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Ke Huang, Ajit C. Balram, Hailong Fu, Chengqi Guo, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Jainendra K. Jain, Jun Zhu

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[Hetero-Orbital Two-Component Fractional Quantum Hall States in Bilayer Graphene](https://mdr.nims.go.jp/datasets/7316d6cb-1ad3-4183-9597-df10162ae841)

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Hetero-Orbital Two-Component Fractional Quantum Hall States in Bilayer GrapheneHetero-Orbital Two-Component Fractional Quantum Hall States in Bilayer GrapheneKe Huang ,1 Ajit C. Balram ,2,3 Hailong Fu ,1,* Chengqi Guo ,1 Kenji Watanabe ,4Takashi Taniguchi ,5 Jainendra K. Jain ,1,† and Jun Zhu 1,6,‡1Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA2Institute of Mathematical Sciences, CIT Campus, Chennai 600113, India3Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai 400094, India4Research 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, Japan6Center for 2-Dimensional and Layered Materials, The Pennsylvania State University,University Park, Pennsylvania 16802, USA(Received 8 January 2025; revised 8 May 2025; accepted 29 May 2025; published 22 July 2025)A two-dimensional electron system exposed to a strong magnetic field produces a plethora of stronglyinteracting fractional quantum Hall (FQH) states, the complex topological orders of which are revealedthrough exotic emergent particles, such as composite fermions, and fractionally charged Abelian and non-Abelian anyons. Much insight has been gained by the study of multicomponent FQH states, where spinand pseudospin indices of the electron contribute additional correlation. Traditional multicomponentFQH states develop in situations where the components share the same orbital states and the resultinginteractions are pseudospin independent; this homo-orbital nature is also crucial to their theoreticalunderstanding. Here, we study “hetero-orbital” two-component FQH states, in which the orbital index ispart of the pseudospin, rendering the multicomponent interactions strongly SU(2) anisotropic in thepseudospin space. Such states, obtained in bilayer graphene at the isospin transition between N ¼ 0 andN ¼ 1 electron Landau levels, are markedly different from previous homo-orbital two-component FQHstates. In particular, we observe strikingly different behaviors for the parallel-vortex and reverse-vortexattachment composite fermion states, and an anomalously strong two-component 2=5 state over a widerange of magnetic field before it abruptly disappears at a high field. Our findings, combined with detailedtheoretical calculations, reveal the surprising robustness of the hetero-orbital FQH effects, significantlyenriching our understanding of FQH physics in this novel regime.DOI: 10.1103/kfn2-qggs Subject Areas: Condensed Matter Physics, GrapheneI. INTRODUCTIONLandau Levels (LLs) of a two-dimensional electron gassupport a plethora of strongly correlated electronic phases,where the form factor of the orbital wave function plays acrucial role in determining the characteristics of the under-pinning Coulomb interactions [1,2]. In GaAs quantumwells, FQH states of the sequence ν ¼ p=ð2p� 1Þ occurin the n ¼ 0 LL and are well described by the integer QHstates of composite fermions (CFs), namely, particlesproduced by binding of electrons and two vortices, con-sisting of p-filled CF Landau levels called Λ levels [1].However, the origin of the FQHE in the n ¼ 1 LL is verydifferent. Here, a softened Coulomb interaction leads to thepairing of CFs at half-fillings to produce putatively non-Abelian FQH states, such as the 5=2 state in GaAs [3].Unconventional FQH states at partial fillings ν̃ ¼ 2=5 and3=8 in the n ¼ 1 LL have also been hypothesized to be non-Abelian [3–12]. Exploring the interplay of the n ¼ 0 andn ¼ 1 LLs through level crossing offers valuable insightsinto the physics of the FQHE [13–19]. For example, it wasreported that the 5=2 state in a wide quantum wellstrengthens briefly before crossing to an n ¼ 0 compress-ible liquid [13,18].The study of multicomponent two-dimensional (2D)systems, where the components—generically referred toas pseudospin—may be the electron spin [1,2,19–27],quantum well or atomic layer [28–32], valley or sublattice*Present address: School of Physics, Zhejiang University,Hangzhou 310058, China.†Contact author: jkj2@psu.edu‡Contact author: jxz26@psu.