Tunable exciton valley-pseudospin orders in moiré superlattices

Excitons in two-dimensional (2D) semiconductors have offered an attractive platform for optoelectronic and valleytronic devices. Further realizations of correlated phases of excitons promise device concepts not possible in the single 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 strongly suppresses the out-of-plane “spin” polarization. Upon increasing vex or applying a small magnetic field of ~10 mT, it transitions into an out-of-plane ferromagnetic (FM-z) spin order that spontaneously enhances the “spin” polarization, i.e., the circular helicity of emission light is higher than the excitation. The phase diagram is qualitatively captured by a spin-1/2 Bose–Hubbard model and is distinct from the fermion case. Our study paves the way for engineering exotic phases of matter from correlated spinor bosons, opening the door to a host of unconventional quantum devices.

The interplay between spin and charge degrees of freedom in strongly correlated systems gives rise to a plethora of exotic phenomena.A notable example is the Hubbard model, where the competition between spin super-exchange interaction and charge hopping leads to a rich phase diagram that may account for the emergence of high temperature superconductivity from a doped antiferromagnetic (AFM) correlated insulator 1,2 .Moiré superlattices recently emerged as a powerful playground to engineer correlated phenomena 3 .Along with the strong lightmatter interaction and unique optical selection rules 4,5 , correlated excitons in semiconducting moiré systems hold promises for novel applications in photonics and valleytronics [6][7][8] .However, while various magnetic orderings and correlated phases of electrons are reported, such as correlated insulator [9][10][11][12][13] , superconductivity [14][15][16] , intrinsic 17,18 or exciton-mediated ferromagnetism 19 , and fractional quantum anomalous Hall states [20][21][22][23] ; correlated phases of excitons remain unexplored until very recently [24][25][26][27] , and exciton "magnets" with "spin" orders have not been demonstrated.
Here we observe an intriguing phase diagram of interlayer-exciton "spin" orders in WSe2/WS2 moiré superlattices near one exciton per lattice site.Spin-up and spin-down exciton "spins" correspond to (and hereafter refer to) the K and K' valleys -two degenerate but inequivalent corners of the hexagonal Brillouin zone -and are related by time-reversal symmetry 4 .Similar to magnetism of real spins, exchange interaction between excitons of different valley pseudospins can lead to spontaneous ordering of the valley degree of freedom.For example, an out-of-plane ferromagnetic (FM-z) spin order polarizes spins to the same out-of-plane direction, which, in the exciton context, corresponds to a state where excitons are spontaneously polarized to the same valley; On the other hand, an in-plane spin is a coherent superposition of spin-up and spin-down.Therefore, excitons in an in-plane (xy) order are each a superposition between the two valleys of equal amplitude 28,29 .To probe exciton "spin" order, we use a pump probe spectroscopy 24 that isolates the low energy excitations of the system (Fig. 1a, see Methods: pump probe spectroscopy).Similar to electrical capacitance measurements 30,31 , the DC pump light controls the background exciton density and maintains the quasi-equilibrium state, while the AC probe light injects a small perturbation of extra excitons and isolates their responses through lock-in detection.Owing to the optical selection rules in transition metal dichalcogenides, left-and right-circularly polarized (LCP and RCP) light selectively couple to spin-up and spin-down excitons 4,32,33 .This allows us to independently control spin of the background excitons and the extra injected excitons through polarization of the pump and probe light, respectively; and obtain spin-resolved system response from polarization-resolved photoluminescence (PL) detection.We can thereby directly create and probe low energy spin excitations of the system.

