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[Benjamin A. Foutty](https://orcid.org/0009-0000-5158-6097), [Vladimir Calvera](https://orcid.org/0000-0002-8888-1759), [Zhaoyu Han](https://orcid.org/0000-0002-1778-6906), Carlos R. Kometter, Song Liu, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), James C. Hone, Steven A. Kivelson, [Benjamin E. Feldman](https://orcid.org/0000-0002-4962-0548)

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[Anomalous Landau Level Gaps Near Magnetic Transitions in Monolayer <math display="inline">  <mrow>    <msub>      <mrow>        <mi>WSe</mi>      </mrow>      <mrow>        <mn>2</mn>      </mrow>    </msub>  </mrow></math>](https://mdr.nims.go.jp/datasets/2a2430d1-27e5-47a1-ac13-af52178d9dc5)

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Anomalous Landau Level Gaps Near Magnetic Transitions in Monolayer WSe2Anomalous Landau Level Gaps Near Magnetic Transitions in Monolayer WSe2Benjamin A. Foutty ,1,2 Vladimir Calvera ,2 Zhaoyu Han ,2 Carlos R. Kometter,1,2 Song Liu,3 Kenji Watanabe ,4Takashi Taniguchi,5 James C. Hone,3 Steven A. Kivelson,2 and Benjamin E. Feldman 1,2,6,*1Geballe Laboratory for Advanced Materials, Stanford, California 94305, USA2Department of Physics, Stanford University, Stanford, California 94305, USA3Department of Mechanical Engineering, Columbia University, New York, New York 10027, USA4Research Center for Electronic and Optical Materials, National Institute for Materials Science,1-1 Namiki, Tsukuba 305-0044, Japan5Research Center for Materials Nanoarchitectonics, National Institute for Materials Science,1-1 Namiki, Tsukuba 305-0044, Japan6Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory,Menlo Park, California 94025, USA(Received 14 February 2024; revised 21 June 2024; accepted 25 June 2024; published 1 August 2024)First-order phase transitions produce abrupt changes to the character of both ground and excitedelectronic states. Here we conduct electronic compressibility measurements to map the spin phase diagramand Landau level (LL) energies of monolayer WSe2 in a magnetic field. We resolve a sequence of first-order phase transitions between completely spin-polarized LLs and states with LLs of both spins.Unexpectedly, the LL gaps are roughly constant over a wide range of magnetic fields below the transitions,which we show reflects spin-polarized ground states with opposite spin excitations. These transitions alsoextend into compressible regimes, with a sawtooth boundary between full and partial spin polarization. Welink these observations to the important influence of LL filling on the exchange energy beyond a smoothdensity-dependent contribution. Our results show that WSe2 realizes a unique hierarchy of energy scaleswhere such effects induce reentrant magnetic phase transitions tuned by density and magnetic field.DOI: 10.1103/PhysRevX.14.031018 Subject Areas: Condensed Matter PhysicsI. INTRODUCTIONElectronic systems with degeneracies arising from inter-nal quantum degrees of freedom are often susceptibleto forming ordered ground states driven by many-bodyinteractions. The quantum Hall regime, in which Landaulevels (LLs) effectively quench kinetic energy, provides amodel platform to study such phases and the transitionsbetween them. In particular, the relative energies ofLLs with distinct spin and/or valley indices can oftenbe modified by experimental tuning knobs which affectboth the many-body ground and excited states in thesesystems [1–16]. However, the nature of charge excitationsnear transitions between competitive LLs depends sensi-tively on details of the LL energetics and can be difficult todirectly probe.Monolayer semiconducting transition metal dichalcoge-nides realize a distinctive LL structure due to theirhierarchy of energy scales. Strong spin-orbit coupling nearthe valence band maxima at valleys K and K0 causes therelevant low-energy bands to be spin-valley locked, so thatonly a single Ising spin orientation is relevant at each valley[17]. This degree of freedom forms a generalized isospin,which we refer to as spin in the rest of the text. Acombination of the large effective mass and the additivecontributions of orbital and Berry-curvature effects pro-duces a single-particle Zeeman splitting of the valence bandE0Z that is large relative