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 15, 031023 (2025)2160-3308=25=15(3)=031023(12) 031023-1 Published by the American Physical Societyhttps://orcid.org/0000-0001-8521-6465https://orcid.org/0000-0002-8087-6015https://orcid.org/0000-0001-5928-2979https://orcid.org/0009-0003-3709-5672https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0003-0082-5881https://orcid.org/0000-0001-8100-967Xhttps://ror.org/04p491231https://ror.org/05078rg59https://ror.org/02bv3zr67https://ror.org/026v1ze26https://ror.org/026v1ze26https://ror.org/04p491231https://crossmark.crossref.org/dialog/?doi=10.1103/kfn2-qggs&domain=pdf&date_stamp=2025-07-22https://doi.org/10.1103/kfn2-qggshttps://doi.org/10.1103/kfn2-qggshttps://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/isospin [33–39], or subband [18,40,41], further enriches theFQHE by establishing different correlations between differ-ent components. Double-layer systems exhibit superfluidbehavior believed to be described by the Halperin (111)wave function [28,30,31,42], where both intralayer andinterlayer interactions lead to the formation of a collectiveν ¼ 1 state. The model of spinful CFs captures the spin-valley singlet, e.g., the equivalent of the Halperin (332)state at ν ¼ 2=5, or partially spin-polarized states in the JainFQH sequence of ν ¼ p=ð2p� 1Þ [1,2,19–27,33,36,39]. Itis interesting to note that all established two-componentstates so far are homo-orbital in nature, i.e., have the sameLL wave function in both components. Hetero-orbital two-component states, while not theoretically prohibited, havenot been observed.The eight closely degenerate LLs of bilayer graphenecentered at the charge neutrality point (E ¼ 0) offer a richplatform to advance the physics of multicomponent FQHEthanks to its inherent spin, valley, and orbital indices andwide tunability [34–37,39]. While monolayer grapheneexhibits the conventional Jain states at ν ¼ p=ð2p� 1Þ inboth N ¼ 0 and 1 LLs [21,22], BLG exhibits the Jain statesin the N ¼ 0 LL and even-denominator FQH states in theN ¼ 1 LL [35,36,39]. FQH states at 2=5 and 3=7 have alsobeen observed in the N ¼ 1 LL, although their natureremains an open question [35,36,39]. An electric displace-ment field (D field) controls the valley Zeeman splitting,which in turn induces the crossing of LLs carrying differentorbital (N ¼ 0, 1) and valley (ξ ¼ þ, −) indices asillustrated in Fig. 1(a) [39]. This crossing point is anexcellent place to explore multicomponent states. Theorbital wave function of the N ¼ 1 LL in BLG comprisesa majority n ¼ 1 orbital and a minority n ¼ 0 orbital inGaAs [Fig. 1(a)], with the weight of the latter increasingsmoothly with increasing magnetic field; this uniqueproperty allows for the exploration of the orbital wave-function-driven topological phase transitions, as predictedrecently [12,16,36].(a)(d)(b) (c)xxxxxxxy(arb. units)FIG. 1. Two-component FQH states at the coincidence of j � 0i and j ∓ 1 i LLs. (a) Schematic EðDÞ diagram showing the electronLLs and the j þ 0i=j − 1i coincidence at positiveD� (black circle). Each LL is labeled by jξNσi, where ξ ¼ þ;−, N ¼ 0, 1, and σ ¼ ↑and ↓ denote the valley, orbital and spin indices, respectively. The cartoon on the right illustrates the atomic-site wave-functiondistribution of the j þ 0i and j − 1i LLs. (b) Upper panel: false color map of RxxðD̃; νÞ. Note that D̃ ¼ Dþ 5 mV=nm adjusts for theoffset in the nominal appliedD field. FQH states appear as vertical dark lines. Lower panel: schematic representation of the data taken atD < 0. The dashed line marks the D� in the range 4=3 < ν < 5=3. New FQH states, which appear as dark spots in the upper panel andare colored orange in the lower panel, emerge at D� for ν ¼ 7=5, 10=7, and 13=9, but not for ν ¼ 4=3, 11=7, 8=5, and 5=3. (c) Verticalscans of RxxðDÞ taken along ν ¼ 7=5, 10=7, and 13=9 (device 011), and 11=7 and 8=5 (device 002), respectively, and centered at theirrespective values of D�. The D� state occupies a similar D-field range at ν ¼ 7=5, 10=7, and 13=9. It is absent at ν ¼ 8=5 and 11=7,similar to what’s shown in the map in panel (b). Here, B ¼ 25 T and T ¼ 0.33 K. See Fig. 5 for additional measurements on the D�states at different magnetic fields and in different devices. (d) Horizontal scans of RxxðνÞ and RxyðνÞ taken at a fixed D̃ ¼ −15 (red) and−16 (blue) mV=nm as indicated by the white dashed lines in panel (b). The former cuts through the D� state of 10=7 and 13=9, and thelatter cuts through the D� state of 7=5 on the negativeD side. The D� state of 7=5 exhibits a well-developed plateau at Rxy ¼ 5=7 h=e2.The D� state of 10=7 shows an incipient plateau at Rxy ¼ 7=10 h=e2. The red Rxx and Rxy traces are offset by 400 Ω and 0.03 h=e2,respectively, for clarity. From device 002, B ¼ 18 T and T ¼ 20 mK.KE HUANG et al. PHYS. REV. X 15, 031023 (2025)031023-2In this work, we report on the observations of hetero-orbital two-component FQH states occurring at the isospintransition of the j � 0i and j ∓ 1i electron LLs in bilayergraphene. Their behavior is underpinned by anisotropicSU(2) interactions and exhibits properties remarkablydifferent from those seen in past homo-orbital systems.A single hetero-orbital two-component state occurs fortwo-vortex CF states at partial fillings 2=5 and 3=7 but notfor reverse vortex CF states at p=ð2p − 1Þ. This case is instark contrast to homo-orbital two-component systemswhere FQH states are observed