Thermodynamics of free and bound magnons in graphene

Symmetry-broken electronic phases support neutral collective excitations. For example, monolayer graphene in the quantum Hall regime hosts a nearly ideal ferromagnetic phase at specific filling factors that spontaneously breaks the spin-rotation symmetry1–3. This ferromagnet has been shown to support spin-wave excitations known as magnons that can be electrically generated and detected4,5. Although long-distance magnon propagation has been demonstrated via transport measurements, important thermodynamic properties of such magnon populations—including the magnon chemical potential and density—have not been measured. Here we present local measurements of electron compressibility under the influence of magnons, which reveal a reduction in the gap associated with the ν = 1 quantum Hall state by up to 20%. Combining these measurements with the estimates of temperature, our analysis reveals that the injected magnons bind to electrons and holes to form skyrmions, and it enables the extraction of free magnon density, magnon chemical potential and average skyrmion spin. Our methods provide a means of probing the thermodynamic properties of charge-neutral excitations that are applicable to other symmetry-broken electronic phases. Although magnons in the quantum Hall regime of graphene have been detected, their thermodynamic properties have not yet been measured. Now, a local probe technique enables the detection of the magnon density and chemical potential.

of about 15-20%. We argue that this gap reduction is a result of magnons binding with electrons or holes to form skyrmions, which, together with estimates of the temperature, allows us to determine the local magnon chemical potential and free magnon density in the system. The method of extracting the thermodynamic properties of magnons introduced in our experiments suggests novel routes towards realizing and probing Bose-Einstein condensation in QHFMs 6 , and is more broadly applicable to other flat-band systems with spontaneously symmetry-broken states.
The device and measurement setup are shown in Fig. 1a,b. Figure 1c shows the two-terminal conductance G 2T between contacts 2 and 3 as a function of back-gate voltage V BG and d.c. bias V d.c. at a magnetic field B = 11 T. Consistent with previous studies 4,5 , for -E Z < V d.c. < E Z , G 2T exhibits a plateau as a result of the quantization of the Hall conductance. However, the quantized Hall plateau disappears as soon as |V d.c. | reaches E Z , signalling the onset of magnon generation and absorption processes.
To study the dependence of the ν = 1 gap on the magnon population, we measure the electron chemical potential µ(ν) as a function of filling factor ν at each value of V d.c. (Methods). Figure 1d shows two representative measurements of the electron chemical potential µ(ν) near ν = 1, with top-gate voltage V TG = 0. The trace at V d.c. = 10 mV (Fig. 1d, red curve) clearly exhibits a reduced gap compared with that at 0 mV (Fig. 1d, blue curve). Similar to the transport behaviour, the gap begins to principally change when V d.c. exceeds E Z (Fig. 1e), initially dropping sharply and reaching suppression of about 20% at the highest biases investigated. The gap reduction shown in Fig. 1d,e, observed at many different locations (Extended Data Fig. 1), demonstrates the remarkable sensitivity of the ν = 1 gap to the presence of magnons.
