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Ke Huang, Hailong Fu, Danielle Reifsnyder Hickey, Nasim Alem, Xi Lin, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Jun Zhu

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[Valley Isospin Controlled Fractional Quantum Hall States in Bilayer Graphene](https://mdr.nims.go.jp/datasets/8e115ba4-887f-47bf-ac72-db3b83ed4758)

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Valley Isospin Controlled Fractional Quantum Hall States in Bilayer GrapheneValley Isospin Controlled Fractional Quantum Hall States in Bilayer GrapheneKe Huang ,1,2 Hailong Fu,1 Danielle Reifsnyder Hickey ,3,4 Nasim Alem,4 Xi Lin,2,5,6Kenji Watanabe ,7 Takashi Taniguchi,8 and Jun Zhu 1,9,*1Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA2International Center for Quantum Materials, Peking University, Beijing 100871, China3Department of Chemistry, The Pennsylvania State University, University Park, Pennsylvania 16802, USA4Department of Materials Science and Engineering, The Pennsylvania State University,University Park, Pennsylvania 16802, USA5Beijing Academy of Quantum Information Sciences, Beijing 100193, China6CAS Center for Excellence in Topological Quantum Computation,University of Chinese Academy of Sciences, Beijing 100190, China7Research Center for Functional Materials, National Institute for Materials Science,1-1 Namiki, Tsukuba 305-0044, Japan8International Center for Materials Nanoarchitectonics, National Institute for Materials Science,1-1 Namiki, Tsukuba 305-0044, Japan9Center for 2-Dimensional and Layered Materials, The Pennsylvania State University,University Park, Pennsylvania 16802, USA(Received 10 January 2022; revised 7 April 2022; accepted 3 June 2022; published 28 July 2022)A two-dimensional electron system placed in a magnetic field develops Landau levels, where strongCoulomb interactions lead to the appearance of many-body correlated ground states. Quantum numberssimilar to the electron spin enable the understanding and control of complex ground state order andcollective excitations. Owing to its spin, valley, and orbital degrees of freedom, Bernal-stacked bilayergraphene offers a rich platform to pursue correlated phenomena in two dimensions. In this work, wefabricate dual-gated Bernal-stacked bilayer graphene devices and demonstrate unprecedented fine controlover its valley isospin degrees of freedom using a perpendicular electric field. Higher sample qualityenables us to probe regimes obscured by disorder in previous studies. We present evidence for a new even-denominator fractional quantum Hall state at filling factor ν ¼ 5=2. The 5=2 state is found to bespontaneously valley polarized in the limit of vanishing valley Zeeman splitting, consistent with atheoretical prediction made regarding the spin polarization of the Moore-Read state. In the vicinity of theeven-denominator fractional quantum Hall states, we observe the appearance of the predicted Levin-Halperin daughter states of the Moore-Read Pfaffian wave function at ν ¼ 3=2 and 7=2 and of the anti-Pfaffian at ν ¼ 5=2 and −1=2. These observations suggest the breaking of particle-hole symmetry inbilayer graphene. We construct a comprehensive valley polarization phase diagram for the Jain sequencefractional states surrounding filling factor 3=2. These results are well explained by a two-componentcomposite fermion model, further demonstrating the SU(2) nature of the valley isospin in bilayer graphene.Our experiment paves the path for future efforts of manipulating the valley isospin in bilayer graphene toengineer exotic topological orders and quantum information processes.DOI: 10.1103/PhysRevX.12.031019 Subject Areas: Condensed Matter Physics, GrapheneI. INTRODUCTIONElectrons occupying a partially filled Landau level (LL)experience strong Coulomb interactions that lead to aplethora of correlated electronic states, generally knownas the fractional quantum Hall (FQH) effect. The FQHeffect hosts complex many-body wave functions, nontrivialtopology, and unconventional quantum exchange statisticsthat are potentially useful for topological quantum comput-ing [1–9]. In particular, even-denominator states occurringat half fillings, such as the ν ¼ 5=2 state in GaAs [1,2,6],have attracted ongoing attention since the 5=2 state ispostulated to be a pþ ip superconductor and harbors non-Abelian excitations potentially useful in the construction ofa topological qubit [3,5,10]. Its fundamental novelty andtechnological appeal have motivated many studies and the*Corresponding author.jzhu@phys.