at both p=ð2pþ 1Þ andp=ð2p − 1Þ fillings [1,2,20,33,36,39]. A systematic studyof the magnetic-field-dependent energy gaps of the threeFQH phases at filling factor 7=5 reveals a number ofunusual observations. The very strong hetero-orbital two-component 2=5 state, denoted as the D� state in measure-ments, develops at a low field of B ∼ 7 T and has the largestgap ΔD�2=5 > ΔN¼02=5 > ΔN¼12=5 , but this state disappears ataround 31 T. The gap of the conventional FQH phase onN ¼ 0 LL follows theffiffiffiffiBpscaling and is characterized byan effective CF mass of 0.13me. The gap of the N ¼ 1phase ΔN¼12=5 increases rapidly with B at lower field butslows down to merge with ΔN¼02=5 at around 28 T, where thetwo-component state begins to disappear. Exact diagonal-ization calculations validate the stabilization of an SU(2)anisotropic two-component state at p=ð2pþ 1Þ at thej� 0i=j∓ 1i coincidence and capture the absence of sucha state at p=ð2p − 1Þ, but they do not explain the collapseof the two-component state at a high B. Our work opens anew avenue to explore multicomponent FQH states inbilayer graphene, a high-quality 2D electron gas with a richisospin structure and immense tunability.II. EXPERIMENTAL RESULTSOur measurements employ high-quality graphite, dual-gated, h − BN-encapsulated, Hall bar devices, the fabrica-tion and characterization of which are described in ourprevious work [39] and in Appendix A. The small D-fieldinhomogeneity (δD < 0.7 mV=nm) realized in our deviceis key to the observation in this work. Figure 1(b) shows afalse color map of the longitudinal resistance RxxðD; νÞnear ν ¼ 3=2 in device 002, where the FQH states appear asvertical dark lines. Many observations of this map werediscussed in Ref. [39]. Here, we focus on features near thecrossing of the j þ 0i and j − 1i LLs, which occurs at a setof ν-dependent D-field values, D�, as illustrated by thedashed line in the schematic diagram below. In its vicinity,fractional states of filling factor range 1 < ν < 2 occupythe j þ 0i LL at low D and transition to the j − 1i LL athigh D in two different ways. For states with ν > 3=2—e.g., 8/5 and 11/7—Rxx exhibits a resistance peak atD� [seeFig. 1(c) and Fig. 5, Appendix C]. This peak is commonlyobserved at LL crossings and corresponds to a brief closingof the gap. In stark contrast, at ν ¼ 7=5, 10=7, and 13=9, anew incompressible state with a deep Rxx minimumemerges at D�. In Fig. 1(d), we verify that the D� stateof 7=5 is a FQH state by observing a Hall conductanceplateau at the correct value. A shoulder also develops fortheD� state of 10=7. Figure 1(c) shows the evolution of Rxxas a function of D at fixed ν ¼ 7=5, 10=7, 13=9, 11=7, and8=5, respectively. Traces at ν ¼ 7=5, 10=7, and 13=9exhibit a clear Rxx minimum at its own D�. The D� state isseparated from the FQH states occupying the j þ 0i andj − 1i LLs by two resistance peaks. The j þ 0i states belongto the conventional Jain FQH sequence while the nature ofthe j − 1i states (at ν ¼ 7=5, 10=7) remains to be clarified[39]. We observe theD� state clearly and consistently at 7=5and 10=7 in multiple devices and in a wide range of B fields.Additional data are given in Fig. 5, Appendix C. Itsappearance supports the development of a new incompress-ible FQH state at the j þ 0i=j − 1i coincidence, andsimilarly at j− 0i=jþ 1i on the negative D side. However,we have not seen this state at fractional fillings ν > 3=2,such as 11=7 and 8=5. Clearly, CF states of parallel-vortex[ν ¼ p=ð2pþ 1Þ] and reverse vortex [ν ¼ p=ð2p − 1Þ]attachment behave very differently here, in contrast to pastspin or valleyful CF systems [1,2,20,33,36,39].We examine the evolution of the D� state in a magneticfield, using ν ¼ 7=5 in device 002 as an example.Figure 2(a) shows a series of D sweeps at ν ¼ 7=5 andrepresentative B fields varying from 7 to 31 T. A full B-fieldset is given in Fig. 6, Appendix C. It is quite remarkablethat the D� state appears stronger than the conventionalFQH phase riding on the N ¼ 0 LL and is already welldeveloped at 7 T. Interestingly, ΔD, the D-field rangeoccupied by the D� state, which is marked by the distancebetween the flanking Rxx peaks in Fig. 2(a), exhibits anonmonotonic magnetic field dependence, as shown inFig. 