–c) aresmaller than the symbol size.Article https://doi.org/10.1038/s41467-024-48725-zNature Communications |         (2024) 15:4254 5Bose–Hubbard model from competitions between the super-exchangeeffect and Nagaoka-type kinetic ferromagnetism19,43. Our pump–probemeasurements establish WSe2/WS2 moiré superlattice as a spin-½Bose–Hubbardmodel, which has been predicted to host a ground stateof FM-xy order at vex = 1 as the virtual hopping of bosons gives rise to anFM in-plane super-exchange interaction J?36,44. Upon further doping,the kinetic energy of extra bosons would favor an FM-z order, similar toNagaoka ferromagnetism in Fermi–Hubbard model19,43,45. We reveal theessential physics near vex = 1 using a phenomenological spin-½ XXZmodel on a triangular lattice, where the effect of adding excitons to thevex = 1 correlated insulator is captured by a z-exchange interaction Jzthat increases with doping (see Methods: “Theoretical phase diagram”and Supplementary Note 1). Figure 5e shows the phase diagram pre-dicted by this phenomenological model, which matches well with theexperimental one. A transition from FM-xy to FM-z order is expectedwith increasing doping (and Jz). In addition, the system is very sensitiveto a Zeeman field Bz near the transition; and a weak Bz would favor theFM-zover the FM-xyorder. Intriguingly, both the FM-xy and FM-zordersare unique consequences of Bose–Einstein statistics—it is well-established that the super-exchange interaction in a Fermi–Hubbardmodel is antiferromagnetic (AFM) along all directions45,46, and theNagaoka FM is also isotropic instead of favoring the z direction43,45.While the phenomenological model here captures the salient featuresfrom the experiment, more quantitative theoretical studies are war-ranted to fully understand the exciton behaviors observed. The recentsuccess of calculatingmoiré excitons fromGW-BSE47 could allow directprediction of the exciton hopping in the Bose–Hubbard model. Fur-thermore, additional effects not captured by the Bose–Hubbardmodel,such as exciton dissociation and decoherence between the two valleys,could also contribute to the exotic exciton responses. Our results pro-vide a valuable experimental reference for future theoretical studies.Our study establishes semiconducting moiré superlattices as anintriguing platform to realize exotic states of excitons, which will alsoopen up novel device concepts in photonics and quantum informationscience. For example, an FM-z order not only stores but also amplifiesvalley polarization, which may serve as a cornerstone of memory anderror correction code48,49. The extremely sensitive magnetic field andexciton filling dependence are consequences of phase transitions andgo beyond the single-particle limit, which may enable efficient lightsource control and optical gates similar to phase-change transistors inelectronics3,50. In addition, the two-component Bose–Hubbard modelcanpotentially support a plethoraofmuchmoreexoticphases beyondthe ferromagnetic orders. For instance, itwas shownnumerically that asupersolid can be realized in the effective XXZ spin-1/2model with oneboson per-site44. Away from the vicinity of one boson per site, the twoflavors of hard-core bosons can be mapped to an SU(4) spin model(with anisotropies, see Supplementary Note 1), which is a system thathas attracted enormous interest in the past few decades and hostsvarious intriguing phases51–57.MethodsDevice fabrication and characterizationThe dual-gated WSe2/WS2 devices were made by layer-by-layer drytransfer method58. Polarization-resolved second harmonic generation(SHG) was used to determine the crystalline angles between mono-layer WSe2 and WS2 before stacking. hBN flakes with a thickness ofaround 20nmwere used as the gate dielectrics and few-layer graphiteflakes were used as the gates and contact electrode. The whole stackwas then released to a 90 nm Si/SiO2 substrate with pre-patterned Aucontacts. Supplementary Fig. 1a shows an optical image of 0-degreealigned WSe2/WS2 device D1. The twist angles were measured by SHGto be within 0 ± 0.5°, limited by experimental uncertainties. Supple-mentary Fig. 1b, c shows the gate-dependent absorption and PL char-acterization of themoiré bilayer at 3 K. At charge neutrality (~−0.05 V),the PL features a single peak at 1.375 eV from interlayer exciton emis-sion, while the absorption shows three peaks from moiré intralayerFig. 