An important piece of evidence that the ν = 1 gap suppression observed for d.c. biases |V d.c. | > E Z results from magnon generation and absorption is its dependence on the local filling factor under the top gate, ν TG . As a consequence of the spin order present in the region under the top gate, magnons freely propagate across when ν TG = ±1, but only weakly for ν TG = 0 and not at all for ν TG = ±2. Figure 2a shows the a.c. non-local voltage V NL measured across contacts 5 and 6, normalized by the a.c. bias V a.c. applied between contacts 2 and 3. In addition to the vanishing non-local voltage for |V d.c. | < E Z , we find that for |V d.c. | > E Z , no appreciable signal is detected for |ν TG | > 2 or for ν TG = 0; on the other hand, a strong non-local voltage is observed for 0 < |ν TG | < 2, in accordance with the expected transport characteristics and energy-splitting hierarchies shown in previous studies 4,5 . Next, we perform gap Thermodynamics of free and bound magnons in graphene Andrew T. Pierce 1,5 , Yonglong Xie 1,2,5 , Seung Hwan Lee 1 , Patrick R. Forrester 1 , Di S. Wei 1 , Kenji Watanabe 3 , Takashi Taniguchi 4 , Bertrand I. Halperin 1 and Amir Yacoby 1 ✉ Letters Nature Physics measurements near contact 5 as a function of ν TG and V d.c. , using the same contacts for magnon generation. Figure 2b shows the reduction in gap at each V d.c., determined by subtracting-from each point-the average of the three traces with |V d.c. | < E Z at each ν TG , which is similar to the transport measurement (Fig. 2a); deviations in the ν = 1 gap are only observed for |V d.c. | > E Z and 0 < |ν TG | < 2. Intriguingly, the bias dependence of the gap and behaviour of V NL appear to define three regimes. First, biases |V d.c. | < E Z result in no magnon generation and thus leave the gap intact. Second, for biases E Z < |V d.c. | ≲ 4E Z , the gap is rapidly suppressed and the magnitude of V NL is large. Finally, for larger biases |V d.c. | ≳ 4E Z , the suppression is more gradual and the magnitude of V NL is vanishingly small. These observations unambiguously establish that gap suppression results from magnon propagation into the bulk.
The first step toward understanding gap suppression is to identify the nature of charge excitation associated with the ν = 1 gap in the absence of magnons. Theoretical studies 2,7-14 have proposed that the lowest-lying charged excitations at ν = 1 are finite-sized skyrmions, consisting of a single charge ±e 'dressed' by one or more extra overturned spins or a valley texture. For skyrmions comprising flipped

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spins (referred to as 'spin skyrmions'), the excitation energy is determined by the competition between E Z and exchange energy. We, therefore, consider a phenomenological model of spin skyrmions with s flipped spins 15 , whose occupation follows a Boltzmann distribution (Methods and Supplementary Information). Extended Data Fig. 2 shows the best fits to the data using this phenomenological model. The satisfactory agreement between the fit and data at many different locations validates our model and allows us to determine the Coulomb energy E C , which sets the overall scale for the skyrmion and magnon energies (Methods), which is around 21.4 meV, consistent with previous local compressibility measurements of the ν = −1 gap 11 . Most notably, we find that <s>, the mean number of extra spins carried by a charge excitation, is less than 6% of an electron spin in the absence of injected magnons, establishing that the lowest-lying charge excitation consists of bare electrons and holes.
The observed gap suppression can be naturally captured by extending the phenomenological spin skyrmion model to incorporate the presence of magnons 16 (Methods), where we describe the magnons by an effective Bose-Einstein distribution with chemical potential µ m (refs. 17,18 ) and electron temperature T. The magnon chemical potential µ m defines an equilibrium between free magnons and those bound as flipped spins in skyrmions, and may be non-zero due to S Z conservation (that is, magnon number conservation) in the bulk, which results from the weak spin-orbit coupling and small number of nuclear spins present in graphene. Since each magnon represents one flipped spin and therefore one unit of E Z , pumping magnons into the system amounts to externally supplying some of the work needed to flip spins. Assuming there is equilibration between the charge excitations and free magnons (Methods), this results in a reduction in the Zeeman free-energy cost by µ m  10 19 18.  Fig. 3a shows the circuit). The centre of the ν = 1 plateau is around V BG = 3.5 V. c, R xx as a function of temperature with no bias applied to contact 3 near ν = 1 using the same circuit as b. d, Temperature of the system as a function of V d.c. extracted from R xx thermometry measurements (Methods). The grey dashed lines mark the Zeeman energy. e-g, Magnon chemical potential µ m /E Z (e), free magnon density per flux n m (f) and the number of extra flipped spins carried by charge <s> (g) extracted from the skyrmion model (Methods). The shaded region corresponds to a medium-bias regime where heating due to magnon injection plays a key role.