2(b). In device 002, the D� state forms betweenapproximately 6 and 30.5 T with a maximum range ΔD ¼2 mV=nm at B ¼ 16 T. In Fig. 2(c), we plot an exper-imental D-B phase diagram for ν ¼ 7=5, where it exhibitsthree FQH phases with distinct orbital wave functions.The disappearance of the D� state at B > 30 T, which alsooccurs for the D� state at ν ¼ 10=7 and 13=9 and in device011 (Fig. 7, Appendix C), is unusual, as FQH states aretypically strengthened at large B due to stronger Coulombinteractions. We note that the disappearance is unlikelycaused by a spin-polarization transition. The significantexchange-enhanced spin Zeeman splitting compared to thesmall valley splitting in the large B and small D regimestudied here makes the spin degrees of freedom nonactivein this problem [35,37,39]. These unprecedented observa-tions and the intricate D and B dependences attest to a richand tunable interaction landscape in BLG, where new FQHstates emerge.To further understand the three FQH phases of 7=5, wehave systematically measured their energy gaps in a widerange of magnetic field using the temperature dependenceHETERO-ORBITAL TWO-COMPONENT FRACTIONAL QUANTUM … PHYS. REV. X 15, 031023 (2025)031023-3of Rxx, an example of which is given in Fig. 3(a) forB ¼ 22 T. It is immediately clear from the data that the D�state is the strongest while the N ¼ 1 state is the weakest.Figure 3(b) shows a few exemplary Arrhenius plotsRxx ∼ expð−Δ=2kBTÞ, from which we determine the gapsize Δ. Values of ΔD�2=5, ΔN¼02=5 , and ΔN¼12=5 are extracted andplotted as a function of B in Fig. 3(c) from 14 to 31 T, whileadditional measurements and fits are given in Figs. 8–10,Appendix C. In our previous work, a noninteractingtwo-component CF model describes the valley isospin-polarization transitions near D ¼ 0 very well [39]. Thus,we have adopted a similar model here to fit the gap of theN ¼ 0 state to ΔN¼02=5 ¼ ℏeBeff=mCFa − Γ, where mCFa ¼αmeffiffiffiffiBp(me is the free electron mass) and Γ is the disorderbroadening. From the fitting, we extract α ¼ 0.13 andΓ ¼ 6.8 K (see analysis in Fig. 10, Appendix C) [1,22,43].The mCFa value is larger than that (mCFa ¼ 0.067meffiffiffiffiBp)obtained for the 2=5 state in monolayer graphene [22],suggesting that LL mixing plays a more important role inBLG; a similar conclusion is reached in Ref. [39]. While wedo not have a quantitative theory, it is truly remarkable thatthe gap of the D� phase exceeds that of the conventionalN ¼ 0 phase by more than one Kelvin. This finding pointsto strong correlations at the j � 0i=j ∓ 1i coincidence.The N ¼ 1 phase of 7=5 exhibits a more complex fieldevolution. Generally, conventional FQH states are notstabilized on the n ¼ 1 LL of GaAs [4,5]. However, theN ¼ 1 LL in BLG consists of a small n ¼ 0 component, theweight of which increases with increasing B [Fig. 1(a)].Experimentally, the N ¼ 1 phase of 7=5 only develops at asubstantial B field. Once formed, ΔN¼12=5 rapidly increaseswith B, reaches the size of ΔN¼02=5 at around 28 T, and thenmerges with the trace of ΔN¼02=5 ðBÞ [Fig. 2(c)]. This Bdependence is consistent with previous capacitance(a) (b) (c)xxFIG. 2. Magnetic field evolution of the ν ¼ 7=5 D� state. (a) R7=5xx ðDÞ sweeps at selected B fields as labeled. Additional B fields aregiven in Fig. 6. (b) Position of D� (left axis, blue triangle) and the phase space ΔD (right axis, red open square) as a function of B. NotethatΔDmeasures from peak to peak, as illustrated in panel (a). Here, jD�jðBÞ follows a linear B dependence given by jD�j ¼ 1.3B − 6:8(dashed line). Data in panels (a) and (b) are taken from the negative D side of device 002. (c) Schematic D-B phase diagram of the 7=5showing the three FQH phases with different orbital wave functions. The phase space of the D� state is exaggerated for clarity.(a)xx xx(b) (c)FIG. 3. Magnetic-field-dependent energy gaps of the three FQH phases at 7=5. (a) R7=5xx ðDÞ sweeps at selected temperatures aslabeled in the plot. Here, B ¼ 22 T. Note that Rxx becomes D independent when the state is fully riding on N ¼ 0 or 1 LLs. Weextract the T-dependent RxxðD�Þ, RxxðN ¼ 0Þ, and RxxðN ¼ 1Þ from similar traces and plot RxxðTÞ in an Arrhenius plot.(b) Exemplary Arrhenius plots for the three states at selected B fields as labeled. Symbols follow the legend in panel (c). Additionalmeasurements and analyses are given in Figs. 8–10. (c) Magnetic field dependence of ΔN¼12=5 (blue triangle), ΔN¼02=5 (red circle), andΔD�2=5 (gray square). The red dashed line is a fit to data: ΔN¼02=5 ðKÞ ¼ 2.0ffiffiffiffiffiffiffiffiffiffiBðTÞp − 6.8. See Fig. 10 for details. Blue and gray dashedlines are guides to the eye. (From device 002.).KE HUANG et al. PHYS. REV. X 15, 031023 (2025)031023-4measurements [36] and may be due to the unique B-fieldevolution of the N ¼ 1 wave function in BLG. Whetherthe N ¼ 1 phase of 7=5 is Abelian or non-Abelian remainsan open question. It is interesting that both the coalescenceof ΔN¼12=5 and ΔN¼02=5 and the disappearance of the D� stateoccur at around 28 T, suggesting a substantial changeof the underpinning interactions that are important toboth phenomena.III. THEORETICAL CALCULATIONSAND DISCUSSIONSWhat is the nature of the new D� state? Given itsoccurrence at the j � 0i=j ∓ 1i level crossing, a two-component state seems a natural candidate. In an earlierwork [39], we showed that a two-component CF modelcomposed of a ðjþ0i; j − 0iÞ spinor can quantitativelydescribe the many valley-isospin-polarization transitionsthat occur near D ¼ 0 [1,2,24–27,44]. There, a state ofν ¼ p=ð2p� 1Þ maps to p-filled CF Λ levels and exhibitsp-level crossings and p − 1 two-component states from−D to þD. A direct generalization of this scheme to aðj ∓ 1i; j � 0iÞ spinor would imply one D� state forν ¼ 2=5 and 2=3, two D� states for 3=7 and 3=5, and soon. Experimentally, however, we only observed a singleD�state at ν ¼ 2=5, 3=7, and 4=9, and none at the reverse-vortex states. This qualitative discrepancy points to thedeficiency of a conventional two-component model. In pasthomo-orbital two-component systems, the diagonal SU(2)interaction matrix elements are identical, i.e., V↑;↑ðrÞ ¼V↓;↓ðrÞ [1,2,24–27,42]. Here, the hetero-orbital nature ofour system leads to anisotropic interaction matrix elements:Vþ0;þ0ðrÞ ≠ Vþ0;−1ðrÞ ¼ V−1;þ0ðrÞ ≠ V−1;−1ðrÞ; thus, thestability of two-component FQH states must bereexamined. It is worth noting that incompressible stateshave also been observed at the crossing of the N ¼ 0 and 1LLs near ν ¼ −5=2 in BLG, although their overall char-acteristics seem to follow the conventional homo-orbitaltwo-component model quite well [36].We quantitatively investigated this problem by obtainingthe exact ground states of the anisotropic two-componentsystem through exact diagonalization and comparingthe solutions to the two-component spin-singlet Ψð1;1Þ2=5 ,Ψð−1;−1Þ2=3 , and partially spin-polarized Ψð2;1Þ3=7 and Ψð−2;−1Þ3=5Jain states that provide near-exact representations for theSU(2) symmetric Coulomb interaction. The interactionsused are given in Appendix B. Note that our model does notinclude LL mixing, lattice-scale anisotropies, or hoppingterms beyond the nearest neighbor (that result in trigonalwarping). Figure 4(a) plots the wave-function overlaps forν ¼ 2=5 and 2=3, which correspond to 2 and −2 filled CFLLs (the latter with reverse vortex attachment). The over-laps for ν ¼ 3=7 and 3=5, corresponding to CF fillings of 3and−3, are given in Fig. 11, Appendix C. We find that for abroad range of B field, the exact two-component groundstates of the anisotropic Hamiltonian at ν ¼ 2=5 and 3=7are almost perfectly described by the isotropic CF states,(a) (b)(c)SphereDiskFIG. 4. Theoretical calculations of anisotropic two-component states at ν ¼ 2=5 and 2=3. (a) Overlaps of the spin-singlet Jain state(jψ triali) with the exact Coulomb ground state (jψexacti) in the E ¼ 0 LL of BLG as a function of the magnetic field for ν ¼ 2=5 (upperpanel) and 2=3 (lower panel). For the Jain states, we have used the Coulomb ground state that arises for B → ∞ in the N ¼ 1 LL ofBLG, which is identical to that in the n ¼ 0 LL. Calculations are performed for n electrons in the spherical geometry using the spherical(red sphere) and disk (blue triangle) pseudopotentials. See Appendix B for the interaction details. The electron number n ¼ 10 forν ¼ 2=5 and 14 for ¼ 2=3. (b) CF Λ level filling diagram for the (2, 0) (N ¼ j − 1i), (1, 1) (two-component), and (0, 2) (N ¼ j þ 0i)states of the 2=5, respectively. (c) Schematic energy diagrams showing the evolution of Eð2; 0Þ, Eð1; 1Þ, and Eð0; 2Þ in aD field and theresulting ground states. Three scenarios are given. In the first diagram, an isotropic interaction results in three ground states, with D�being the (1, 1) state. In the second diagram, we show how an anisotropic interaction shifts the location of D� and changes the width ofthe (1, 1) state. The third diagram depicts a scenario where the (1, 1) state never becomes a ground state, which could potentially explainthe disappearance of the D� state at very large B.HETERO-ORBITAL TWO-COMPONENT FRACTIONAL QUANTUM … PHYS. REV. X 15, 031023 (2025)031023-5implying that the ground state is SU(2) symmetric to anexcellent approximation even though the microscopicinteraction is not. In contrast, the agreement is relativelypoor for ν ¼ 2=3 and ν ¼ 3=5, indicating that two-com-ponent FQHE is not robust and may not be realizedexperimentally at these filling factors. These findings agreeremarkably well with the experiment. In the case ofν ¼ 2=5, our results suggest that the V0 and V1 Haldanepseudopotentials are positive and dominant for each of theVσ;σ0 interactions (σ; σ0 ¼ þ0;−1) since Ψð1;1Þ2=5 is the exactground state of a model with only V0 and V1 terms. We donot have similar qualitative arguments for why the state at3=7 is well described by the Jain wave function but those at3=5 and 4=7 are clearly not.Further insights into the hetero-orbital two-componentstates can be gained by comparing the phase diagrams ofthe isotropic and anisotropic interactions. Figures 4(b)and 4(c) illustrate the key physics at ν ¼ 2=5. Thecalculated ground-state energies of the single- and two-component states Eðn↑; n↓Þ, where n↑;↓ is the filling factorof the N ¼ j ∓ 1 i and j � 0i orbitals, respectively, evolvelinearly with the D field, and their competition determineswhich state manifests. In the isotropic case, Eð2; 0Þ ¼Eð0; 2Þ at ΔD ¼ D −D�s ¼ 0, where D�s is the single-particle LL crossing point and Eð1; 1Þ is below both Eð2; 0Þand Eð0; 2Þ; all three