Spin-½ Bose-Hubbard model
Figure 1b shows the pump probe PL spectrum of a 0-degree-aligned WSe2/WS2 moiré device D1 with unpolarized pump, probe light and PL detection (the electron density is kept at 0 throughout this study).Unpolarized light couples to the total population of excitons 4,32,34 .The measurement therefore directly obtains the energy to remove one exciton from the system, i.e., its chemical potential.A sudden jump of exciton chemical potential is observed at one exciton per moiré site (vex = 1, nex =1.9 × 10 12 cm -2 , see Methods: Calibration of background exciton density), corresponding to a bosonic correlated insulator state 24 .The low and high energy peaks, labelled peak I and II, correspond to PL emissions from a singly occupied site and a doublon site (site with two excitons).Their energy shift of ~30 meV provides a direct measurement of exciton-exciton on-site repulsion and indicates the strong correlation between excitons 24 .
We then switch experimental configurations to inspect spin excitations of excitons.The minimum model to account for exciton "spins" is a two-component Bose Hubbard model 35 , given by Here the =1,2 label K and K' valley pseudo-spin of excitons, t is hopping between nearest neighboring sites, and the interactions between excitons consist of intra-species repulsion U and inter-species repulsion V. To establish such a model and separately determine U and V, we use linearly polarized pump light to generate equal population of two "spins" in the background, an LCP probe light to selectively inject extra K valley ("spin"-up) excitons, and monitor "spin"resolved responses by separately collecting RCP (K' valley) and LCP (K valley) PL from the probe light only (Fig. 1, c and d).The K' and K responses are rather similar at vex<1 (Fig. 1e), which can be understood in the single exciton picture from a short valley lifetime that quickly relaxes valley polarization.We directly capture such relaxation process by time-resolved pump probe PL measurements (Fig. 1g).The pulsed probe light selectively injects K excitons at time zero, and the K' response remains unchanged.Afterwards the valley polarization quickly disappears over time, resulting in similar overall responses from the two valleys (Fig. 1e).
In contrast, the two valleys' responses become dramatically different above vex = 1 (Fig. 1f).Most strikingly, their responses have opposite signs for peak I.The negative K' response indicates that adding extra K excitons will decrease the number of singly occupied K' sites.Such behavior is incompatible with the single exciton picture where adding K excitons always increase both K and K' exciton populations 34,36 and is instead a unique consequence of exciton correlation.Our observation can be naturally understood from the Bose Hubbard model: the K excitons injected by the probe form doublons at vex > 1, which will decrease the number of singly occupied sites by converting them into doublon sites.The decrease in K' sites therefore indicates that K excitons selectively form doublon sites with K' excitons (Extended Data Fig. 2a).This is further confirmed by the perfect valley balance in doublon emission (peak II) regardless of experimental configuration (Extended Data Fig. 2b), which requires K and K' excitons to be symmetric within any doublons.
Our results thus unambiguously establish a "spin"-dependent on-site repulsion between excitons.The ~30 meV jump of exciton chemical 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 form by two excitons of opposite "spins" like electrons in a Fermi-Hubbard model, which offers a rare realization of spin-½ Bose-Hubbard model.

Inter-site spin-dependent interactions between excitons
Next, we investigate inter-site spin interactions that may lead to spin orders.We vary the pump light polarization and keep the probe LCP.Different pump polarization maintains background excitons of different spins, while the LCP probe light always injects spin-up excitons.Any difference in the measured probe response can therefore directly reflect spin-dependent interaction between excitons.Fig. 2 a-c show the probe-induced spin imbalance (the difference between K and K' emission induced by the probe light) as a function of exciton fillings for LCP, RCP and linear pump respectively.See Extended Data Fig. 4 for results on another device D2.
Peak II always shows zero spin-imbalance signal, as expected for doublon emission (Extended Data Fig. 2).Peak I is insensitive to pump polarization at low exciton density, suggesting negligible spin interaction effects.At increasing exciton density, in contrast, the signals vary dramatically with pump polarization.Under both LCP and RCP pump (Fig. 2, a and b), we observe a sharp signal 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 the strongest spin interaction in the system is the on-site AFM exchange, it cannot account for the symmetric behaviors between the LCP and RCP pump or the sensitive filling dependence (see Methods: Spin-1/2 Bose Hubbard model).These features therefore indicate rapidly changing inter-site spin interactions when doping slightly away from a bosonic correlated insulator.
We first investigate the feature at vex = 1.1 and monitor its evolution under an out-of-plane magnetic field Bz.Fig. 2 d-f shows the magnetic field-dependent spin imbalance spectra for fixed vex = 1.1 and different pump polarization.Surprisingly, the probe-induced spin imbalance under CP pumps or linear pump are either suppressed or enhanced by an order of magnitude, respectively, upon applying a tiny magnetic field of 5 mT.Such sensitive magnetic field dependence and low saturation field (~20 mT) of exciton spin polarization have not been reported before [37][38][39][40] , and generally indicates adjacent phase transitions with strong spin fluctuations 41 .
To further confirm the phase transition, we also perform pump-only PL measurement.Such measurement collects PL from all background excitons in the system and is therefore less sensitive to spin interactions than the pump-probe measurement.However, a spin order will affect not only low energy excitations but all excitons in the system and should therefore be observable in such measurements.Fig. 3a shows example spectra of K and K' PL at vex = 1.39 with RCP pump, from which we obtain the PL raw helicity  PL =  ,PL −  ′ ,PL  ,PL +  ′ ,PL .  ,PL and   ′ ,PL are the PL emission intensity from  and  ′ excitons (peak I in Fig. 3a), which are proportional to the number of singly occupied K and K' sites, respectively.Fig. 3b summarizes  PL under different pump polarization and exciton fillings.To reveal spin orders, we introduce generalized helicity GH =  PL /pump, where pump is the helicity of pump light (see Methods: Data analysis and Extended Data Fig. 9).In the case of no spin order, the LCP and RCP components of pump light should independently contribute to PL emission, and therefore GH will be a constant over pump polarization.A spin order, on the other hand, generates a mean field that depends on all exciton spins in the system and thus on the pump polarization.Consequently, GH will change with pump helicity pump.
Figure 3, c and d show GH and normalized GH over pump polarization (helicity) at different exciton fillings.Since GH is not well-defined at  pump=0 (linear pump), only data at |pump|>0.02 are obtained experimentally (symbols); and GH at pump=0 can be extrapolated from the limit of pump→0 (see Methods: Data analysis).At low exciton density GH is indeed a constant over pump polarization.In contrast, GH becomes a "Λ" shape at vex = 1.1 and quickly transitions into a "V" shape at vex > 1.25, echoing the two features in the pump probe PL spectra (Fig. 2, a and b).When we further apply an out-of-plane magnetic field, the constant GH at low exciton filling remains intact (Fig. 3e).The "Λ" shape GH at vex = 1.1, on the other hand, changes dramatically and becomes a "V" shape under both ±30 mT field (see Extended Data Fig. 7a for data at 30 mT).Such sensitive and symmetric magnetic field dependence is consistent with the pump-probe results and again signifies an adjacent phase transition.We have also measured normalized GH at 60 K as a reference (Fig. 3f), which is flat over the whole exciton density range.This further confirms the origin of nontrivial shapes in GH from exciton spin orders.