to the cyclotron energy Ecyc inmonolayer WSe2 (E0Z=Ecyc ≈ 2) [18]. The large effectivemass also enhances the relative importance of interactions,such that the dimensionless parameter rs is of the order of5–10 at achievable carrier densities [19].Prior work has shown that these interactions drive adensity-dependent exchange enhancement of the effectiveZeeman energy EZ [9,18,20]. This increases the spinsplitting of the valence bands as the hole density decreasesand leads to preferential occupation of fully spin-polarizedLLs at low densities [Fig. 1(a)] [19]. At higher densities,both “majority” and “minority” spin Zeeman-split LLs areoccupied, causing alternating LL gap sizes dominated byeven or odd integers [9,18,20–22]. Recent studies havenoted the possibility of first-order phase transitions at thecrossover between these limits in monolayer WSe2 and*Contact author: bef@stanford.eduPublished by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.PHYSICAL REVIEW X 14, 031018 (2024)2160-3308=24=14(3)=031018(8) 031018-1 Published by the American Physical Societyhttps://orcid.org/0009-0000-5158-6097https://orcid.org/0000-0002-8888-1759https://orcid.org/0000-0002-1778-6906https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-4962-0548https://ror.org/00f54p054https://ror.org/00hj8s172https://ror.org/026v1ze26https://ror.org/026v1ze26https://ror.org/05gzmn429https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevX.14.031018&domain=pdf&date_stamp=2024-08-01https://doi.org/10.1103/PhysRevX.14.031018https://doi.org/10.1103/PhysRevX.14.031018https://doi.org/10.1103/PhysRevX.14.031018https://doi.org/10.1103/PhysRevX.14.031018https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/related systems, but hysteresis has not been observed and adetailed understanding of the LL energetics as the systemtransitions from fully to partially spin polarized has untilnow been lacking [9,23,24].In this work, we use a scanning single-electron transistorto measure the inverse electronic compressibility dμ=dn ofvalence band holes in monolayer WSe2 in a perpendicularmagnetic field. At the crossover between fully spin-polar-ized LLs and the lowest-energy minority spin LL beingfilled, we resolve first-order phase transitions, includinghysteresis in the LL gaps and adjoining sharp tails ofnegative compressibility that extend outward into nearbycompressible electronic states. Surprisingly, the LL gapsare roughly constant over a wide range of magnetic fieldsbelow these phase transitions. Through high-resolutionmeasurements of the thermodynamic LL gaps and thefirst-order phase transitions, together with supportingtheoretical calculations, we systematically characterizethe nature of low-energy charge excitations throughoutthe phase diagram. Collectively, these indicate multiplereorderings of the LL structure as we vary carrier densityand magnetic field. Our results provide a straightforwardway, in the correlation-dominated regime, to understand thespin character and energies of ground and excited elec-tronic states.II. FIRST-ORDER SPIN TRANSITIONSIn Fig. 1(b), we present a Landau fan of dμ=dn as afunction of carrier densityn and perpendicularmagnetic fieldB. Across data from three distinct devices, we observequalitatively similar behavior, though we focus on data froma single sample through most of the main text (seeSupplemental Material Sec. II for a detailed comparison[25]). Most of the incompressible features can be identifiedvia their slope in the n-B plane as integer quantumHall gaps,which occur at all integer filling factors ν. Additionally, wenote fractional quantum Hall states in the lowest LL con-sistent with previous reports [9,26] (Supplemental MaterialSec. III [25]).FIG. 1. First-order Landau level (LL) phase transitions in monolayer WSe2. (a) Schematic of the valence band LL structure inmonolayer WSe2. The two relevant bands are spin up at valley K and spin down at valley K0, which are split by a density-dependenteffective Zeeman energy EZ. Ecyc is the cyclotron energy. (b) Inverse electronic compressibility dμ=dn in monolayer WSe2 as a functionof hole density n and perpendicular magnetic field B. Black circles mark sharp drops in the magnitude of dμ=dn along incompressibleLL gaps. Broad vertical features are artifacts due to long ac charging time of