become ground states as a function ofΔD, as shown in the first diagram of Fig. 4(c). In theanisotropic case, however, the energy ordering of thesestates is qualitatively different. We have calculated Eð2; 0Þ,Eð1; 1Þ, and Eð0; 2Þ as a function of the magnetic field byextrapolating the finite system results to the thermody-namic limit, and we have plotted the results in Fig. 12,Appendix C. Our calculations show that Eð2; 0Þ > Eð0; 2Þand (1, 1) lies in between. If Eð1; 1Þ is closer to Eð0; 2Þ thanto Eð2; 0Þ, (1, 1) will become the ground state for somerange of ΔD, as depicted in Anisotropic I of Fig. 4(c). Ourcalculations corroborate this scenario, and we identify theexperimental D� state with the two-component (1, 1) state.If Eð1; 1Þ is closer to Eð2; 0Þ instead, the system willtransition directly from the (0, 2) state to the (2, 0) statewithout going through the D� state. We suspect that thissituation, called Anisotropic II, corresponds to the exper-imental situation of B > 30 T, though this disappearance isnot captured by our calculations. Concomitantly with thedisappearance of the D� state, the slope of ΔN¼12=5 ðBÞ alsochanges near 28 T. These observations, together with thelarge value of ΔD�2=5 and the substantial renormalization ofΔN¼02=5 , point to the need for a more refined theoreticaltreatment to capture the energetics of BLG quantitatively.Similar considerations can also be applied to ν ¼ 3=7 toexplain the formation of a single D� state at the levelcrossing. In an isotropic two-component system, fourground states including two partially polarized states areexpected and indeed observed at ν ¼ 11=7 ¼ 2 − 3=7 nearD ¼ 0 [39]. Here, anisotropic interactions lead to thesplitting between Eð3; 0Þ and Eð0; 3Þ and between Eð2; 1Þand Eð1; 2Þ. Exact diagonalization calculations performedfor the 3=7 state are given in Fig. 11, Appendix C, where weshow how the movement of the energies allows thepossibility of eliminating the partially polarized (2, 1) stateso that only the (1, 2) state survives in the vicinity of thecrossing. This case is likely the experimental D� state weobserved. Its disappearance at high B likely originates froma similar interaction change to the 2=5 case. Indeed, the D�state at 7=5, 10=7, and 13=9 disappears around 30 T in ourmeasurement (Fig. 7, Appendix C), which we hope thatfuture calculations may be able to capture.IV. SUMMARY AND OUTLOOKThemultiple isospin degeneracy and strong correlations inthe E ¼ 0 Landau level of graphene and bilayer grapheneoffer an excellent platform to explore the physics of many-body coherent, multicomponent states, such as a cantedantiferromagnet and a Kekulé phase at ν ¼ 0 [45–48]. In thiswork, we combined experiment and theory to reveal theproperties of unconventional two-component FQH states,occurring at the coincidence of j ∓ 1i and j � 0i Landaulevels in bilayer graphene. We show that even if theintracomponent and intercomponent interactions are alldifferent, two-component states can still be stabilized, albeitwith a set of characteristics distinguished from previousisotropic systems. Our measurement of the energy gapsof the three FQH phases at ν ¼ 7=5 also highlight theimportance of understanding the impact of Landau-levelmixing in bilayer graphene, in order to reach agreement withexperiments. Finally, the complex magnetic field evolutionof fractional states riding on theN ¼ 1 Landau level, and thepossibility of other topological orders [12], remains an openquestion for future experiment and theory.ACKNOWLEDGMENTSWe thank Herbert. A. Fertig, Liang Fu, BertrandHalperin, Udit Khanna, Ganpathy Murthy, Zlatko Papic,Edward H. Rezayi, and Efrat Shimshoni for helpfuldiscussions. We thank Lu Li and Kuan-Wen Chen forproviding guidance and a capacitance thermometer used tocalibrate the temperature readings in cell 9 (31 T, 0.3 K) ofthe National High Magnetic Field Laboratory. The experi-ment is supported by the Department of Energy throughGrant No. DE-SC0022947 and by the National ScienceFoundation through Grant No. NSF-DMR-1904986. TheNational Science Foundation grant supported the devicefabrication. The Department of Energy grant supported themeasurements, data analysis, and manuscript writing.A. C. B. acknowledges the financial support from theScience and Engineering Research Board (SERB) ofthe Department of Science and Technology (DST) viathe Mathematical Research Impact Centric SupportKE HUANG et al. PHYS. REV. X 15, 031023 (2025)031023-6(MATRICS) Grant No. MTR/2023/000002. Computationalportions of this work were conducted using the Nandadeviand Kamet supercomputers maintained and supported bythe Institute of Mathematical Science’s High-PerformanceComputing Center. Some numerical calculations wereperformed using the DiagHam package [49], for whichwe are grateful to its authors. J. K. J. was supported in partby the U.S. Department of Energy, Office of Basic EnergySciences, under Grant