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 12, 031019 (2022)2160-3308=22=12(3)=031019(14) 031019-1 Published by the American Physical Societyhttps://orcid.org/0000-0001-8521-6465https://orcid.org/0000-0002-8962-1473https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0001-8100-967Xhttps://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevX.12.031019&domain=pdf&date_stamp=2022-07-28https://doi.org/10.1103/PhysRevX.12.031019https://doi.org/10.1103/PhysRevX.12.031019https://doi.org/10.1103/PhysRevX.12.031019https://doi.org/10.1103/PhysRevX.12.031019https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/discoveries of even-denominator FQH states with poten-tially similar origins in other 2D systems such as bilayergraphene [11–13], ZnO [14], and WSe2 [15]. Whilethermal conductance measurements have provided goodevidence on the non-Abelian nature of the 5=2 state [16], itsexchange statistics has not been explicitly verified ininterferometry studies. The particle-hole symmetry of theeven-denominator states is fundamental to its understand-ing, as ground states of different symmetries belong todifferent topological orders and harbor different edgemodes [2,17–22]. Furthermore, the Moore-Read wavefunction of the 5=2 state must be a spin triplet and is infact expected to be spontaneously spin polarized in the limitof zero Zeeman splitting [3,10,23,24]. Experimental studiesof these important issues remain ongoing [6].The electron spin and spinlike degrees of freedom innew materials and artificial structures play a fundamentalrole in the formation of correlated phenomena throughexchange interactions and the symmetry requirement of amany-body wave function [25–28]. Owing to its spin,orbital, and valley isospin degrees of freedom, the E ¼ 0octet of bilayer graphene (BLG) supports a rich varietyof spontaneously broken symmetries and correlations[11,12,29–31]. In particular, a perpendicular electric dis-placement field D generated through dual gating controlsthe electrical valley Zeeman splitting Ev between statesoccupying different valleys or layers, thus offering apowerful experimental knob to control the character ofthe LLs and the nature of the interactions they support. Thistuning is compatible with device scaling and is independentof the magnetic field B, which controls the strength of theCoulomb interactions. It is a distinct experimental controlof the BLG platform, which can be deployed to constructand elucidate many-body phenomena in a correlated,multicomponent 2D system.In this work, we make ultra-high-quality BLG devicesthat enable fine control of the valley Zeeman splitting Ev,and we explore its profound impact in realizing FQH stateswith different ground state orders and valley isospin (VIS)polarizations. In the regime of low D field, an unprec-edented even-denominator FQH state emerges at fillingfactor 5=2 and is found to be spontaneously valleypolarized in the limit of vanishing Ev. We observe clearLevin-Halperin daughter states of either the Moore-ReadPfaffian or the anti-Pfaffian in the vicinity of ν ¼ 7=2, 3=2(Pfaffian) and ν ¼ 5=2, −1=2 (anti-Pfaffian), with theappearance of a Hall resistance plateau at ν ¼ 1þ 7=13.These results suggest that the even-denominator FQH statesin BLG break particle-hole symmetry, and the brokensymmetry sensitively depends on the underlying inter-actions. We construct a comprehensive experimental phasediagram of the VIS polarization for odd-denominator FQHstates in the range 0 < ν < 2. These measurements trulyestablish the valley isospin in BLG as a spinlike electronicdegree of freedom and pave the pathway to future effortsexploiting its utility in controlling the ground state orderand topology of correlated electronic states.II. DEVICE FABRICATIONOur dual-graphite-gated, h-BN encapsulated Hall bardevices are made using dry van der Waals transfer and sidecontact techniques largely following methods introducedin the literature [11,12,32]. A different etching protocol isused in our fabrication process, which led to higher-qualityelectrical contacts compared to prior studies [11]. The detailsFIG. 