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per flipped spin for the spin skyrmion, favouring the formation of skyrmions over bare electrons or holes and thus suppressing the overall charge gap. To compare the predictions of this model with our experiments, we compute the ν = 1 gap as a function of µ m and T with the parameters obtained by fitting the measured zero-bias µ(ν) curves ( Fig. 3a and Methods). The results of these calculations indicate that considerable enhancements of µ m and T are required to achieve the measured gap suppression at large biases (as shown by the constant-gap contours in Fig. 3a). To use our model to extract the magnon chemical potential µ m , an independent estimate of electron temperature T as a function of d.c. bias is required. Such an estimate is furnished by a measurement of longitudinal resistance R xx in the presence of magnon pumping using a circuit configuration that is insensitive to magnon absorption at the contacts (Methods). Strikingly, we find that the measured R xx displays a sudden increase for |V d.c. | > E Z , indicative of its magnon origin. The comparison of the bias-dependent R xx measurement with the measurement of R xx at zero bias as a function of temperature ( Fig. 3c) suggests that injecting magnons into the system results in the electron temperature heating up to approximately 3 K. By finding the best-fit temperature for each V d.c. , we extract the quantitative values of electron temperature T (Fig. 3d), which spans three distinct regimes. In the low-bias regime of |V d.c. | < E Z , no magnons are generated and T remains at the base temperature. Between E Z and approximately 4E Z , T rapidly increases as a function of bias. Finally, above approximately 4E Z , T saturates and once again remains constant to the highest biases investigated. We have performed similar estimates using a variety of circuit configurations, both two-and four-terminal configurations (Extended Data Figs. [3][4][5], which point to a similar range of temperatures. Estimates of T(V d.c. ) and the results of our model calculations allow us to relate the measured gap values to µ m . Specifically, we determine µ m (V d.c. ) (Fig. 3e) by matching our measured gap values and T to the simulation results (Fig. 3a,d; Extended Data Fig. 6 provides the analysis at another location). As in the case of T, the measured gap and V NL , we find that µ m (V d.c. ) exhibits three separate regimes. At low bias, that is, |V d.c. | < E Z , we have µ m = 0 in accordance with the general properties of the Bose-Einstein distribution. At intermediate bias, namely, E Z < |V d.c. | ≲ 4E Z , we observe no increase in µ m , despite the presence of magnon transport signatures in V NL (Extended Data Fig. 7). Thus, the behaviour of the measured gap in the intermediate-bias regime can be explained as a result of heating due to the injected magnons without invoking the possibility of skyrmion formation. At high bias, that is, where T is approximately 3 K, we extract the values of µ m in excess of zero, as expected in the presence of magnon pumping. We emphasize that gap suppression observed in this regime cannot be explained by heating alone, as this would require the temperature to continue to linearly increase beyond V d.c. = ±5 mV and reach as high as 6 K at V d.c. = ±10 mV, in direct contradiction to the temperature estimated from our zero-bias R xx measurements (Fig. 3b,c and Extended Data Fig. 3).