No. DE-SC0005042. K.W. andT. T. acknowledge support from the JSPS KAKENHI(Grants No. 21H05233 and No. 23H02052), CREST(JPMJCR24A5), JST, and World Premier InternationalResearch Center Initiative (WPI), MEXT, Japan. Workperformed at the National High Magnetic Field Laboratorywas supported by the NSF through Grant No. NSF-DMR-1644779 and the State of Florida.The authors declare no competing interests.DATA AVAILABILITYThe data that support the plots in the main text areavailable from Harvard Dataverse [50], and other findingsof this study are available from the corresponding authorsupon reasonable request.APPENDIX A: DEVICE FABRICATION ANDMEASUREMENT SETUPSDevices 002, 011, and 015 used in this work were alsoused in Ref. [39], where detailed fabrication and charac-terization are given. Briefly, we use a polypropylenecarbonate (PPC) stamp to pick up thin h − BN/BLG/h − BN/graphite sequentially. After annealing the stackin an O2=Ar mixture at 450 °C, we transfer another layer ofgraphite flake exfoliated on the PPC stamp onto the top ofthe stack. The Hall bar structure is patterned by e-beamlithography and reactive ion etching. The edge contact ismade by the two-step etching protocol described inRef. [39] and deposited with Cr=Au.Electrical transport measurements were done usingstandard low-frequency lock-in techniques. All measure-ments in Fig. 3(c) and open circles in Fig. 8(e),Appendix C, were taken in cell 9 of the National HighMagnetic Field Laboratory (NHMFL) using a He-3 cryo-stat. Solid circles in Fig. 8(e), Appendix C, were taken inSCM-1 of the NHMFL. A small discrepancy between thetwo sets of data is attributed to the lower electron temper-ature in SCM-1. Blue squares (device 015) in Fig. 9,Appendix C, were taken in SCM-4 (dilution fridge, up to28 T) of the NHMFL.APPENDIX B: INTERACTIONS USED IN THEEXACT DIAGONALIZATION STUDIESFor the isospin that combines the valley and orbitalindices ðj ∓ 1i; j � 0iÞ, the Coulomb interaction is notSU(2) symmetric because it depends on the orbital degreeof freedom. It can be fully characterized by the Haldanepseudopotentials [51], which are the energies of twoelectrons in a definite relative angular momentum m.The general form in the planar disk geometry is givenby (the magnetic length lB is set to unity)VN;N0m ðθÞ ¼ZdqFN;N0 ðqÞe−q2Lmðq2Þ; ðB1Þwhere q is the wave number, LkðxÞ is the kth Laguerrepolynomial, and FN;N0 ðqÞ is called the form factor, with Nand N0 being the orbital indices of the two electrons. Theform factors are given byF0;0ðqÞ ¼ 1F1;1ðθ; qÞ ¼�sin2ðθÞL1�q22�þ cos2ðθÞL0�q22��2F0;1ðθ; qÞ ¼ L0�q22��sin2ðθÞL1�q22�þ cos2ðθÞL0�q22��;ðB2Þwhere the B-dependent parameter θ can be estimated astanðθÞ ≈ tlB=ffiffiffi2pℏvF, where t is the hopping integral andvF is the Fermi velocity. For graphene, the typical Fermivelocity is vF ¼ 106 m=s, and taking t ¼ 350 meV, asobtained from DFT calculations at zero field [52],the magnetic field B and θ are related throughB ¼ 93.06½cotðθÞ�2 ½T�. In Eq. (B2), it suffices to considerthe range 0 ≤ θ ≤ π=2 since the form factor only dependson sin2ðθÞ. For θ ¼ 0, or equivalently B → ∞, we recoverthe form factor for the n ¼ 0 LL of GaAs.The three sets of pseudopotentials are given byV0;0m ¼ Γðmþ 12Þ2Γðmþ 1ÞV1;1m ðθÞ ¼�sin2ðθÞL1�q22�þ cos2ðθÞL0�q22��2V0;1m ðθÞ ¼ffiffiffiπp32�162F1�12;−m; 1; 1�− 82F1�32;−m; 1; 1�sin2ðθÞþ 32F1�52;−m; 1; 1�sin4ðθÞ�V0;1m ðθÞ ¼ffiffiffiπp32�162F1�12;−m; 1; 1�− 82F1�32;−m; 1; 1�sin2ðθÞþ 32F1�52;−m; 1; 1�sin4ðθÞ�; ðB3ÞHETERO-ORBITAL TWO-COMPONENT FRACTIONAL QUANTUM … PHYS. REV. X 15, 031023 (2025)031023-7where ΓðxÞ is the Gamma function and 2F1 is the Gauss hypergeometric function. The analogous sphericalpseudopotentials can be obtained by following the derivation outlined in Refs. [53–56]. These pseudopotentials areused in the exact diagonalization computations of the ground state.APPENDIX C: ADDITIONAL FIGURES(a) (b) (c) (d) (e)(f) (g) device 011(arb. units)device 002device 015FIG. 5. (a)–(d) RxxðDÞ traces taken along fixed filling factors and in different magnetic fields from 22 to 31 T as labeled. Partial fillingsν̃ ¼ ν − 1 ¼ 1=3 (black), 2=5 (red), 3=7 (green), 4=7 (violet), and 3=5 (brown). Traces are vertically stacked for clarity. (e) CorrespondingRxyðDÞ traces taken at 22 T and 31 T. The dashed lines indicate the expected Hall resistance value at each filling factor. All filling factorsexhibit FQH states on the j − 0i and j þ 1i LLs. TheD� state (marked by the * symbol) forms at 2=5 and 3=7 but not at other filling factors.TheD� state disappears completely at B ¼ 31 T and T ¼ 0.33 K. (From device 002.) (f), (g) False color map of RxxðD; νÞ in devices 015and 011, respectively, showing the presence of theD� states at ν ¼ 7=5 and 10=7. The white dashed line traces the trueD ¼ 0 locations ineach device. B ¼ 8.9 T and T ¼ 0.33 K in panel (f). B ¼ 18 T and T ¼ 20 mK in panel (g).device 002FIG. 6. R7=5xx ðDÞ sweeps at fixed B fields as labeled in the plot. (From device 002.).KE HUANG et al. PHYS. REV. X 15, 031023 (2025)031023-8(a) (b)device 002device 015device 002FIG. 9. Energy gap of the N ¼ 1 phase of 7=5. (a) Arrheniusplots and fits for the N ¼ 1 state at different B fields as labeled.