1. Valley isospin controlled fractional quantum Hall states in bilayer graphene. (a) Optical micrograph of device 002. The BLGand the top graphite gate are etched into a Hall bar shape outlined in black. The bottom graphite gate is outlined in white. The thicknessof the top and bottom BN sheets is 28 nm and 23 nm, respectively. See Appendix A for fabrication details. (b) Wave functions of LLsjξ; Ni. Here, ξ ¼ þ, − and N ¼ 0, 1 denote the valley and orbital indices. (c) False color map of Rxx (D, ν) at B ¼ 18 Tand T ¼ 20 mK. We label the parent LL of each region. D� transitions are marked by red dashed lines. The black dashed linesmark the true D ¼ 0 locations. States occupying the j � 0i LLs exhibit the two-flux CF Jain sequence shown in Fig. 10 inAppendix B. (d) Energy-level diagram that captures the qualitative features of panel (c). Even-denominator states at ν ¼ 3=2 and 7=2 arereported in Refs. [11,12].KE HUANG et al. PHYS. REV. X 12, 031019 (2022)031019-2of the fabrication are given in Appendix A. Figure 1(a)shows an optical micrograph of device 002. The carrierdensity n and displacement field D are, respectively,given by n ¼ gBGðVBG − VBG0Þ þ gTGðVTG − VTG0Þ andD ¼ ½gBGðVBG − VBG0Þ − gTGðVTG − VTG0Þ�e=2ε0, withthe gating efficiencies gBG ¼ 7.3 × 1011 V−1 cm−2 andgTG ¼ 5.9 × 1011 V−1 cm−2 in device 002. Measurementsused to characterize the devices, as well as parameters ofdevices 011 and 015, are also given in Appendix A.III. RESULTS AND DISCUSSIONA. Even-denominator fractional quantumHall state at ν = 5=2Figure 1(c) shows a false color map of the longitudinalresistance Rxx (D, ν) in the filling factor range 1 < ν < 4 atB ¼ 18 T. Integer and fractional quantumHall states appearas dark lines. They occupy the j � 0i and j � 1i LLs ofthe BLG, the wave functions of which are illustrated inFig. 1(b) [29]. Figure 1(d) gives an energy level diagram ofthis regime. The valley Zeeman energy E�v is defined asE�v ¼ �1=2 gvD, where �1=2 corresponds to the quantumnumbers of the two valley isospins and gv is the barevalley g-factor. The valley Zeeman splitting Ev increaseswith increasing D following Ev ¼ gvD, with gv ¼1.43 K=ðmV=nmÞ. We obtain the value of gv fromFig. 8 in Appendix B and Ref [30]. The j þ 0i andj − 1i levels become degenerate at D¼D�, whereEv¼E10 [11,12,29,30]. This level crossing leads to theclosing of gaps for states occupying these two LLs,resulting in an Rxx increase in our measurements. We marktheD� transitions in Fig. 1(c) using four red dashed lines. Inregimes of D > D�, we observe the same LL orbitals as inprevious studies, where two even-denominator FQH statesat ν ¼ 3=2 and 7=2 have been identified [11,12]. They alsooccur in our devices [see Fig. 1(c)]. In this work, we focuson the regime of D < D�, where disorder has obscuredprevious studies [11,12].In this small-D regime, states in the range of 2 < ν < 4occupy the j � 1i LL levels. Remarkably, a strong Rxxminimum develops at ν ¼ 5=2 in our devices. This Rxxminimum occurs in multiple devices and is accompanied bya well-quantized Rxy plateau as shown in Fig. 9(a) ofAppendix B. Thus, the state at ν ¼ 5=2 is an even-denominator FQH state. Its appearance on the j � 1i LLlevels, while somewhat intuitive, is only observed nowthanks to the high quality of our devices. As the LLdiagram in Fig. 1(d) shows, the 5=2 state resembles itscounterpart in GaAs but with the spin index now replacedby the VIS index in BLG. A Moore-Read wave functionrequires the 5=2 state to be polarized in any spin and isospinsectors while numerical simulations of a zero-thickness 2Dsystem further support its spontaneous polarization inthe limit of Ez ¼ 0 [10,23,24]. Numerous experimentshave examined the spin polarization of the 5=2 state inGaAs [6,33–40]. While the state is generally thoughtto be spin polarized in a sufficiently large magnetic field,measurements conducted in the low-field range of B ≤ 1 Thave uncovered potential phase transitions [36,37]. In thesmall-Ez regime, the interpretation of measurements iscomplicated by the finite wave-function thickness in GaAsand the changes of the interaction energies when themagnetic field changes. In comparison, the electrical tuningof Ev, together with the high quality and near-zero thick-ness of BLG, presents a clean approach to probe the valleyisospin polarization in BLG, in advance of theoreticalcalculations.We measure the gap of the 5=2 stateΔ5=2 as a function ofthe D field using thermally