A further insight can be gained by examining the density n m of the equilibrated free magnons obtained from our calculations and the mean number of overturned spins per skyrmion <s> as a function of V d.c. . Figure 3f shows the extracted n m (V d.c. ) in units of magnons per flux quantum ϕ 0 . In the range of E Z < |V d.c. | ≲ 4E Z , the finite V NL and gap suppression measurements demonstrate that magnons are at work (Extended Data Fig. 7). However, we find that µ m does not increase in this range and n m , therefore, remains negligibly small. We speculate that two possible scenarios may explain this apparent contradiction. One hypothesis is that for E Z < |V d.c. | ≲ 4E Z , there is an additional population of magnons, possibly of a very long wavelength, which is not in thermal equilibrium with the electrons and thus is not captured in the computed n m values despite contributing to G 2T and V NL . A comparison of the lower bound on the magnon lifetime obtained from our transport measurements and device geometry with theoretical estimates of the magnon-electron scattering time shows that magnons with k less than approximately 0.01l B -1 , where l B is the magnetic length, may scatter with skyrmions slowly enough to fail to equilibrate. A second, more exotic possibility is that, in fact, only a very small number of magnons are present in this bias regime, which would imply that a highly efficient mechanism of transport is responsible for the changes in G 2T and V NL . On the other hand, for 4E Z ≲ |V d.c. |, a finite population of equilibrated free magnons emerges, which appears to linearly scale with V d.c. . We note that for a 100 µm 2 sample at 11 T, the highest equilibrium magnon density of about 3 × 10 -3 per ϕ 0 corresponds to a total number of equilibrated magnons only of the order of 300. It is possible that a population of non-equilibrated magnons also persists in this regime. In any case, these observations suggest that the absorption rate of magnons at the contacts may be outpaced by the finite population of free magnons, causing V NL to weaken and G 2T to level off at high biases (Extended Data Fig. 7). Finally, the corresponding <s> (Fig. 3g) displays a similar trend as n m and is three excess overturned spins at the highest biases, consistent with our overall mechanism of gap suppression. The change in behaviour as V d.c. exceeds 4E Z may be related to the large increase in   Letters Nature Physics specific heat as µ m increases (Supplementary Information) and/or to higher-energy valley-spin excitations hosted by a valley-polarized charge-density-wave ground state (Methods) 19 .
In a low-density electron system, correlation effects induced by the Coulomb repulsion between carriers can result in negative (inverse) electronic compressibility dµ/dn (refs. 20,21 ), which can be observed at ν = 0 + ε, 1 ± ε and 2 ± ε (Fig. 4a,b). The associated correlation energy scale, approximated in our model to the leading order by E WC ~ √ ε, governs the magnitude of negative compressibility. Intriguingly, we find that the negative-compressibility features at ν = 1 ± ε respond differently to V d.c. than those at ν = 0 + ε and 2 ± ε, with those at ν = 1 ± ε being greatly diminished at high bias voltages. The pronounced reduction with increasing bias for E Z < |V d.c. | ≲ 4E Z is presumably due to heating, but the reduction with increasing bias beyond this point-where the electron temperature is found to be constant-signals that the strength of correlations is suppressed by the presence of magnons. A possible explanation is that the formation of skyrmions may decrease the magnitude of correlation energy, because the electric charge of a skyrmion is more spread out than a bare electron or hole in the lowest Landau level. A comparison of the average negative compressibility for ν = 0 + ε, 1 ± ε and 2 ± ε (Fig. 4c,d) shows that it is sensitive to V d.c. only near ν = 1, providing additional evidence for the magnon origin. Further study is required to fully establish the microscopic mechanism of these effects.
Looking ahead, the methods of measuring µ m demonstrated here can be used to map out this important quantity over extended spatial regions. As the gradient of µ m is the driving force of magnon currents, such studies may provide further new insights into the nature of magnon transport in the system 22 , including the ν = 0 state in monolayer graphene, which may support spin superfluidity 6,23 . The ability to tune µ m in situ raises the possibility of dynamical control of quantum phases analogous to recent pump-probe experiments 24 , but using magnetic excitations instead of terahertz frequencies. Finally, our combined ability of manipulating and probing the magnon chemical potential is immediately applicable to intriguing correlated insulating states recently reported in moiré superlattice systems, which are expected to support electrically addressable neutral excitations similar to the ν = 1 QHFM 25-27 .

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Methods Sample preparation. The device consists of monolayer graphene encapsulated by two layers of hexagonal boron nitride (hBN) on a p-doped Si substrate with a 285 nm layer of SiO 2 , and was fabricated using a dry transfer technique. A gold top gate was defined using electron-beam lithography and thermally evaporated Cr/Au. The final device geometry was defined by electron-beam lithography and reactive-ion etching. Edge contacts were made by thermally evaporating Cr/Au while rotating the sample using a tilted rotation stage.