Tunable transient exciton spin orders
To unravel the nature of the exciton spin orders, we performed detailed magnetic field dependence at vex = 1.1 and 1.3.Figure 4a and 4b show the PL raw helicity and GH at vex = 1.1 from 0 to 30 mT.Intriguingly, the GH at Bz >20 mT exceeds unity in a wide pump polarization range.A GH > 1 means that the spin polarization of the system is higher than the pump.This cannot be explained by field-induced symmetry breaking between the two spins, which would favor one spin over the other and lead to asymmetric GH between RCP and LCP pump.In contrast, the observed GH is symmetric against pump=0 and exceeds unity on both sides (Fig. 4b).If excitons in the system were not decaying over time -equivalently, all PL emissions are re-absorbed by the system -the system would keep amplifying the spin polarization.A tiny initial spin-up/down injection would then eventually develop into a close to fully spin-up/down state.Such spontaneous spin polarization is the hallmark of an FM-z order.On the other hand, because here excitons are in a quasi-equilibrium between decaying and pumping, any system memory is lost over the exciton lifetime and there should not be hysteresis.Hence all orders identified experimentally are of transient nature at the timescale of exciton decay.
Our observation indicates a transient FM-z order of excitons at vex = 1.1 and magnetic field Bz >20 mT.The zero-field state at vex = 1.1 is more exotic.Phenomenologically, it shows opposite behaviors from the high field FM-z order in both pump probe and GH measurements (Fig. 2a-c and Fig. 4b), indicating distinct exciton spin states.Its extremely sensitive magnetic field dependence and low saturation field (~20 mT) is particularly surprising.The most wellknown effect from an out-of-plane magnetic field is the Zeeman splitting that lifts the degeneracy between the two exciton spins.Similarly, the dominant effect of magnetic field on excitons is an energy splitting between the two valleys, termed "valley Zeeman effect" 37,38 , with a g-factor of 4 in monolayer TMD.However, the splitting should be <~0.1 meV under 10 mT 34 , which is too small compared to the expected energy scale of spin interaction (~1 meV, see Methods: Discussions on magnetic field dependence).In addition, applying positive and negative Bz should lead to opposite Zeeman splitting.Instead, in our experiment they have mostly symmetric effects and eventually result in a transition into the same FM-z order (Fig. 2 d-f, Fig. 3e and Extended Data Fig. 7a).Both observations exclude a simple linear coupling between the Zeeman field and the order parameter and suggest a finite in-plane component in the zero-field order, i.e., an xy order (For more discussions, see Methods: xy order).
Indeed, such phase transition can fully explain our experimental observations.Under linear pump, the xy spin order creates an in-plane mean field, which will efficiently mix up and down spins and suppress spin polarization.On the other hand, spins under CP pumps are initialized to be along the z direction and the in-plane mean field is weaker.We therefore expect a stronger suppression of spin polarization near linear pump and a weaker suppression near CP pumps, i.e., a "Λ" shape GH (Fig. 4b).The high field FM-z order, on the contrary, amplifies spin polarization.This leads to a sharp rise of  PL with pump and a large GH>1 when |pump| is small.At large | pump|,  PL saturates since it cannot exceed 1 (Fig. 4a); and GH is always smaller than 1 when |pump| =1 (CP pumps).We therefore expect a "V" shape GH as observed experimentally (Fig. 4b).The transition region between the xy and FM-z orders at intermediate field is more complicated, where GH becomes asymmetric between positive and negative pump or Bz (Fig. 4b).This indicates extrinsic symmetry breaking between the two spins by the magnetic field and thus no well-defined order (see Methods: Discussions on magnetic field dependence).
We now turn to the feature at vex >1.25.Its zero field behaviors are qualitatively similar to the high field behaviors at vex = 1.1 in all measurements: pump probe measurement (Fig. 2) shows a stronger spin imbalance signal under linear pump compared to CP pump; GH shows a "V" shape with GH > 1 over a wide range of pump helicity.Upon applying an out-of-plane magnetic field, these behaviors are qualitatively unchanged and quantitatively enhanced.For example, GH is enhanced to a giant value of 2.3 near linear pump (Fig. 4d), corresponding to a rapid increase and saturation of spin polarization as pump 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 is already in an FM-z order without magnetic field, i.e., suggesting a filling-controlled transition between xy to FM-z order at vex ~1.25.
The pump-probe measurement results (Fig. 2) are also naturally explained by the competition between the in-plane and out-of-plane spin interactions of excitons.At vex =1.1, the dominant in-plane spin interactions under a linear pump rapidly quench out-of-plane spin imbalances, while a CP pump reduces such quench by forcing exciton spins to be out-of-plane.We therefore observe stronger probe-induced spin imbalance signals under both LCP and RCP pumps compared to the linear pump case.Upon increasing exciton filling and/or magnetic field Bz, the FM-z spin interactions dominate and enhance spin imbalances under linear pump as the system is not fully polarized.Once all spins are polarized under CP pump, the system is not susceptible to spin excitations anymore and the probe-induced spin imbalance becomes vanishingly small.
We next measure the temperature dependence of these orders.Figure 5 a and b show normalized GH at vex = 1.1 and 1.3, respectively, for temperatures from 3 to 60 K.The "Λ" shape GH at vex = 1.1 and "V" shape GH at vex = 1.3 disappear at around 35 and 50 K, respectively, indicating melting of the associated spin orders.To quantify the temperature dependence, we define  GH = GH(CP pump)−GH(linear pump)