activated transport. Figure 2(a)plots the D-field sweeps of R5=2xx ðD̃Þ measured at differenttemperatures, from which we obtain data points for RxxðTÞand plot them in an Arrhenius plot in Fig. 2(b). Fits toRxxðTÞ ∼ expð−Δ=2kBTÞ yield the gaps Δ5=2ðD̃Þ at differ-ent D fields. Figure 2(c) plots Δ5=2ðD̃Þ for both positiveand negative values in the range of jD̃j > 1–2 mV=nm,where the effect of potential disorder can be neglected.Here, D̃ ¼ Dþ 12.5 mV=nm adjusts for a small offsetin the applied D field. The bottom x axis of Fig. 2(c)plots the normalized Ev=Ec, where Ec ¼ ðe2=4πϵϵ0lBÞ ¼217ffiffiffiffiffiffiffiffiffiB½T�pK is the Coulomb energy scale in BLG usingϵ ¼ 3 for the dielectric constant of h-BN. The gap Δ5=2ðD̃Þsaturates at 0.3 and 0.35 K, respectively, at large positiveand negative D fields and extrapolates to a finite value ofapproximately 0.2 K at D̃ ¼ 0 from both sides. The trendΔ5=2ðD̃Þ exhibits, together with a sizable gap at Ev ¼ 0,provides strong experimental evidence for a spontane-ously valley-polarized 5=2 state. As we demonstrate inFig. 4, the valley isospin in BLG indeed behaves as anSU(2) spin. The spontaneous valley polarization of the5=2 state is consistent with numerical simulations per-formed for the real electron spin of the Moore-Readstate [23,24]. Prior work in BLG finds that the even-denominator state at ν ¼ 3=2 is spin polarized at 17 T[11]. Given the similar Zeeman splitting in both cases, weexpect the 5=2 state examined here to also be spinpolarized, thus satisfying the triplet requirement of theMoore-Read wave function in any spin or isospin [3].Figure 2(d) shows the behavior of R5=2xx ðD̃Þ andΔ5=2ðD̃Þclose to D̃ ¼ 0, where the valley polarization ofthe 5=2 state is expected to switch abruptly in samples freeof disorder. Instead, we observe a percolation transitionfrom the j þ 1i LL to the j − 1i LL with a full width ofapproximately δD ∼ 0.7 mV=nm or δEv ∼ 1 K. This tran-sition is signaled by two approximately T-independentcrossover points atDc ∼�0.35 mV=nm in Fig. 2(d), whereΔ5=2 apparently drops to zero. We emphasize that the Tdependence of Rxx close to D̃ ¼ 0 is not related to theVALLEY ISOSPIN CONTROLLED FRACTIONAL QUANTUM HALL … PHYS. REV. X 12, 031019 (2022)031019-3physics of the 5=2 state but rather reflects the generalconduction behavior of a percolation network; theincreased bulk conduction in the presence of mixed valleydomains leads to Rxx ∼ σ−1xx , hence, an opposite dR=dTnear D̃ ¼ 0. The low disorder of our devices allows us toprobe the intrinsic behavior of the 5=2 state down to thesmall valley Zeeman splitting of Ev=Ec ∼ 10−3, where the5=2 state remains valley polarized.B. Particle-hole symmetry of the even-denominatorfractional quantum Hall statesThe particle-hole (p-h) symmetry of a half-filled LL isanother open question of keen interest to the quantum Hallcommunity [2,16–22,41–45]. The Moore-Read Pfaffianand its p-h conjugate, the anti-Pfaffian, explicitly breakthe p-h symmetry. Their energy difference is small, andeach state has gained support in numerical calculationsperformed in GaAs [17,18,41–43]. Theory also suggeststhe possibility of a third ground state, known as the p-hPfaffian, which preserves the p-h symmetry and can bestabilized when both disorder and LL mixing are included[44,45]. The different Majorana neutral modes of the threedistinct topological orders allow them to be differentiated inthermal conductance and noise measurements, and recentexperimental results have favored the p-h Pfaffian [16,22].However, the situation is far from being settled. In thehierarchical theory, quasiparticles and quasiholes awayfrom the half-filled N ¼ 1 LL condense into incompress-ible fractional quantum Hall states known as the Levin-Halperin states [46]. The first daughter states of the Pfaffianoccur at partial fillings ν̃ ¼ 7=13 and 8=17, while the firstdaughter states of the anti-Pfaffian occur at ν̃ ¼ 6=13 and9=17 [46]. Transport studies in GaAs have revealed theFIG. 