the sample (Supplemental Material Sec. I [25]). (c) Spinsusceptibility χeff as a function of n, determined from the densities of the LL transitions [such as those highlighted in (b)] from threedistinct samples. We also present data from Ref. [9] and theoretical predictions based on quantum Monte Carlo calculations shaded ingray for comparison. (d),(e) Enlargements of dμ=dn in the white box in (b), with the density swept in opposite directions (large whitearrows). (f) Difference between (d) and (e), demonstrating pronounced hysteresis from a first-order phase transition. (g) Schematics ofLL energies that respectively correspond to the starred positions in (d) and (e); note that due to the hole carriers, states are filled from thetop downward. N denotes the LL orbital index and EF is the Fermi level.BENJAMIN A. FOUTTY et al. PHYS. REV. X 14, 031018 (2024)031018-2Following constant integer ν, we observe abrupt drops inthe magnitude of the LL gaps [marked by black circles inFig. 1(b)] that occur when the first minority spin LL crossesthe highest occupied majority spin LL, corresponding to atransition from full to partial spin polarization [9]. Wedetermine the effective spin susceptibility χeff by identify-ing the number of polarized LLs at the phase transitions[9,10,27]. Note that this is actually a response to a finitefield, and can strictly be identified as the “susceptibility” inthe noninteracting limit. We find that χeff extends up to 13,corresponding to a g factor of roughly 35 at the lowestdensity transition that we can resolve [Fig. 1(c)]. Theobserved increase of χeff with decreasing hole density isbroadly consistent with previous quantum Monte Carlocalculations of a 2D electron gas, shaded in gray [27,28](Supplemental Material Sec. IX [25]).We resolve hysteresis in both the LL gaps and adjacentnegative compressibility features (discussed in detailbelow) as shown in Figs. 1(d)–1(f), direct evidence offirst-order phase transitions. As the hole density is swept inopposite directions, the sharp drop in gap size occurs atdistinct magnetic fields (we also observe signatures ofhysteresis upon sweeping B; see Supplemental MaterialSec. II [25]). The behavior reflects spontaneous polariza-tion switching of the last occupied LL [Fig. 1(g)] [23,24].Specifically, the large gap is stabilized when sweeping froma fully polarized phase, where exchange interactions favormaintaining maximum spin polarization and enhance theeffective g factor [29]. We only resolve hysteresis as thefirst (orbital index N ¼ 1) minority spin LL becomescompetitive with the valence majority spin level; at lowermagnetic fields at the same density where LLs with higherindices cross, there are no sharp changes in the measuredgaps (Supplemental Material Sec. IV [25]). This indicatesthat the additional g-factor enhancement is suppressedwhen both majority and minority spin LLs are occupied,destroying the expected first-order transition or rendering itundetectably weak.III. ANOMALOUS LL GAP SCALINGOur measurements encode information not only aboutchanges in the occupied LLs, but also about excited statesimmediately above the Fermi level. To study the LLenergetics in the vicinity of the spin phase transitions,we integrate dμ=dn to obtain the thermodynamic gapsΔν atinteger filling factors ν (Fig. 2). For each integer quantumHall gap, we observe three distinct behaviors as a functionof magnetic field. We relate these behaviors to distinctground states and their lowest-energy charge excitations, asdetailed below.At fixed filling factor and low fields, highlighted by a red“P” in Figs. 2(a)–2(e), the LLs are fully spin polarized. Weobserve a linear field dependence of these gaps, indicatingthat they are set by the cyclotron energy Ecyc ¼ ðℏeB=m�Þto the next spin-majority LL. We extract an effective massm� ≈ 0.31me from the linear slope, where ℏ is the reducedPlanck’s constant and e and me are the electron charge andmass (m� depends weakly on sample; see SupplementalMaterial Sec. II [25]). At fixed filling factor and sufficientlyhigh field, highlighted by a gray “M” in Fig. 2, the LLs arein a “mixed” regime where both spins are occupied.Individual gaps grow and shrink as the density-dependenteffective Zeeman energy changes the relative spacingbetween LLs of different spin, but the pairwise sum