GH(CP pump)
. A positive and negative  GH correspond to a "Λ" shape and "V" shape GH and indicates xyand zspin order, respectively.Fig. 5 c and d show the phase diagram of  GH at 0 and -30 mT (see Extended Data Fig. 7b for 30 mT data).We also mark regions where GH (linear pump) >1 with dotted texture, which is the hallmark of an FM-z order.At zero field, the system first enters an xy order upon increasing exciton filling to vex ~1 and then transitions into an FM-z order at vex~1.25.At magnetic field of -30 mT the xy order is suppressed, and the FM-z order is favored over the filling range of vex>1.1.Its melting temperature keeps increasing with the exciton filling.

Discussions and outlook
Our observations provide strong evidence of phase transitions from a (transient) xy order to FM-z order driven by both exciton filling and magnetic field.This can be naturally understood in a spin-½ Bose-Hubbard model from competitions between the super-exchange effect and Nagaoka-type kinetic ferromagnetism 18,42 .Our pump-probe measurements establish WSe2/WS2 moiré superlattice as a spin-½ Bose Hubbard model, which has been predicted to host a ground state of FM-xy order at vex = 1 as the virtual hopping of bosons gives rise to an FM in-plane super-exchange interaction  ⊥ 35,43 .Upon further doping, the kinetic energy of extra bosons would favor an FM-z order, similar to Nagaoka ferromagnetism in Fermi Hubbard model 18,42,44 .We reveal the essential physics near vex=1 using a phenomenological spin-½ XXZ model on a triangular lattice, where the effect of adding excitons to the vex=1 correlated insulator is captured by a z-exchange interaction Jz that increases with doping (see Methods: Theoretical phase diagram and Supplementary Materials). Figure 5e shows the phase diagram predicted by this phenomenological model, which matches well with the experimental one.A transition from FM-xy to FM-z order is expected with increasing doping (and Jz).In addition, the system is very sensitive to a Zeeman field Bz near the transition; and a weak Bz would favor the FM-z over the FM-xy order.Intriguingly, both the FM-xy and FM-z orders are unique consequences of Bose-Einstein statistics -it is well established that the super-exchange interaction in a Fermi Hubbard model is antiferromagnetic (AFM) along all directions 1,44 , and the Nagaoka FM is also isotropic instead of favoring the z direction 42,44 .
Our study establishes semiconducting moiré superlattices as an intriguing platform to realize exotic states of excitons, which will open up novel device concepts in photonics and quantum information science.For example, an FM-z order not only stores but also amplifies valley polarization, which may serve as a cornerstone of memory and error correction code 45,46 .The extremely sensitive magnetic field and exciton filling dependence are consequences of phase transitions and go beyond the single particle limit, which may enable efficient light source control and optical gates similar to phase-change transistors in electronics 47,48 .In addition, the two-component Bose-Hubbard model can potentially support a plethora of much more exotic phases beyond the ferromagnetic orders.For instance, it was shown numerically that a supersolid can be realized in the effective XXZ spin-1/2 model with one boson per-site 43 .Away from the vicinity of one boson per site, the two flavors of hard-core bosons can be mapped to an SU(4) spin model (with anisotropies, see Supplementary Materials), which is a system that has attracted enormous interest in the past few decades and hosts various intriguing phases 49- 55 .