2. The gap of the 5=2 state. (a) R5=2xx ðD̃Þ sweeps at selected temperatures as labeled in the plot. Here, both gates are sweptsimultaneously to stay on the ν ¼ 5=2 minimum. Data points are read from smoothed traces, an example of which is shown for the77-mK trace (violet dashed line). (b) T-dependent Rxx obtained from traces in panel (a) and plotted on an Arrhenius plot for selected Dfields as labeled. Solid lines are fits to expð−Δ5=2=2kBTÞ, from which we extractΔ5=2: Upper and lower bounds of Δ5=2 from the fits areshown as error bars in panel (c). Here, we obtain D̃ ¼ Dþ 12.5 mV=nm using the black dashed line crossing ν ¼ 5=2 in Fig. 1(c). (c) D̃dependence of Δ5=2: Dashed lines extrapolate to 0.2 K at D̃ ¼ 0. Note that Δ5=2 is larger on the negativeD side due to weaker screeningfrom the top graphite gate, which is farther away. (d) Upper panel: T dependence of Rxx at very small D̃. As the insets illustrate, thesample is comprised of domains of opposite valley polarizations near D̃ ¼ 0, and the dominance of bulk conduction leads to Rxx ∼ 1=σxxand a negative dR=dT. The transition to edge conduction at large D̃ produces roughly T-independent crossover points at Dc ∼�0.3 mV=nm labeled by the arrows. The lower panel of (d) plots the extracted Δ5=2. Note that Δ5=2 drops precipitously towards zero atDc. Similar percolation transitions accompany other valley polarization transitions that occur at both integer and fractional fillings in oursamples, such as shown in Figs. 4 and 12.KE HUANG et al. PHYS. REV. X 12, 031019 (2022)031019-4appearance of a clear FQH state at ν̃ ¼ 6=13, suggestingan anti-Pfaffian ground state [21]. The understandingand reconciling of these measurements remain ongoing.BLG exhibits multiple even-denominator FQH statesincluding the new 5=2 state. Using the Levin-Halperinstates as indicators, our transport studies point to thePfaffian order at ν ¼ 3=2, 7=2 and the anti-Pfaffian orderat ν ¼ 5=2, −1=2.The p-h asymmetry appears to be most prominent atν ¼ 3=2, as shown in Fig. 3(a) for device 015 and Fig. 11(a)for device 002. We see clear Rxx minima at ν̃ ¼ 7=13 and8=17, together with a Hall plateau at Rxy ¼ 0.65 h=e2corresponding to the 20=13 filling. Strong Rxx minima atthese fractional fillings are also found in device 002 andshown in Fig. 11(a). In the literature, a peak at 7=13was alsofound in the capacitance measurements of Ref. [12]. Incontrast, the newly observed 5=2 state appears to follow theanti-Pfaffian order, as suggested by the appearance of Rxxminima at ν̃ ¼ 6=13 and 9=17, as labeled in Fig. 3(b).Figures 3(b) and 3(e) show their appearance in both devices015 and 002. The Rxx minima, though shallow, appear atthe same fractions over a wide range of the D field andare robust in thermal cycling. It is useful to compare theirweak but robust appearance with other accidental andirreproducible minima in Rxx, a number of which can beseen in Figs. 3(c)–3(e). Following the same reproducibilitycriteria, we identify the ν ¼ 7=2 state to be Pfaffian[Fig. 3(c)] and the ν ¼ −1=2 state to be anti-Pfaffian[Fig. 3(d)]. In the vicinity of all four even-denominatorstates, we find the daughter states of either the Pfaffian orthe anti-Pfaffian, but not both, and the daughter statesalways appear in tandem on both sides of the half-filling.In addition, we see a weak but robust appearance of theν̃ ¼ 6=13 state near ν ¼ −5=2 [Fig. 11(b)], suggestingthat the −5=2 state is likely anti-Pfaffian also. Table Isummarizes the broken p-h symmetries of five even-denominator states in BLG. Subtle differences of theinteraction at different filling factors play a clear rolein the resulting asymmetry, and calculations capable ofexplaining all of them self-consistently will shedFIG. 3. Particle-hole asymmetry at half-filled N ¼ 1 LLs in BLG. Panels (a)–(e) plot Rxx as a function of the partial fillingν̃ ¼ ν − ½ν� near ν ¼ 3=2 (a), 5=2 (b), 7=2 (c), −1=2 (d), and 5=2 (e), respectively. Panel (a) also plots Rxy measured concomitantly.Panels (a) and (b) are from device 015, and (c)–(e) are from device 002. The data are obtained at fixed D fields as labeled, andB ¼ 18 T unless otherwise noted. Here, T ¼ 20 mK. In panel (a), Rxy plateaus are observed at ν̃ ¼ 2=5, 1=2, 7=13, and 3=5 andquantized to the correct values given by their full filling factors. Weak but reproducible Rxx minima occur at ν̃ ¼ 8=17; 3=7, and 4=7.In panels (a)—(e), dashed lines mark fractional fillings calculated from the positions of 1=3 and 2=3. Only reproducible minima aremarked. Blue lines mark ν̃ ¼ 8=17 and 7=13, and magenta lines mark ν̃ ¼ 6=13 and 9=17. Their differences, though small, arediscernable in our data. Additional data on ν ¼ 3=2 and −5=2 from device 002 are given in Fig. 11 of Appendix B.VALLEY ISOSPIN CONTROLLED FRACTIONAL