5 | Phase diagram of exciton spin orders. a, b Temperature dependence ofnormalized GH at νex = 1.1 (a) and νex = 1.3 (b). The “Λ” shape and “V” shape featuremelt at around 35K and 50K, respectively. c, d Phase diagrams of ΔGH at Bz = 0 mT(c) and −30 mT (d). A positive (negative) ΔGH corresponds to a “Λ” (“V”) shape GHand indicates xy (z) spin order. White dotted texture marks regions with GH > 1 atlinear pump, which is the hallmark of an FM-z spin order. e Theoretical phasediagram from a phenomenological spin ½ XXZ model, where h is the out-of-planemagnetic field and Jz is z-direction exchange interaction. The in-plane exchange J?is fixed to be 1/6. ϕx (ϕz) is the expectation value of Sx (Sz). The color representsorientation of the order parameter and the opacity represents its amplitude. Effectsfrom adding extra excitons to a vex = 1 correlated insulator are captured by Jz thatincreases with exciton filling. A transition from the FM-xy to FM-z order is expectedupon both increasing Jz (exciton filling) andmagnetic field, which is consistent withour experimental observations.Article https://doi.org/10.1038/s41467-024-48725-zNature Communications |         (2024) 15:4254 6excitons. At νe = 1 and νh = 1 (white arrows), the emission peak blue-shifts suddenly and the absorption peaks show a kink, indicating theemergence of correlated insulator of charges. All measurements areperformed at a base temperature of 3 K in 0°-alignedWSe2/WS2 deviceD1 unless specified.Pump–probe spectroscopyThe samples were mounted in a closed-cycle cryostat (QuantumDesign, OptiCool). A continuous wave 660nm diode laser was used asthe pump light with beam size of around 100 μm2. The large pumpbeam size ensures a homogeneous intensity in the center region that isinspected by the probe beam. A pulsed 680 nm light from a super-continuum laser (YSL Photonics, 300 ps pulse duration, variablerepetition rate) were used as the probe light. The beam size ofprobe light was around 4 μm2. The probe intensity was kept below30 nW/μm2, while the pump intensity ranged from 0 to 3 μW/μm2.To isolate the response from probe-created excitons, the probe lightwasmodulated by an optical chopper at frequency of 10Hz. The signalwas detected by a liquid-nitrogen-cooled CCD camera coupled with aspectrometer (Princeton Instruments), which was externally triggeredat 20Hz and phase locked to the chopper. The spectra with andwithout the probe light were thereby obtained, and their differencegives the signals from the probe light only (see Fig. 1b for an example).To help isolate the probe light response, we also implement a spatialfilter at a conjugate image plane of the sample, which only allows lightfrom the probe-covered region to go through. For polarization-resolved measurements, the polarization of pump/probe/PL is con-trolled by broadband polarizers and half-wave plates. In time-resolvedPL measurements (TRPL), the signals are collected by an avalanchephotodiode (MPD PDM series) and analyzed by a time-correlated sin-gle photon counting module (ID Quantique ID1000). Since the pumplight is CW while the probe light is pulsed, their contribution can bedirectly separated in the time domain and no AC modulation is nee-ded. We thereby directly track the system’s dynamical response toextra transient excitons at certain background exciton filling.Calibration of background exciton densityWe precisely calibrate the exciton density and filling at each pumpintensity through time-resolved PL measurements. This is done in twosteps. We first perform time-resolved PL (TRPL) measurement usingthe CW pump light as excitation light (Supplementary Fig. 10a). PLemission rate is a constant over time, as expected from CW excitation.This allows us to determine the emission rate at each pump intensity(Supplementary Fig. 10c). Next, we establish the relation betweenemission rate and exciton density by replacing the CW pump with apulsed pump light (300ps pulse duration, 1MHz repetition rate) withthe same wavelength (660nm) and beam profile. All other experi-mental configurations are also kept identical. Supplementary Fig. 10bshows the time-resolved PL using the pulsed excitation light of dif-ferent pulse fluences. The decay dynamics changes with pulse fluencebut is always much slower than the instrumental response function(IRF, Supplementary Fig. 10b inset). Therefore, the emission rateimmediately after time-zero corresponds to the exciton density cre-atedby thepulsed excitation lightwithout any relaxation,which canbedirectly obtained from the pulse fluence.The above procedure allows us to reliably determine excitondensity without complications from the exciton lifetime or relaxationdynamics. Since the system reaches quasi-equilibrium in a short time(<1 ps), each measured emission rate uniquely corresponds to oneexcitondensity at quasi-equilibrium,whether the excitation light isCWor pulsed. For example, at charge neutrality we identify vex = 1 at pumpintensity of 0.406 μW/μm2, which corresponds to an emission rate of562 (Supplementary Fig. 10c). The same emission rate is achieved bythe pulsed pump lightwithfluence F =0.25 J/m2 immediately after timezero (Supplementary Fig. 10d). The exciton density is directly obtainedfrom the pulse fluence