Methods:
Device fabrication and characterization: The dual-gated WSe2/WS2 devices were made by layer-by-layer dry transfer method 56 .Polarization-resolved second harmonic generation (SHG) was used to determine the crystalline angles between monolayer WSe2 and WS2 before stacking.hBN flakes with a thickness of around 20 nm were used as the gate dielectrics and few-layer graphite flakes were used as the gates and contact electrode.The whole stack was then released to a 90 nm Si/SiO2 substrate with pre-patterned Au contacts.Extended Data Fig. 1a shows an optical image of 0-degree aligned WSe2/WS2 device D1.The twist angles were measured to be within 0 ± 0.5°, limited by experimental uncertainties.Extended Data Fig. 1 b and c show the gate-dependent absorption and PL characterization of the moiré bilayer at 3 K.At charge neutrality (~-0.05V), the PL features a single peak at 1.375eV from interlayer exciton emission, while the absorption shows three peaks from moiré intralayer excitons.At νe=1 and νh=1 (white arrows), the emission peak blueshifts suddenly and the absorption peaks show a kink, indicating the emergence of correlated insulator of charges.All measurements are performed at a base temperature of 3 K in 0°-aligned WSe2/WS2 device D1 unless specified.

Pump probe spectroscopy:
The samples were mounted in a closed-cycle cryostat (Quantum Design, OptiCool).A continuous wave 660 nm diode laser was used as the pump light with beam size of around 100 μm 2 .The large pump beam size ensures a homogeneous intensity in the center region that is inspected by the probe beam.A pulsed 680 nm light from a supercontinuum laser (YSL Photonics, 300 ps pulse duration, variable repetition rate) were used as the probe light.The beam size of probe light was around 4 μm 2 .The probe intensity was kept below 30 nW/μm 2 , while the pump intensity ranged from 0 to 3 μW/μm 2 .To isolate the response from probe-created excitons, the probe light was modulated by an optical chopper at frequency of 10 Hz.The signal was detected by a liquid-nitrogen-cooled CCD camera coupled with a spectrometer (Princeton Instruments), which was externally triggered at 20 Hz and phase locked to the chopper.The spectra with and without the probe light were thereby obtained, and their difference gives the signals from the probe light only (see Fig. 1b for an example).To help isolate the probe light response, we also implement a spatial filter at a conjugate image plane of the sample, which only allows light from the probe-covered region to go through.For polarization-resolved measurements, the polarization of pump/probe/PL is controlled by broadband polarizers and half-wave plates.In time-resolved PL measurements (TRPL), the signals are collected by an avalanche photodiode (MPD PDM series) and analyzed by a time-correlated single photon counting module (ID Quantique ID1000).Since the pump light is CW while the probe light is pulsed, their contribution can be directly separated in the time domain and no AC modulation is needed.We thereby directly track the system's dynamical response to extra transient excitons at certain background exciton filling.

Calibration of background exciton density:
We precisely calibrate the exciton density and filling at each pump intensity through time-resolved PL measurements.This is done in two steps.We first perform time-resolved PL (TRPL) measurement using the CW pump light as excitation light (Extended Data Fig. 10a).PL emission rate is a constant over time, as expected from CW excitation.This allows us to determine the emission rate at each pump intensity (Extended Data Fig. 10c).Next, we establish the relation between emission rate and exciton density by replacing the CW pump with a pulsed pump light (300 ps pulse duration, 1 MHz repetition rate) with the same wavelength (660nm) and beam profile.All other experimental configurations are also kept identical.Extended Data Fig. 10b shows the time-resolved PL using the pulsed excitation light of different pulse fluences.The decay dynamics changes with pulse fluence but is always much slower than the instrumental response function (IRF, Extended Data Fig. 10b inset).Therefore, the emission rate immediately after time-zero corresponds to the exciton density created by the pulsed excitation light without any relaxation, which can be directly obtained from the pulse fluence.
The above procedure allows us to reliably determine exciton density without complications from the exciton lifetime or relaxation dynamics.Since the system reaches quasi-equilibrium in a short time (<1ps), each measured emission rate uniquely corresponds to one exciton density at quasi-equilibrium, whether the excitation light is CW or pulsed.For example, at charge neutrality we identify vex =1 at pump intensity of 0.406 μW/μm 2 , which corresponds to an emission rate of 562 (Extended Data Fig. 10c).The same emission rate is achieved by the pulsed pump light with fluence F= 0.25 J/m 2 immediately after time zero (Extended Data Fig. 10d).The exciton density is directly obtained from the pulse fluence though nex = αF/(hν) = (1.9 ± 0.2) × 10 12 cm -2 , where F is the pulse fluence, α=(0.023± 0.002) is absorption of WSe2 at 660nm using its dielectric function and considering the multi-layer structure of our device, hν is the photon energy of 660nm light.nex matches well with the expected exciton density n0 = 2 √3  2 =1.9x10 12 cm -2 at vex =1, where aM ~ 8 nm is the moiré periodicity considering 4% lattice mismatch and 0-degree twist angle.We calibrate the exciton density at all pump intensity and polarization following the above procedure.