QUANTUM HALL … PHYS. REV. X 12, 031019 (2022)031019-5much theoretical light on this fundamental question.Finally, we briefly note the appearance of unconventionalFQH states at ν̃ ¼ 2=5, 3=5, 3=7, and 4=7 in Fig. 3(a),especially the well-developed 2=5 and 3=5 statesexhibiting quantized Rxy plateaus. These observationsoffer the future possibility of exploring the proposednon-Abelian orders and topological phase transitions inBLG [4,47].C. Valley isospin polarization transitions ofodd-denominator fractional quantum Hall statesThe understanding and control of the spin or isospinconfiguration of a FQH state is a fundamental and key steptowards the generation of parafermions [2,7,26,48,49].The ease of tuningEv and ν continuously in a single deviceenables us to systematically study the ground state VISpolarization of FQH states. Figures 4(a) and 4(b) showmaps of RxxðD; νÞ near D ¼ 0 for 4=3 < ν < 5=3 and1=3 < ν < 2=3, respectively. In Fig. 4(a), we observenumerous gap closing points reminiscent of spin orpseudospin transitions observed in other 2D systems[2,26,48,49]. Similar gap closing points, also in thevicinity of D�, were observed in Ref. 12 and interpretedusing an effective single-particle-like model [12].Here, we examine our data in the theoretical frameworkof two-flux composite fermions (CFs) with SU(2)spins or isospins. In this model, the fractional filling νnear 3=2 maps to ν� filled CF Λ levels throughν ¼ 2 − ν�=ð2ν� � 1Þ. As illustrated in Fig. 4(c), VIS(partial) polarization transitions occur when Λ levels ofopposite valley indices cross one another. This conditioncorresponds to Ev ¼ ½1 − ν�; 3 − ν�;…; ν� − 1�ℏω�c , for atotal number of ν� transitions for the ν�th Λ level. Here,ω�c ¼ ðeBeff=m�3=2Þ is the cyclotron frequency of CFs,Beff ¼ 3ðB − B3=2Þ, and m�3=2 is the effective polarizationFIG. 4. Valley isospin polarization of Jain FQH states. (a,b) False color maps of Rxx (D, ν) similar to Fig. 1(c) in the filling factorranges of 4=3 < ν < 5=3 (a) and 1=3 < ν < 2=3 (b). The map in panel (a) expands upon the low-D region between the red dashed linesin Fig. 1(c). The top axis in panel (a) labels the corresponding CF Λ level filling factor ν�. The side panel of panel (a) plots a line cutthrough ν ¼ 4=3. The red circles mark D̃ ≥ 0 VIS transitions for ν ¼ 4=3, 7=5, 10=7, and 13=9. Data taken at B ¼ 18 T andT ¼ 20 mK. No transitions are observed in 1=3 < ν < 2=3. (c) Free CF Λ level fan diagram including valley isospin. The blue and reddashed lines represent Λ levels of the “þ” and “−” valley polarizations, respectively, and ðn−; nþÞ labels the filling factor of each valley.Here, ν� ¼ n− þ nþ. The slope of the red dashed lines is obtained through a fit to Ev ¼ ðν� − 1Þℏω�c. This fit yields m�3=2 ¼ 2.6me,which determines the rest of the diagram. Open circles correspond to transitions marked by the same symbol in panel (a). A similaranalysis is performed for states with ν > 3=2 and yields a smaller m�3=2 ¼ 1.9me. See Fig. 13 in Appendix B for the fits in both regimes.(d) Normalized critical valley Zeeman energy αcrit for FQH states on both sides of ν ¼ 3=2 at fixed magnetic fields as labeled in the plot.The solid violet lines plot calculations from Ref. [50] divided by 4.TABLE I. Ground state wave function symmetry of the even-denominator states in BLG.ν 3=2 5=2 7=2 −1=2 −5=2P-H symmetry Pf aPf Pf aPf aPf*(likely)KE HUANG et al. PHYS. REV. X 12, 031019 (2022)031019-6mass at ν ¼ 3=2 [2]. With a single fitting parameterm�3=2 ¼ 2.6me, where me is the free electron mass invacuum, we can capture all eight transitions markedby red circles in Fig. 4(a). In stark contrast to4=3 < ν < 5=3, no valley polarization transitions areobserved for states in 1=3 < ν < 2=3 in magneticfields ranging from 14 T to 31 T, suggesting all two-fluxCFs in the last LL are spontaneously valley polarized.Following a similar argument, we surmise that theeven-denominator FQH states at ν ¼ 7=2 and −1=2 mayalso be spontaneously valley polarized [Fig. 1(c) andRefs. [11,12] ], though measurements performed inFig. 