though nex = αF/(hν) = (1.9 ± 0.2) × 1012 cm−2,where F is the pulse fluence, α = (0.023 ± 0.002) is absorption of WSe2at 660nm using its dielectric function and considering the multi-layerstructure of our device, hν is the photon energy of 660nm light. nexmatches well with the expected exciton density n0 = 2ffiffi3pa2M= 1.9 ×1012cm−2 at vex = 1, whereaM ~ 8 nm is themoiréperiodicity considering4% latticemismatch and 0-degree twist angle. We calibrate the excitondensity at all pump intensity and polarization following the aboveprocedure.Data analysisTo ensure the reliability of GH = ηPL=ηpump, we carefully calibrate theuncertainties in both ηpump and ηPL. In our experiment, the pump lightgoes through a half-wave plate (HWP) and a quarter-wave plate (QWP)before impinging on the sample. The pump helicity ηpump is controlledby the HWP angle θ and should ideally follow a simple sine function ofηpump = sinðπ θ�θ045°Þ. On the other hand, imperfect optics and/oralignment may result in deviation from such relation. To calibrate theuncertainty in ηpump, we directly measure the LCP and RCP compo-nents in the sample-reflected pump light using identical experimentalconfiguration as measuring the LCP and RCP PL, as shown in Supple-mentary Fig. 9a. This allows us to determine θ0 when the LCP and RCPcomponents are equal. The extracted ηpump (Supplementary Fig. 9b)shows a near-perfect match with the ideal sine relation (gray curve)and a relative standard deviation Δηpump/ηpump < 2%.To calibrate the uncertainty in ηPL, we measure ηPL twice underidentical experimental configurations at each exciton filling andextract the deviation between the twomeasurements. SupplementaryFig. 9c shows the results for three representative exciton fillingsvex = 0.02, 1.12, and 1.39, from which we obtain a standard deviationΔηPL of 0.21%, 0.17%, and 0.17%. Such a small uncertainty correspondsto an error bar smaller than the symbol size in PL raw helicity (Fig. 3band Supplementary Fig. 9c). On the other hand, the uncertainty in GHwill bedramatically amplified at small |ηpump| sinceGH=ηPL=ηpump, andGH becomes nominally ill-defined at ηpump = 0. We therefore extra-polate GH at ηpump = 0 from the limit of ηpump→0. As exemplified inSupplementary Fig. 9d, the uncertainty in GH becomes reasonablysmall (<5%) when |ηpump| > 0.05 (outside green shaded region), wherethe GH curve is already flat with pump polarization. This indicates thatGH has a well-defined value in the ηpump→0 limit, and our extrapola-tion is reliable. Another way to understand the reliability of GH at small|ηpump| is that it is simply the slope between ηPL and ηpump. As one candirectly see in the PL raw helicity (Fig. 3b), ηPL has a well-defined slopenear |ηpump| = 0.Spin-½ Bose–Hubbard modelTo account for the two species of excitons related by time-reversalsymmetry, the simplest form of the Bose–Hubbard model readsH =X<i,j>,α�tbyi,αbj,α +h:c:+Xi,αU ni,α � 1=2� �2 +XiVni,1ni,2Here the α = 1, 2 label K and K’ valley pseudospin of excitons, t ishopping between nearest neighboring sites, and the interactionsbetween excitons consist of intra-species repulsionU and inter-speciesrepulsion V. Our flavor-resolved pump–probe results indicate U >V >0, i.e., an on-site AFM interaction. Consequently, doublon sites alwaysform between one K valley and one K’ valley exciton, and the chemicalpotential jump at vex = 1 directly measures V ~ 30meV.On the other hand, the on-site interactions cannot account for thedistinctive spin imbalance responses between LCP/RCP pump andlinear pump at vex > 1 (Fig. 2). The on-site AFM interaction can indeedinduce different probe responses between different pump polariza-tions: an LCP/RCP pumpwill generatemore K/K’ background excitons,Article https://doi.org/10.1038/s41467-024-48725-zNature Communications |         (2024) 15:4254 7whichwill suppress/assist the formationof doublon siteswith the extraK excitons from an LCP probe. Such effect should therefore beopposite under LCP and RCP pump compared to linear pump, which isincompatible with the symmetric behaviors of features in Fig. 2 underboth LCP and RCP pumps. In addition, the strength of the on-siteinteraction effects scales linearly with the doublon site density andshould continuously increase with vex; while experimental featuresshow sensitive and non-monotonic filling dependence. Last but notleast, the effect from the AFM on-site interaction remain largely intactup to 60K (Supplementary Fig. 5). However, all the observed featuresdisappear at 60K (Fig. 5 and Supplementary Fig. 6). These pieces ofevidence indicate a dominant role of inter-site spin interactions tofeatures in Figs. 2–5.Theoretical phase diagramInter-site spin interactions naturally emerge in the spin-½Bose–Hubbard model. One well-established mechanism is the super-exchange effect JSE ~ t2/V (see Supplementary Note 1 and ref. 36). In aFermi–Hubbard model, JSE is isotropically AFM. In a spin-½Bose–Hubbard model, in contrast, its in-plane components JSE,xy areFM. Quantitatively, we estimate JSE,xy to be ~1meV using typical hop-ping energy of moiré excitons t ~ 5meV (refs. 7,59) and the measuredinter-species on-site interaction energy V ~ 30meV. We can also inde-pendently estimate JSE,xy ~ 1meV from the temperature dependenceusing 6JSE,xy = kBTc, where Tc = 35 K is themelting temperature of the xyorder. The consistency between two independent methods furtherconfirms the validity of our estimation. The out-of-plane component ofthe super-exchange effect JSE,z ~ (2t2/U- t2/V) can be either FM or AFM36.SinceU >V >0 guarantees | JSE,z |<| JSE,xy |, an FM-xy order is expected attheMott insulator state of vex = 1. On the other hand, further doping ofexcitons will lead to additional effective FM interaction JN similar toNagaoka ferromagnetism in Fermi–Hubbard model43,45 (see Supple-mentaryNote 1).While JN is isotropic in the Fermi–HubbardmodelwithSU(2) spin rotation symmetry, the symmetry is explicitly broken in thespin-½ Bose–Hubbard model when U ≠ V; and JN favors FM-z. As JNgrows with exciton filling above vex = 1, it eventually makesJz = JSE,z + JN > JSE,xy, leading to a transitionbetween FM-xy to FM-z order.We encapsulate competition between the super-exchange effect andthe Nagaoka-type ferromagnetism in a phenomenological XXZ model(see Supplementary Note 1), which successfully captures all salientfeatures of the experimental observation.The two-orbital Bose–Hubbard model can potentially support aplethora of much more exotic phases beyond the ferromagneticorders being discussed here. For example, it was shown numericallythat a supersolid can be realized in the effective XXZ spin-1/2 modelwith one boson per-site44. In the Supplementary Note 1, we will alsobriefly discuss the potential exotic phases when we go beyond thevicinity of one bosonper site, as two flavors of hard corebosons can bemapped to an SU(4) spin model (with anisotropies), which is a systemthat has attracted enormous interests in the past few decades as it canbe engineered in transition metal oxides with both spin and orbitaldegrees of freedom, graphene-based moiré systems, as well as coldatoms51–57. Various exotic phases of spin systems with exact orapproximate SU(4) symmetries have been discussed in literature60–67.Discussions on magnetic field dependenceThe observed magnetic field dependence at vex = 1.1 (Fig. 4a, b) can beintuitively understood through an interplay between the molecularfield and the externalfield for an xy pseudospin order. At zero externalfield, the xy pseudospin order leads to finite jϕx j and generates an in-planemean field that aligns pseudospins to in-plane.With a sufficientlylarge external field, on the other hand, the total field becomes out-of-plane. As a result, xy orders relying on the in-plane mean field aresuppressed and z orders are favored. Once the system enters the zorder (|Bz| > 20 mT), the Zeeman field applied will not significantlybreak symmetry between the two pseudospins since the Zeemanenergy scale (~0.1meV at 20 mT) is much smaller than the exchangeinteraction (~1meV). Indeed, we observe largely symmetric behaviorsunder positive and negative field for |Bz | >20 mT (Fig. 3e and Supple-mentary Fig. 7a). The intermediate field regime (|Bz| < 20 mT) showsasymmetric responses for positive and negative Bz, indicating externalsymmetry breaking from the Zeeman field. As a result, there is no well-defined spontaneous symmetry breaking or pseudospin order. Theasymmetric behaviors can be qualitatively understood from the factthat jϕx j remains finite in this regime. The total field is therefore tiltedand can still mix the two pseudospins and induce rapid switchingbetween them, during which the Zeeman splitting favors the flavorwith lower energy.xy orderThe observed PL helicity is always zero at linear pump, indicating zeroaverage out-of-plane “spin”. There are two possible scenarios: all siteshave in-plane “spin”; or domains of up and down spins. The secondscenario applies to the case of vex > 1.25, which shows rapid increase ofPL helicity with pump helicity and a “V” shape GH, as discussed in themain text. The distinctively different “Λ” shape GH at vex ~ 1.1 thusexcludes this scenario and indicate in-plane “spin”. In addition, thesymmetric and sensitive magnetic field dependence (Fig. 2d–f) indi-cates afinite in-planemeanfield at linear pump. Therefore, the in-planespin directions cannot be completely random and the system shouldhave at least a finite-range, transient xy order. On the other hand, ourobservation does not require a long-range xy order and provides noinformation on the correlation length of the order. A long-range