Data analysis:
To ensure the reliability of GH =  PL / pump, we carefully calibrate the uncertainties in both pump and  PL .In our experiment, the pump light goes through a halfwave plate (HWP) and a quarter-wave plate (QWP) before impinging on the sample.The pump helicity  pump is controlled by the HWP angle  and should ideally follow a simple sine function of pump = sin (π . On the other hand, imperfect optics and/or alignment may result in deviation from such relation.To calibrate the uncertainty in pump, we directly measure the LCP and RCP components in the sample-reflected pump light using identical experimental configuration as measuring the LCP and RCP PL, as shown in Extended Data Fig. 9a.This allows us to determine  0 when the LCP and RCP components are equal.The extracted pump (Extended Data Fig. 9b) shows a near-perfect match with the ideal sine relation (grey curve) and a relative standard deviation Δpump /pump <2%.
To calibrate the uncertainty in  PL , we measure  PL twice under identical experimental configurations at each exciton filling and extract the deviation between the two measurements.Extended Data Fig. 9c shows the results for three representative exciton fillings vex = 0.02, 1.12 and 1.39, from which we obtain a standard deviation Δ PL of 0.21%, 0.17% and 0.17%.Such a small uncertainty corresponds to an error bar smaller than the symbol size in PL raw helicity (Fig. 3b and Extended Data Fig. 9c).On the other hand, the uncertainty in GH will be dramatically amplified at small |pump| since GH =  PL /pump, and GH becomes nominally illdefined at pump = 0. We therefore extrapolate GH at pump = 0 from the limit of pump→0.As exemplified in Extended Data Fig. 9d, the uncertainty in GH becomes reasonably small (<5%) when |pump|>0.05 (outside green shaded region), where the GH curve is already flat with pump polarization.This indicates that GH has a well-defined value in the pump→0 limit, and our extrapolation is reliable.Another way to understand the reliability of GH at small |pump| is that it is simply the slope between  PL and  pump .As one can directly see in the PL raw helicity (Fig. 3b),  PL has a well-defined slope near |pump|=0.

Spin-½ Bose-Hubbard model:
To account for the two species of excitons related by timereversal symmetry, the simplest form of the Bose-Hubbard model reads Here the =1,2 label K and K' valley pseudo-spin of excitons, t is hopping between nearest neighboring sites, and the interactions between excitons consist of intra-species repulsion U and inter-species repulsion V. Our flavor-resolved pump-probe results indicate U>V>0, i.e., an on-site AFM interaction.Consequently, doublon sites always form between one K valley and one K' valley exciton, and the chemical potential jump at vex=1 directly measures V~30 meV.
On the other hand, the on-site interactions cannot account for the distinctive spin imbalance responses between LCP/RCP pump and linear pump at vex>1 (Fig. 2).The on-site AFM interaction can indeed induce different probe responses between different pump polarizations: an LCP/RCP pump will generate more K/K' background excitons, which will suppress/assist the formation of doublon sites with the extra K excitons from an LCP probe.Such effect should therefore be opposite under LCP and RCP pump compared to linear pump, which is incompatible with the symmetric behaviors of features in Fig. 2 under both LCP and RCP pumps.In addition, the strength of the on-site interaction effects scales linearly with the doublon site density and should continuously increase with vex; while experimental features show sensitive and non-monotonic filling dependence.Last but not least, the effect from the AFM on-site interaction remain largely intact up to 60 K (Extended Data Fig. 5).However, all the observed features disappear at 60 K (Fig. 5 and Extended Data Fig. 6).These pieces of evidence indicate a dominant role of inter-site spin interactions to features in Fig. 2-5.

Theoretical phase diagram:
Inter-site spin interactions naturally emerge in the spin-½ Bose-Hubbard model.One well-established mechanism is the super-exchange effect JSE~t 2 /V (see Supplementary Materials and Ref. 35 ).In a Fermi Hubbard model, JSE is isotropically AFM.In a spin-½ Bose-Hubbard model, in contrast, its in-plane components JSE,xy are FM.Quantitatively, we estimate JSE,xy to be ~1 meV using typical hopping energy of moiré excitons t~5 meV (Ref. 6,57) and the measured inter-species on-site interaction energy V~30 meV.We can also independently estimate JSE,xy ~1 meV from the temperature dependence using 6JSE,xy=kBTc, where Tc=35 K is the melting temperature of the xy order.The consistency between two independent methods further confirms the validity of our estimation.The out-ofplane component of the super-exchange effect JSE,z~ (2t 2 /U-t 2 /V) can be either FM or AFM 35 .Since U>V>0 guarantees | JSE,z |<| JSE,xy |, an FM-xy order is expected at the Mott insulator state of vex=1.On the other hand, further doping of excitons will lead to additional effective FM interaction JN similar to Nagaoka ferromagnetism in Fermi Hubbard model 42,44 (see Supplementary Materials).While JN is isotropic in the Fermi Hubbard model with SU(2) spin rotation symmetry, the symmetry is explicitly broken in the spin-½ Bose-Hubbard model when U≠V; and JN favors FM-z.As JN grows with exciton filling above vex=1, it eventually makes Jz=JSE,z+JN>JSE,xy, leading to a transition between FM-xy to FM-z order.We encapsulate competition between the super-exchange effect and the Nagaoka-type ferromagnetism in a phenomenological XXZ model (see Supplementary Materials), which successfully captures all salient features of the experimental observation.
The two-orbital Bose-Hubbard model can potentially support a plethora of much more exotic phases beyond the ferromagnetic orders being discussed here.For example, it was shown numerically that a supersolid can be realized in the effective XXZ spin-1/2 model with one boson per-site 43 .In the Supplementary Materials we will also briefly discuss the potential exotic phases when we go beyond the vicinity of one boson per site, as two flavors of hard core bosons can be mapped to an SU(4) spin model (with anisotropies), which is a system that has attracted enormous interests in the past few decades as it can be engineered in transition metal oxides with both spin and orbital degrees of freedom, graphene-based moiré systems, as well as cold atoms [49][50][51][52][53][54][55] .Various exotic phases of spin systems with exact or approximate SU(4) symmetries have been discussed in literature [58][59][60][61][62][63][64][65] .