2 for ν ¼ 5=2 are required to confirm this hypothesis.The different behaviors of the FQH states in the tworegimes connected by ν to 2 − ν transformation point to astrong LL mixing effect [2].Extending similar measurements and analysis to highermagnetic fields (18–31 T), we plot in Fig. 4(d) thenormalized critical valley Zeeman energy αcrit ¼ Ev=Eccorresponding to the onset of full valley polarization(see Fig. 12 in Appendix B for raw data). Here, Ec ¼ðe2=4πϵϵ0lBÞ ¼ 217ffiffiffiffiffiffiffiffiffiB½T�pK is the Coulomb energy scalein BLG using ϵ ¼ 3 for the dielectric constant of h-BN.Results from different magnetic fields collapse quitewell, suggesting an approximateffiffiffiffiBpscaling of m�3=2.The tentlike solid lines in Fig. 4(d) represent exactdiagonalization calculations performed for spin polariza-tion transitions in a zero-thickness 2D system [50], whichis applicable to any SU(2) isospins. The qualitativeagreement between our data and theory supports theSU(2) character of the VIS in BLG. Quantitatively,our results of αcrit are more symmetric around 3=2and are approximately 4–5 times smaller than theory.Extrapolation to 3=2 yields m�3=2 ∼ 0.50meffiffiffiffiBp, in com-parison to the theoretical value of 0.13meffiffiffiffiBpin graphene[50]. Measurements on device 011 yield nearly identicaltransitions (Fig. 14), indicating that the underlying phys-ics is insensitive to sample details. We attribute the smallαcrit to the effect of LL mixing, the inclusion of which isnecessary to accurately capture the energetics of corre-lated phenomena in BLG [51].IV. CONCLUSIONIn summary, we report the observation of an even-denominator FQH state at filling factor 5=2 in Bernal-stacked bilayer graphene. The state remains polarized inthe limit of vanishing valley isospin splitting, offeringindirect support to a spontaneously spin-polarizedMoore-Read state. The even-denominator states areparticle-hole asymmetric, with the asymmetry consistentwith a Pfaffian order at filling factors 3=2 and 7=2 and theanti-Pfaffian order at 5=2 and −1=2. The valley isospinbehaves like an SU(2) spin with excellent experimentalmaneuverability. We demonstrate the control of theground state valley polarization of a large family ofFQH states and envision the manipulation of valleyisospin to be a powerful tool in elucidating othercorrelated electronic phenomena and constructing quan-tum information devices.ACKNOWLEDGMENTSK. H., H. F., and J. Z. are supported by the NationalScience Foundation through Grants No. NSF-DMR-1904986 and No. NSF-DMR-1708972 and by theKaufman New Initiative Research Grant No. KA2018-98553 of the Pittsburgh Foundation. H. F. acknowledgessupport from the Penn State Eberly Research Fellowship.D. R. H. and N. A. acknowledge support from the NSFCAREER program (DMR-1654107) and the Penn State 2DCrystal Consortium-Materials Innovation Platform (2DCC-MIP) under NSF Cooperative Agreements DMR-1539916and DMR-2039351. FIB/SEM and TEM measurementswere performed in the Materials CharacterizationLaboratory of the Materials Research Institute at thePennsylvania State University. X. L. acknowledges thesupport of Beijing Natural Science Foundation(Grant No. JQ18002) and the National Key Researchand Development Program of China (GrantNo. 2017YFA0303301). K.W. and T. T. acknowledgesupport from JSPS KAKENHI (Grants No. 19H05790,No. 20H00354, and No. 21H05233). Work performed atthe National High Magnetic Field Laboratory was sup-ported by the NSF through Grant No. NSF-DMR-1644779and the State of Florida. We thank Jainendra Jain, Ajit C.Balram, William Faugno, Zlatko Papic, Bertrand Halperin,Allan H. McDonald, and Gabor Csathy for helpful dis-cussions. We thank Dr. Elizabeth Green and Wenkai Zhengfor assisting in measurements performed during theCOVID-19 pandemic.The authors declare no competing financial interest.APPENDIX A: DEVICE FABRICATION ANDCHARACTERIZATION1. Device fabricationWe fabricated our devices using the processesdescribed below. The top/bottom h-BN sheet thicknessis 28 nm=23 nm, 24 nm=27 nm, 35 nm=35 nm, and25 nm=25 nm, respectively, for devices 002, 011, 012,and 015. We performed transport measurements on devices002, 011, and 015 and microscopy studies on device 012.Figure 5(a) shows a scanning electron microscopy (SEM)image of device 012. Figures 5(b) and 5(c) show cross-sectional scanning transmission electron microscopy(STEM) or TEM images of two slabs lifted from theVALLEY ISOSPIN CONTROLLED FRACTIONAL QUANTUM HALL … PHYS. REV. X 12, 031019 (2022)031019-7two locations marked in Fig. 5(a) using a focused ion beam(FIB). The sample preparation and microscopy measure-ments follow that of Ref. [32]. TEM studies were per-formed using an FEI Titan3 G2 