Measurements. All the measurements were carried out in a 3 He cryostat with a base temperature of approximately 500 mK. The transport measurements were performed using standard lock-in techniques with a 100 µV excitation with frequencies ranging from 17 to 40 Hz. The temperature-dependent measurements were recorded by applying current to a resistive heater located at the 3 He stage. The SET tips were fabricated using the procedure described elsewhere 28 . The diameter of the SET is approximately 100 nm, and it was held about 300 nm above the encapsulated graphene. Compressibility measurements were performed using d.c. and a.c. techniques similar to those described elsewhere 28 . The SET serves as a sensitive detector of changes in electrostatic potential δφ, which is related to the chemical potential of the graphene flake by δµ = -eδφ when the system is in equilibrium. In the a.c. scheme used to measure dµ/dn, an a.c. voltage is applied to the Si back gate to weakly modulate the carrier density of graphene, and the corresponding changes in SET current are converted to chemical potential by normalizing the signal by that from a small a.c. bias directly applied to the sample. For d.c. measurements, an analogue proportional-integral-derivative controller is used to maintain the SET current at a fixed value by changing the tip-sample bias. The corresponding change in sample voltage provides a direct measure of µ(n).
Spin skyrmion model. We briefly summarize the skyrmion model used for estimating the magnon chemical potential µ m . Assuming that both density of overturned spins and deviation from ν = 1 is small, the densities of electron-like and hole-like skyrmions with s overturned spins, denoted as n e s and n h s , respectively, follow the Boltzmann distributions given by where E e s and E h s are the energy of elementary charged electron-like and hole-like excitations at ν = 1, respectively; µ is the electron chemical potential; T is the temperature; δµ WC (ν) is a Wigner crystal-like energy functional; a e and a h are fit parameters; and µ m is the magnon chemical potential (Supplementary Information provides complete details of the parameters used). The energies E e s and E h s as well as δµ WC (ν) are parametrized by an overall phenomenological Coulomb energy scale E C that is treated as a fit parameter to be obtained by fitting the zero-bias µ traces. Extended Data Fig. 2 shows examples of the zero-bias fit results, which are in excellent agreement with the experimental traces, along with the fit parameters E C , a e , a h and a Gaussian density-broadening parameter Δν. These fit parameters, along with our independent measurements of the ν = 1 gap and T, can then be combined with the distributions n e s and n h s to determine μ m and other thermodynamic properties ( Fig. 3e-g, Extended Data Fig. 6b-d and Supplementary Information). R xx and G 2T thermometry. To obtain an estimate of the electron temperature T independent of our compressibility measurements, we perform R xx measurements in the presence of magnon generation using the circuit shown in Extended Data Fig. 3. Keeping contact 2 grounded, we apply an a.c. bias between contacts 1 and 4 and measure the longitudinal a.c. voltage V xx across contacts 5 and 6. We emphasize that this measurement of R xx is different than the non-local voltage and is not expected to be directly sensitive to contributions from magnon generation and absorption, and we expect the phonon contribution to R xx to be small in the temperature range of interest (Supplementary Information provides further details). To generate magnons, a d.c. bias is applied to contact 3; in this case, no a.c. modulation is applied to the magnon generation contacts. Strikingly, the measured R xx as a function of d.c. bias (Extended Data Fig. 3) displays an abrupt change when the applied d.c. bias exceeds the Zeeman energy, reminiscent of the response observed in the magnon transport experiments with a.c. modulation applied to contact 3 (Fig. 1). However, we emphasize that the change in R xx is not caused by magnon absorption events as in the case of the V NL signal discussed earlier or by other hot-carrier effects, because the a.c. modulation used for monitoring R xx is not applied to the d.c.-biased contacts used for magnon generation. Extended Data Fig. 3d,e shows R xx measured using the same circuit with no d.c. bias applied to contact 3 as a function of temperature. Under these conditions, the electron temperature is expected to be well equilibrated with the lattice temperature 29 . Remarkably, we find good agreement between an R xx trace measured at a given d.c. bias and that at a given temperature (Fig. 3e, where the error bar is estimated by matching the R xx value with a d.c. bias to the temperature-dependent R xx with up to 5% error), suggesting that the change in R xx at a d.c. bias greater than the Zeeman energy is equivalent to an increase in the temperature of the system. We note that although the phonon temperature is expected to remain near the base temperature in the biased case, we do not expect phonons to play an important role in our measurements (Supplementary Information). A comparison of these two R xx measurements, therefore, allows us to determine the temperature of the system when magnons are pumped into the system and uniquely determine μ m .