Discussions on magnetic field dependence:
The observed magnetic field dependence at vex=1.1 (Fig. 4a and b) can be intuitively understood through an interplay between the molecular field and the external field for an xy pseudospin order.At zero external field, the xy pseudospin order leads to finite |  | and generates an in-plane mean field that aligns pseudospins to in-plane.With a sufficiently large 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 are suppressed and z orders are favored.Once the system enters the z order (|Bz|>20 mT), the Zeeman field applied will not significantly break symmetry between the two pseudospins since the Zeeman energy scale (~0.1 meV at 20 mT) is much smaller than the exchange interaction (~1 meV).Indeed, we observe largely symmetric behaviors under positive and negative field for |Bz|>20 mT (Fig. 3e and Extended Data Fig. 7a).The intermediate field regime (|Bz|<20 mT) shows asymmetric responses for positive and negative Bz, indicating external symmetry breaking from the Zeeman field.As a result, there is no well-defined spontaneous symmetry breaking or pseudospin order.The asymmetric behaviors can be qualitatively understood from the fact that |  | remains finite in this regime.The total field is therefore tilted and can still mix the two pseudospins and induce rapid switching between them, during which the Zeeman splitting favors the flavor with lower energy.
xy order: The observed PL helicity is always zero at linear pump, indicating zero average outof-plane "spin".There are two possible scenarios: all sites have in-plane "spin"; or domains of up and down spins.The second scenario applies to the case of vex >1.25, which shows rapid increase of PL helicity with pump helicity and a "V" shape GH, as discussed in the main text.The distinctively different "Λ" shape GH at vex ~1.1 thus excludes this scenario and indicate in-plane "spin".In addition, the symmetric and sensitive magnetic field dependence (Fig. 2df) indicates a finite in-plane mean field at linear pump.Therefore, the in-plane spin directions cannot be completely random and the system should have at least a finite-range, transient xy order.On the other hand, our observation does not require a long-range xy order and provides no information on the correlation length of the order.A long-range FM-xy order corresponds to a global phase coherence between the two valleys; therefore, PL from the system should be linearly polarized.We did not observe linear helicity in PL (Extended Data Fig. 8), indicating that there is no true long-range FM-xy order.This can be naturally understood as there can only be at most a quasi-long range xy order in 2D at finite temperature 66 ; and all orders observed in this work should be of transient nature.Besides intrinsic spin fluctuations and exciton decay, coupling to phonons and other quasiparticles may further introduce decoherence channels acting as a random in-plane magnetic field, thereby limiting the time-and length-scale of the xy order.A transition from the FM-xy to FM-z order is expected upon both increasing Jz (exciton filling) and magnetic field, which is consistent with our experimental observations.

Extended Data Figures:
Extended Data

Fig. 2 :
Fig. 2: Inter-site spin-dependent exciton interaction.a-c, Probe-induced spin imbalance as a function of background exciton fillings for LCP (a), RCP (b) and linear (c) pump respectively.For both LCP and RCP pump, a sharp enhancement of signals at νex~1.1 followed by a quick drop at νex~1.2 are observed (black arrows).In contrast, these features are missing under linear pump, indicating rapidly changing exciton spin interaction when doping slight away from the correlated insulator state.d-f, Evolution of probe-induced spin imbalance at νex~1.1 under outof-plane magnetic field Bz for LCP (d), RCP (e) and linear (f) pump, respectively.The signals show sensitive and symmetric change under a small |Bz|~5 mT.All the measurements are performed at base temperature 3 K unless specified.

Fig. 3 :
Fig. 3: Evidence of exciton spin orders from generalized helicity (GH).a, pump-only PL from K (black) and K' (red) valley with RCP pump at vex = 1.39.Peak I 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 I under different pump helicity ηpump and exciton fillings ex.GH does not depend on ηpump at low exciton density, consistent with the single-particle picture with no spin order.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.e, Normalized GH at Bz = -30 mT (see Bz = 30 mT in Extended Data Fig.7a).The "Λ" shape at νex~1.1 becomes "V" shape under ±30 mT field.The sensitive and symmetric Bz dependence echoes with the pump probe measurement results.f, Normalized GH at 60 K. GH remains flat over the whole exciton filling range.GH and normalized GH at νex = 0.03, 1.12, 1.39 are highlighted in c-f.Data are shown as lines and symbols for νex = 1.12 and 1.39; and only lines are shown at other fillings for visual clarity.Error bars represent standard deviation in PL helicity, GH and normalized GH (b-f, see Methods: Data analysis).