ofgaps Δν þ Δνþ1 at a given density matches the cyclotronenergy with a similar effective mass to that of the polarizedLLs [Fig. 2(k); see Supplemental Material Sec. V [25] ].This is in agreement with prior measurements of LLgap sizes in monolayer WSe2 at higher hole densities,though we measure a slightly lower effective mass [18](Supplemental Material Sec. II [25]). Both the polarizedand mixed regimes can be well described by the previouslyconsidered model in which LLs are affected by a smoothdensity-dependent Zeeman enhancement but are otherwiseunchanged energetically [9]. Our experiments, however,demonstrate a more complicated behavior at the transitionbetween these regions.Between the polarized and mixed regimes, we observethat each LL gap plateaus as the field is increased precedingits first-order phase transition. Remarkably, the range ofmagnetic fields over which the gaps are flat, highlighted bya purple “T” in Fig. 2, can extend over several Tesla (e.g.,between 6.5 and 11 T for ν ¼ −8). At a given density, twogaps (the final two LL gaps that are not in the mixedregime) are approximately field independent and divergefrom the cyclotron energy scale. This is best illustratedby plotting the LL gaps within a fixed density range[Fig. 2(k)]. It is surprising that for any given hole density,multiple LL gaps are set by a scale comparable to, butsmaller than, the cyclotron energy. Similar plateaus persistacross all three samples which were fabricated independ-ently and exfoliated from bulk WSe2 crystals from differentsources, as well as a fourth device of Bernal bilayer WSe2(Supplemental Material Secs. 2, 5, and 6 [25]). Thisconsistency indicates intrinsic and generic behavior unre-lated to disorder or details of dielectric screening.The gap we measure is equivalent to the particle-holeexcitation energy [30]. Mapping the field dependence ofthese gaps thus allows us to determine the spin character ofcharge (hole) excitations. To address how different exci-tations evolve in a field, we consider the effects of Coulombinteractions on the spin-split LLs in WSe2, beyond thegeneral effect of a density-dependent spin susceptibility.Using Hartree-Fock calculations with RPA-screened inter-actions which take into account the large LL mixing in thismaterial, we study the charge gaps from the highest-energyfilled spin-majority LL [identified by its orbital index Nand spin as (jνj − 1;↑)] to both the subsequent spin-majority (jνj;↑) and lowest-energy spin-minority (1;↓)LLs (Supplemental Material Secs. 7 and 8 [25]). Theresulting gaps are plotted in Figs. 2(f)–2(j), with aANOMALOUS LANDAU LEVEL GAPS NEAR MAGNETIC … PHYS. REV. X 14, 031018 (2024)031018-3schematic illustration of how the LL energies evolve withmagnetic field in Fig. 2(l).The gap to the next unoccupied spin-majority LL ismostly set by the cyclotron energy, as exchange interactionsaffect both majority-spin LLs similarly. The gap to thelowest spin-minority LL (the “spin-flip gap”) is determinedby both single-particle and exchange interactions; thelatter strongly renormalize this gap because they havedifferent effects on the particle and hole excitations.The relative balance of these two contributions at a givenfilling factor will vary as the magnetic field (and there-fore carrier density) is tuned. At low magnetic fields,exchange interactions are comparatively stronger anddisfavor minority spin occupation, increasing the spin-flipgap. At higher magnetic fields, the kinetic energy domi-nates the behavior, leading to a linear decrease with B fromthe large orbital mismatch between the relevant LLs. Theresult is a nonmonotonic dependence of the spin-flip gapwith B so that it becomes competitive (and is eventuallyfavored) compared with the cyclotron gap (SupplementalMaterial Sec. VIII [25]).Our numerical calculations [Figs. 2(f)–2(j)] indicate thatfor the LLs we probe in our experiment, the curvature of thespin-flip gap is quite low at the crossover field BT. Thisqualitatively matches the plateaus we observe over anintermediate field range in our measurements. We thereforeFIG. 