(From device 002.) (b) ΔN¼12=5 as a function of B from the fits inpanel (a) (circle) and from device 015 (square) using similarmeasurements. Both devices show that ΔN¼12=5 is very small at lowand intermediate B fields but increases rapidly with B once thefield reaches 20 T.(a) (b)device 002FIG. 10. Energy gap of the N ¼ 0 phase of 7=5. (a) Arrheniusplots and fits for the N ¼ 0 state at different B fields as labeled.We normalized RxxðTÞ by Rxx measured at T ¼ 1.33 K to bettercompare the data taken at different magnetic fields. (b) ΔN¼02=5 as afunction of B extracted from panel (a). The best fit to data is givenby the red solid line, which corresponds to ΔN¼02=5 ¼ 2.0ffiffiffiffiBp − Γ,where Γ ¼ 6.8 K is the disorder broadening energy scale con-sistent with our previous assessment of the bulk disorder level inthis device [39]. Also shown are the high and low bounds ofpossible fits. They correspond to 2.5ffiffiffiffiBp − 8.9 (blue dashed line)and 1.6ffiffiffiffiBp − 5.2 (green dashed line), respectively. Using the CFmodel, we write Δ7=5 ¼ ℏeBeff=m�a − Γ, where Beff ¼ 3ðB7=5 −B3=2Þ ¼ B7=5=5 is the effective magnetic field at ν ¼ 7=5 andm�a ¼ αmeffiffiffiffiBpis the effective activation CF mass. The fitscorrespond to α ¼ 0.13� 0.03.(a)(d) (e)(b) (c)device 002FIG. 8. Energy gap of the D� phase of 7=5. (a)–(c), T-dependent R7=5xx ðDÞ traces taken at B ¼ 28, 29, and 29.5 T,respectively. As the phase space of the D� state narrows, it isalso gradually destabilized, likely due to an increasing mixtureof N ¼ 0 and N ¼ 1 domains. (d) Arrhenius plots and fits thatproduced the gray open squares shown in Fig. 3(c). (e) ΔD�2=5 as afunction of B, including results obtained from panel (d) (opencircle) and additional measurements taken in a differentcryostat (solid circle). A small systematic difference betweenthe two sets is due to the different temperature readout schemes.(From device 002.).(a) (b)(c) (d)device 011FIG. 7. Magnetic field evolution of the D� state at ν ¼ 7=5,10=7, and 13=9 in device 011. (a)–(c) R7=5xx ðDÞ, R10=7xx ðDÞ, andR13=9xx ðDÞ sweeps at selected B fields as labeled in the plot. TheD� state is marked by a * symbol. Note that ΔD is as defined inFig. 2. (d) ΔDðBÞ for ν ¼ 7=5 (square), 10=7 (circle), and 13=9(triangle). All three vanish together near 30 T. The dashed linesare guides to the eye.HETERO-ORBITAL TWO-COMPONENT FRACTIONAL QUANTUM … PHYS. REV. X 15, 031023 (2025)031023-9[1] J. K. Jain, Composite Fermions (Cambridge UniversityPress, Cambridge, England, 2007).[2] B. Halperin and J. K. Jain, Fractional Quantum HallEffects: New Developments (World Scientific, Singapore,2020).[3] R. L. Willett, The quantum Hall effect at 5=2 filling factor,Rep. Prog. Phys. 76, 076501 (2013).[4] J. S. Xia, W. Pan, C. L. Vicente, E. D. Adams, N. 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(a) Overlaps of the partially polarized Jainstate (jψ triali) with the exact Coulomb ground state (jψexacti) in the E ¼ 0 LL of BLG as a function of the magnetic field for ν ¼ 3=7(upper panel) and 3=5 (lower panel). Calculations are performed for n electrons in the spherical geometry using the spherical (redsphere) and disk (blue triangle) pseudopotentials. The electron number n ¼ 11 for ν ¼ 3=7 and 14 for ν ¼ 3=5. (b) CF Λ level fillingdiagram for the 3=7 state, showing four potential isospin configurations. The j − 1i state corresponds to (3, 0). The j þ 0i statecorresponds to (0, 3). The (2, 1) and (1, 2) are partially isospin-polarized states. (c) Schematic energy diagrams showing the evolution ofthe four configurations in a D field and the resulting ground states. In an isotropic interaction, both the (2, 1) and the (1, 2) manifest,which is indeed observed in Ref. [39], where the two-component spinor is (j þ 0i ; j − 0i). Similar to the case of 2=5, anisotropicinteractions change the energies of the four states. The middle diagram depicts a scenario where only the (1, 2) state manifests, which islikely the D� state observed in experiment. The last diagram shows how the system may transition from j þ 0i to j − 1i directly, whichmay correspond to the disappearance of the 3=7 D� state at large B.FIG. 12. Magnetic-field-dependent thermodynamic energies ofthe various candidate states at filling factor 2=5. The calculationsare performed for the zeroth LL of bilayer graphene in thespherical geometry. The two isospin components correspond toj − 1i and j þ 0i.KE HUANG et al. PHYS. REV. 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SUMMARY AND OUTLOOK ACKNOWLEDGMENTS DATA AVAILABILITY APPENDIX A: DEVICE FABRICATION AND MEASUREMENT SETUPS APPENDIX B: INTERACTIONS USED IN THE EXACT DIAGONALIZATION STUDIES APPENDIX C: ADDITIONAL FIGURES References