FM-xyorder corresponds to a global phase coherence between the two val-leys; therefore, PL from the systemshould be linearly polarized.Wedidnot observe linear helicity in PL (Supplementary Fig. 8), indicating thatthere is no true long-range FM-xy order. This can be naturally under-stood as there can only be at most a quasi-long range xy order in 2D atfinite temperature68; and all orders observed in this work should be oftransient nature. Besides intrinsic spin fluctuations and exciton decay,coupling to phonons and other quasiparticles may further introducedecoherence channels acting as a random in-plane magnetic field,thereby limiting the time- and length-scale of the xy order.Comparison with spin orders of electrons in moiré systemsSpin orderings of electrons have been widely observed in bothTMD11,19,23,24,69,70 and graphene moiré systems71,72. They can be dividedinto two classes:1. The spinordering is associatedwithband topology. These systemsinvolve two or more orbitals, and the magnetism is intimatelyconnected to topological states such as Chern insulator. Examplesystems includemagic-angle twist bilayer graphene71, ABC trilayergraphene/hBN72, AB stackedMoTe2/WSe2 heterobilayers69 and AAstacked MoTe2 homobilayers23,24.2. The underlying band is non-topological. These systems can bedescribed by a single band Fermi–Hubbard model and the spinexchange/orders can be understood from doping a Mottinsulator. Examples include frustrated magnetic interaction inWSe2/WS211,70 and kinetic magnetism in MoSe2/WS219.The exciton spin orders reported here belong to a category withdistinctive different physics. Since the single-particle excitonbands arenon-topological, the physics here can be captured by a Hubbardmodel. Nevertheless, the bosonic nature of excitons gives rise to dis-tinct behaviors from the Bose–Hubbard model, as compared to theFermion case. In particular, a Bose–Hubbard model predicts in-planeFM super-exchange interaction at v = 136 and out-of-plane FM interac-tion from Nagaoka-type mechanism at v > 1; while a Fermi–Hubbardmodel predicts antiferromagnetic (AFM) super-exchange interactionalong all directions at v = 145,46, and the Nagaoka FM is also isotropicArticle https://doi.org/10.1038/s41467-024-48725-zNature Communications |         (2024) 15:4254 8instead of favoring the zdirection43,45. Our experimental observation oftransition between an xy order to an FM-z order is therefore a uniqueconsequence of Bose–Einstein statistics.Discussions on doublon emissionIdeally, peak II (emission from doublons) should only appear at vex ≥ 1in all measurement configurations. In Fig. 1c, d, peak II emerge belowvex = 1. This ismainlydue to the large intensity of the pulsedprobe lightwe used for time-resolved measurements, which can transientlyincrease the exciton density by a significant amount to form doublons.Another important practical factor is sample inhomogeneity. Becauseexciton density depends on exciton lifetime, its spatial inhomogeneityis very sensitive to defects, strain etc. and is expected to be muchlarger than the charge case. At an average vex < 1, there could alreadybe regions in the sample with vex ≥ 1, which leads to doublon emission.These issues could be addressed or alleviated by using a weaker probeand/or a more homogeneous sample. 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C.X. is supported bythe Simons Investigator program.Author contributionsC.J. conceived and supervised the project. R.X., J.H.N. and Z.Z. fabri-cated the devices. R.X. performed the optical measurements. R.X. andS.L.B. analyzed the data. K.S. and C.X. performed theoretical calcula-tions on the spin model. R.B., H.R. and S.T. grew the WSe2 and WS2crystals. K.W. and T.T. grew the hBN crystals. C.J. and R.X. wrote themanuscript with the input from all the authors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-48725-z.Correspondence and requests for materials should be addressed toChenhao Jin.Peer review informationNatureCommunications thanks the anonymousreviewers 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/s41467-024-48725-zNature Communications |         (2024) 15:4254 10https://doi.org/10.1038/s41467-024-48725-zhttp://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Tunable exciton valley-pseudospin orders in moiré superlattices Results Spin-1/2 Bose–Hubbard�model Inter-site spin-dependent interactions between excitons Tunable transient exciton spin�orders Discussion Methods Device fabrication and characterization Pump–probe spectroscopy Calibration of background exciton density Data analysis Spin-½ Bose–Hubbard�model Theoretical phase diagram Discussions on magnetic field dependence xy�order Comparison with spin orders of electrons in moiré systems Discussions on doublon emission Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information