2. Magnetic field dependence of LL gaps. (a)–(e) Experimentally measured LL gapsΔν as a function of B for fixed integer valuesof the filling factor ν. We observe three distinct behaviors: fully polarized LLs withΔν ¼ Ecyc (labeled by “P”), a transition region whereΔν plateaus (labeled by “T”), and a “mixed” regime where both spins are occupied and Δν þ Δνþ1 ¼ Ecyc (labeled by “M”). Verticaldotted lines indicate boundaries separating these behaviors. (f)–(j) Corresponding theoretically predicted LL gaps from Hartree-Fockcalculations with RPA-screened interactions. We show Δcyc (dotted lines), the gap to the next unoccupied majority spin LL, and Δsf(dashed lines), the gap to the N ¼ 1minority spin LL. The LL gap is the smaller of these (thick colored lines). (k) Measured gaps Δ as afunction of B in the range −1.2 × 1012 < n < −1.0 × 1012 cm−2. Color indicates the filling factors of different gaps: Polarized and“transition” gaps (from ν ¼ −4 to ν ¼ −10) are plotted individually (Δ≡ Δν), while the pairwise sum of gaps are plotted for evenν ≤ −11. The dashed black line is a linear fit (excluding the transition gaps). (l) Illustration of the LL energies relevant to Δν, given interms of Ehole ¼ −E, so LLs are filled from the bottom up. At BT, the lowest-energy LL above the gap switches from majority tominority spin, while at BC, the system undergoes a first-order phase transition to the mixed regime.BENJAMIN A. FOUTTY et al. PHYS. REV. X 14, 031018 (2024)031018-4interpret the transition region as a spin-polarized ground statethat favors occupation of an opposite spin LL upon doping.At higher magnetic fields (B > BC), the system undergoes afirst-order transition to a mixed regime where the ð1;↓Þ LLjumps to lower energy than the ðjνj − 1;↑Þ state. The preciseordering of LLs [for example, whether the ð1;↓Þ LL alsojumps below the ðjνj − 2;↑Þ state] is sensitive todetails of theapproximation in our theoretical calculations (SupplementalMaterial Sec. VIII [25]) and is ambiguous in experiment, sowe restrict our quantitative comparison to B < BC in Fig. 2.IV. REENTRANT MAGNETISM AND FULL SPINPHASE DIAGRAMThe close competition between distinct phases alsoaffects the behavior of the system at partial LL filling.Our measurements near each LL phase transition reveala sharp negative compressibility feature emanating out-ward toward lower hole density as the field decreases[Figs. 3(a)–3(c)]. We interpret this behavior, indicative of afirst-order isospin phase transition [31–34], as an extensionof the LL reordering into compressible states of a partiallyfilled LL [Fig. 3(d)].As holes are initially depleted from integer filling [pinkregion, Fig. 3(d)], holes are removed from the highest-energy majority spin LL [Fig. 3(e)]. This depletion willdecrease the exchange interactions, and the partially filed LLwill be pushed toward the unoccupied minority spin LL. Asadditional holes are removed and the sample enters the grayregion in Fig. 3(d), there is an abrupt reordering of spins andthe minority spin LL is instead occupied [Fig. 3(f)]. Similarphenomenology was also suggested by recent transportmeasurements in a related system [24]. These transitionsat partial LL filling, along with the observation of multipleLL gap plateaus at fixed n (varying B), imply reentrant spinpolarization as the system sequentially fills, depletes, andagain fills holes into theN ¼ 1minority spin LL. This leadsto a “sawtooth” boundary between fully and partially spin-polarized phases in the n-B plane, which we show in the fullspin phase diagram in Fig. 3(g).FIG. 3. Spin transitions within partially filled LLs and full phase diagram. (a)–(c) High-resolution measurements of dμ=dn near phasetransitions at ν ¼ −9;−10;−11, highlighting the negative compressibility which extends into the adjacent LL. Panel (a) is measured attemperature T ¼ 1.6 K, while panels (b) and (c) are at T ¼ 0.35 K. The small shift in ν of the incompressible peak that occurs across thephase transition in (b) and (c) likely reflects asymmetry in the broadening of the crossing LLs (Supplemental Material Sec. X [25]).