Alternatively, the two-terminal conductance G 2T may be used as a proxy for the temperature instead of the four-terminal R xx . Extended Data Fig. 4a,b shows the bias-and temperature-dependent two-terminal conductance G 2T , respectively, measured with an a.c. voltage between contacts 1 and 4 using the circuit shown in Extended Data Fig. 3a. As in the case of R xx measurements, once the system has heated beyond about 5 K, the principal signatures of the quantum Hall effect vanish (in this case, the plateau), thus placing an overestimated but crucial upper bound for the temperature of our system. Extended Data Fig. 4c shows G 2T at a d.c. bias of -10 mV compared with a selection of zero-bias traces taken at various temperatures, which points to a temperature of about 3 K-in good agreement with the results obtained by analysing R xx . We have verified this behaviour in numerous circuit configurations-both two-terminal and four-terminal configurationswhich consistently point to the same range of temperatures (Extended Data Fig.  5, except for the positive biases shown in Extended Data Fig. 5h, which is likely due to a bad contact). We regard the R xx measurements as a more reliable indicator of temperature, as G 2T is more susceptible to effects stemming from the contact resistance. Nevertheless, our observation that both R xx and G 2T thermometry techniques yield approximately the same electron temperature leads us to conclude that reliable estimates can be derived from either technique.

Thermalization of magnons and electrons.
To understand the degree of equilibration between the skyrmions and free magnons and why there could be an additional population of magnons that are not in equilibrium with the electrons, we note that the degree of thermalization between a magnon and the skyrmion population depends on its momentum k. Moderate-to-short-wavelength magnons with k l −1 B are equivalent to well-separated electron-hole pairs and therefore may be expected to thermalize with the skyrmion population very quickly-at a rate nearly equal to that of a single free electron or hole. On the other hand, long-wavelength magnons with momenta k ≪ l −1 B are equivalent to tightly bound electron-hole pairs, which only carry a small electric dipole moment and therefore are expected to couple more weakly to skyrmions and thermalize more slowly. In general, for a given magnon to equilibrate with the skyrmion population, its lifetime must exceed the mean magnon-electron scattering time τ me (k). The magnon-electron scattering rate Γ k = τ −1 me has been calculated within the context of bilayer QHFMs to be 30 where δν is the difference in the Landau-level filling factor from the nearest integer (taken as 0.01 here) and v(k) is the magnon velocity. To obtain a lower bound for the magnon lifetime, we note that in the absence of magnetic impurities, we expect magnon absorption events at the contacts 4 to be the dominant mechanism by which magnons are removed from the bulk. The requirement that magnons survive long enough to travel the approximately 10 μm distance between the contacts in the device, therefore, allows us to estimate a lower bound on the lifetime τ min ≈ d/v(k), where d = 10 μm. Extended Data Fig. 8 plots the two timescales τ me (k) and τ min (k) as a function of k, and shows that only very-long-wavelength magnons with k 0.01l −1 B are expected to scatter slowly enough with skyrmions to fail to come into equilibrium.

Role of valley skyrmions.