Fig. 4 :
Fig. 4: Tunable exciton spin orders with magnetic field and exciton filling.a,b, Magnetic field dependence of PL raw helicity ηPL (a) and GH (b) at νex=1.1 from 0 to 30 mT.The GH transforms from a "Λ" shape to a "V" shape and saturates at ~20 mT, suggesting a phase transition.At high field, GH exceeds 1 in a wide range of pump helicity, indicating spontaneous increase of spin polarization, which is the hallmark of an FM-z spin order.c,d, Same as a,b for νex=1.3.The zero field GH is similar to the high field GH at νex=1.1, signifying an FM-z spin order at zero field.The "V" shape GH is further enhanced by Bz.Data are shown as lines and symbols for Bz = 0 and 30 mT; and only lines are shown at other magnetic fields for visual clarity.Error bars in a and c are smaller than the symbol size.

Fig. 5 :
Fig. 5: Phase diagram of exciton spin orders.a,b, Temperature dependence of normalized GH at νex=1.1 (a) and νex=1.3(b).The "Λ" shape and "V" shape feature melt at around 35 K and 50 K, respectively.c,d, Phase diagrams of  GH at Bz=0 mT (c) and -30 mT (d).A positive (negative)  GH corresponds to a "Λ" ("V") shape GH and indicates xy (z) spin order.White dotted texture marks regions with GH >1 at linear pump, which is the hallmark of an FM-z spin order.e, Theoretical phase diagram from a phenomenological spin ½ XXZ model, where h is the out-of-plane magnetic field and Jz is z-direction exchange interaction.The in-plane exchange  ⊥ is fixed to be 1/6.x (z) is the expectation value of S x (S z ).The color represents orientation of the order parameter and the opacity represents its amplitude.Effects from adding extra excitons to a vex=1 correlated insulator is captured by Jz that increases with exciton filling.A transition from the FM-xy to FM-z order is expected upon both increasing Jz (exciton filling) and magnetic field, which is consistent with our experimental observations.

Fig. 1 : 2 :Extended Data Fig. 7 :Extended Data Fig. 8 :Extended Data Fig. 9 :
Basic characterizations of devices.a, optical image of a representative dual-gated 0-degree-aligned WSe2/WS2 device D1.Yellow and green solid lines denote contours of the monolayer WSe2 and WS2 flakes, respectively.b,c, Electron dopingdependent absorption (b) and PL (c) spectrum of device D1 at zero pump intensity.White arrows indicate one filling of electrons and Extended Data Fig.Response of a bosonic correlated insulator to transient extra K excitons.a, (1), CW linear pump light injects equal number of K valley (red) and K' valley (grey) background excitons that form an exciton lattice.(2), Pulsed circular polarized probe light transiently injects two extra K valley excitons, which takes two K' sites to form doublon sites (yellow circles).(3), Doublon sites have equal probabilities to emit K(K') excitons and leave a single site of K'(K).(4), After the doublons decay the system will have one more K single site and one less K' single site, giving rise to negative K' response and positive K response of peak I with equal amplitude.b, Probe-induced TRPL measurements of doublon emissions under linear pump and LCP probe configuration.The two valleys show identical amplitude and dynamics, consistent with the expectation of doublon emission.Extended Data Fig. 6: Probe-induced spin imbalance in device D1 at 60 K.No pump polarization dependence is observed across all exciton fillings, indicating vanishing spindependent interactions at 60 K. Normalized generalized helicity and phase diagrams of   at B z = 30 mT.Linear polarization-resolved PL at v ex = 1.1.No linear helicity is observed between vertical (VV) and horizontal (VH) PL detection, indicating that there is no global long-range FM-xy order.The pump light is linearly polarized along the vertical direction.Estimation of measurement uncertainties.a, LCP (blue) and RCP (black) components of the sample-reflected pump light under identical experimental configuration as polarization-resolved PL measurements.b, Pump helicity (black symbols) and PL helicity (red symbols) at νex=1.39 and zero magnetic field with different HWP angles.The pump helicity shows near-perfect match with theoretical curve (grey line) with a relative standard deviation of 1.7%.Inset shows the deviation between the measured and theoretical pump helicity.c, The deviation in  PL between two successive measurements, from which we calculate the standard deviation in  PL to be 0.21%, 0.17% and 0.17% for νex=0.02,1.12 and 1.39, respectively.d, Generalized helicity (GH) at νex=1.39 and zero magnetic field as a function of HWP angles.The standard deviation in GH becomes reasonably small (<5%) when |pump|>0.05 (outside of the green shaded region).ExtendedData Fig. 10: Calibration of exciton density via TRPL measurement.a, TRPL using a 660 nm CW pump light of different pump intensity.b, Same as a but with a 660 nm pulsed pump light (300 ps duration, 1 MHz repetition rate).Inset: Comparison between IRF and PL dynamics indicates negligible exciton relaxation immediately after time zero.c, Emission rates from the CW pump light of different intensities.d, Emission rates from the pulsed pump light of different fluences.