(d) Schematic showing regions of distinct partially occupied LLs as a function of Bwhen jνj ≤ N for a given integerN. The pink shadedregion corresponds to a fully polarized phase, with filling ðjν↑j; jν↓jÞ ¼ ðN − ϵ; 0Þ, where ν↑ð↓Þ is the filling factor of holes in the spin-↑(spin-↓) sector and 0 < ϵ < 1. The light gray region is a mixed state with ðjν↑j; jν↓jÞ ¼ ðN − 1; 1 − ϵÞ; i.e., a minority spin LL ispartially filled. (e) and (f) Schematic of the LL orderings in each compressible phase, with red spin majority and blue spin minority LLslabeled by their respective orbital indices. (g) Schematic depiction of the full spin phase diagram as a function of n and B. Lines indicatethe LL gap behavior, while shading indicates the spin character of the filled states.ANOMALOUS LANDAU LEVEL GAPS NEAR MAGNETIC … PHYS. REV. X 14, 031018 (2024)031018-5Finally, we discuss how these phase transitions dependon temperature, which provides further insight into therelative free energies of distinct states. In Figs. 4(a) and4(b), we show dμ=dn as a function of ν and temperature Tat a constant magnetic field B ¼ 4.85 T [near the ν ¼ −10transition shown in Fig. 3(b)]. The negative compressibilityfeature shifts to lower hole density as the system coolsbetween T ¼ 1.5 K and T ¼ 0.35 K, indicating that themixed phase is favored at higher temperatures and thuscarries higher relative entropy.The incompressible LL gap significantly strengthens atlower temperatures, as expected. In contrast, the negativecompressibility weakens at the lowest temperatures ofour measurement, displaying a nonmonotonic magnitudeas a function of temperature. This contrasts with measure-ments of isospin transitions in distinct systems, wheresuch features sharpen at lower temperatures [32–36]. InFigs. 4(c) and 4(d), we compare dμ=dn around the ν ¼ −8transition at T ¼ 1.6 K and T ¼ 0.35 K, demonstratingthat negative compressibility is barely visible at the lowesttemperatures of our measurement. To explain this behavior,we use a Sommerfeld expansion to obtain a phenomeno-logical model for the free energy around the phasetransition. We find that at sufficiently low T, the relativeslopes of the free energy as a function of density will becloser together due to an asymmetry in the density of statesof the two phases, suppressing the negative dμ=dn at thephase transition (Supplemental Material Sec. X [25]).V. OUTLOOKIn conclusion, our experiments reveal singular changesin the interaction-induced renormalization of LL energies atthe crossover between spin-polarized and mixed states.The intertwined electronic and magnetic structure in thissystem enables gate control over macroscopic changes inmagnetization at the Fermi level. Our results are relevant toa wider class of systems where many-body effects areeven more prominent. While the sizable single-particleZeeman energies characteristic of monolayer WSe2 makethe system susceptible to spin polarization even withoutinteractions, related systems have displayed exchangedriven polarization in the absence of a large Zeeman energy[37,38]. Interaction-induced polarization (and related phasetransitions) is also relevant within moiré heterostructures, inwhich flat moiré bands quench the kinetic energy akin toLLs and complete spin polarization can be favored atboth zero and finite magnetic field [34,35,39–41]. Ourcomprehensive understanding of the relative kinetic andinteraction effects at transitions between full and partialspin polarization provides a framework for both exper-imental and theoretical study of energetics within these stillmore strongly interacting platforms.ACKNOWLEDGMENTSWe acknowledge helpful conversations with Allan H.MacDonald and Brian Skinner. Device fabrication andscanning single-electron transistor measurements wereprimarily supported by NSF-DMR-2103910. Supportingdata from bilayer WSe2 was supported by the U.S.Department of Energy, Office of Basic Energy Sciences,Award No. DE-SC0023109. V. C., Z. H., and S. A. K.were supported by U.S. Department of Energy, Office ofBasic Energy Sciences, under Contract No. DE-AC02-76SF00515. Synthesis of WSe2 (S. L., J. H.) was supportedby NSF MRSEC program at Columbia through theCenter for Precision-Assembled Quantum Materials(DMR-2011738). K.W. and T. T. acknowledge supportfrom the JSPS KAKENHI (Grants No. 20H00354 andNo. 23H02052) and World Premier International ResearchCenter Initiative (WPI), MEXT, Japan. B. A. F. acknowl-edges support from a Stanford Graduate Fellowship. Part ofthis work was performed at the Stanford Nano SharedFacilities (SNSF), supported by the National ScienceFoundation under Award No. ECCS-2026822.[1] S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H. 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