A number of theoretical studies have considered the nature of the lowest-lying charged excitations in the ν = 1 QHFM [10][11][12][13][14] . Although valley skyrmions may be favoured under ideal conditions, the presence of a boron nitride substrate in encapsulated devices may result in the breaking of sublattice symmetry and therefore disfavour the formation of valley skyrmions. Although we do not find direct evidence for a gap at the charge-neutrality point (CNP) in our device, we observe a robust incompressible state at ν = 5/3, with an incompressible peak comparable in magnitude to those occurring at ν = 1/3 and 2/3 (Extended Data Fig. 9). The conspicuous absence of this state in previous local compressibility measurements on suspended devices was attributed to low-lying valley-skyrmionic excitations with energy less than that of a Laughlin quasiparticle 11,28 . Thus, the observation of robust incompressible states at ν = 5/3 strongly suggests that valley skyrmions are disfavoured in our sample. Furthermore, within the spin skyrmion model of gap suppression, we do not expect the presence of magnons to alter the energy cost of adding a valley skyrmion. These conclusions also hold for valley-coherent ground states, in which the spin excitation spectrum is expected to be the same as the valley-polarized case considered above 19 . Hence, we conclude that valley skyrmions are unlikely to play an important role in the observed ν = 1 gap reduction.

Discussion of a possible gap at the CNP.
To search for evidence of sublattice symmetry breaking, we performed high-resolution local compressibility measurements near the CNP at zero magnetic field, which was compared with a Letters Nature Physics model that considers the sublattice-gapped Dirac form μ (n) = √ ∆ 2 0 4 + πv 2 F n h 2 , where Δ 0 and ν F are the sublattice gap and Fermi velocity, respectively, and ℏ is the reduced Planck constant. Extended Data Fig. 10 shows two fits of the measured inverse compressibility at zero magnetic field to the sublattice-gapped Dirac model, one with disorder broadening and the other one without broadening. The unbroadened fit favours a scenario in which the sublattice gap is zero. The broadened fit, however, yields a mean squared error approximately one-half that of the unbroadened fit, as well as favours a scenario in which the sublattice gap is approximately 12.3 meV with a broadening of 7 × 10 9 cm -2 , consistent with that extracted from our fit to the ν = 1 gap at a high magnetic field. These considerations suggest that sublattice symmetry is likely broken by the boron nitride substrate, disfavouring the formation of valley skyrmions, despite the compressibility signature of the gap being obscured by disorder broadening at zero magnetic field.

Data availability
Source data are provided with this paper. All other data that support the findings of this paper are available from the corresponding author upon reasonable request.

code availability
The code that supports the findings of this study is available from the corresponding author upon reasonable request. Fig. 1 | Additional examples of the ν=1 gap suppression by the presence of magnons. a, Optical image of the device indicating the circuit used for magnon generation and the locations where the gap measurements were taken. b, Bias-dependent energy gap measured at location 2. The grey dotted lines mark ±E Z . While the origin of the small asymmetry for |V DC |< E Z is unclear, its magnitude is much smaller than overall suppression observed at higher bias, and the top gate dependence shows that the onset consistently occurs near E Z (see Fig. 2). c-g, Chemical potential µ near ν = 1 measured with V DC = 0 mV (blue) and V DC = 10 mV (red) at 6 different locations. Although the local value of the ν = 1 gap varies, its reduction by the presence of magnons is clearly reproduced in all the data sets. Fig. 3 | R xx thermometry. a, Circuit used for R xx measurements. Contacts 2 and 3 are used to generate magnons. Contacts 1, 6, 5, and 4 are used to measure R xx . The white arrows indicate the chirality of the current flow. b, R xx as a function of V DC at V BG = 3.8 V. The value of R xx tends to saturate for |V DC | greater than ~4E Z , suggesting that the temperature saturates. c, Individual R xx traces measured at base temperature with various values of V DC applied to contact 3 near ν=1 using the circuit shown in a. The center of the ν=1 plateau is around a back gate voltage of 3.5 V. d, Individual R xx traces measured at various temperatures with zero DC bias applied to contact 3 near ν=1 using the circuit shown in a. e, Individual R xx traces measured at base temperature with 10 mV applied to contact 3 (blue dots) and at various temperatures with 0 mV applied to contact 3 (orange, yellow and purple lines). The close agreement between the blue dotted line and the red line suggests that the effect of magnon generation on the R xx measurement is primarily due to heating. These measurements also demonstrate that the increase in temperature due to magnon generation does not exceed 4.5 K at V DC = -10 mV.