operating at 200 kVand anFEI Talos F200X operating at 80 kV. SEM imaging andFIB TEM sample preparation were performed on a ThermoScientific Scios 2 DualBeam analytical FIB-SEM using ionbeam voltages ranging from 30 kV down to 5 kV forlamella thinning.To make a device we first build a h-BN/graphene/h-BN/bottom graphite gate stack and transfer it to SiO2=Sisubstrate using a dry transfer method [28,30]. This processis followed by annealing in an Ar=O2 atmosphere at 450 °Cfor 3 hours to remove polymer residue from the transfer.We then exfoliate a graphite sheet (3–4 nm thick) on aPPC (polypropylene carbonate) stamp and transfer it tothe stack. This will be the top gate. Figure 6(a) shows aschematic of the finished stack.A sequence of e-beam lithography and reactive ion etch(RIE) steps illustrated in Figs. 6(b)–6(f) is used to shape thestack into the Hall bar structure shown in Fig. 1(a). We firstetch the top graphite sheet into an area that is slightly largerthan the bottom graphite gate to expose the h-BN/BLG areareserved for making contacts later [Fig. 6(b)]. We thenuse e-beam lithography to define the Hall bar structure[Fig. 6(c)] and three sequential etching steps to pattern thetop gate/h-BN/BLG stack. The three steps are illustrated inFigs. 6(d)–6(f). The CHF3=O2 etching step in Fig. 6(d) wasdone unintentionally for device 002 and repeated in devices011, 012, and 015. In this step, the top graphite gateprotects the h-BN/BLG underneath from being etched, andthe etching time is not long enough to fully remove the toph-BN sheet in areas outside the top graphite sheet. Device703 was made without this step, and we compare theirperformances in Fig. 7. We then use O2 plasma to shapethe top graphite gate [Fig. 6(e)] followed by a CHF3=O2plasma to etch the h-BN/BLG stack into the Hall barstructure [Fig. 6(f)]. Table II shows the parameters used forthe two types of plasma. Etching times for device 002 areshown in Fig. 6 and adjusted for other devices.FIG. 6. The etching steps used in the fabrication of device 002.(a) Illustration of the finished graphite/h-BN/BLG/h-BN/graphitefive-layer stack. (b) Illustration of the shape of the top graphitesheet after the O2 etch. It extends over the bottom graphite gate(white dashed lines) by about 200 nm on the left and right sides.(c) Etching mask used in the next three etching steps on panel (b).(d) CHF3=O2 etch defining the shape of the terminals outside thetop graphite sheet. The h-BN/BLG underneath the top graphitesheet remains intact. The entire BLG sheet is not exposed in thisstep. (e) O2 etch, which defines the shape of the top graphite gate.(f) Finished Hall bar device.FIG. 5. Microscopy studies. (a) SEM image of device 012. The bottom graphite gate and the bottom h-BN sheet are outlined in cyanand magenta dashed lines, respectively. The orange dashed line outlines the Hall bar profile of the top h-BN/BLG stack. (b) High-resolution bright-field STEM image of the cross section cut through the white dashed line in panel (a). (c) Conventional TEM image ofthe cross section cut through the black dashed line in panel (a), showing the Au=Cr side contact. The etching profiles resemble those ofprevious studies.KE HUANG et al. PHYS. REV. X 12, 031019 (2022)031019-8Finally, we pattern and deposit one-dimensional Au=Crside contacts using e-beam lithography and physical vapordeposition. The substrate is cooled to 5 °C and rotatedduring the deposition. The deposition rate and thicknessfor each metal are as follows: Cr: 0.5 Å=s, 5 nm andAu: 1.5 Å=s, 45 nm.2. Device characterizationContacts in our devices reside outside the top and bottomgraphite gates and are doped by the Si backgate. We applyVSi ¼ 60 V to ensure good performance in a large mag-netic field. While STEM images in Figs. 5(b) and 5(c) donot reveal visible topographic differences compared topreviously reported devices [32], electrical contacts inour devices appear to perform better compared to Rxxtraces shown in the literature, as indicated by less noisy,non-negative values when Rxx vanishes at integer andfractional fillings [11,52]. More robust contacts may havefacilitated the observations of the even-denominator stateat ν ¼ 5=2 and the valley polarization transitions of thefractional quantum Hall states, which are absent in previoustransport studies [11,12]. A second distinguishing featuredevices 002 and 011 exhibit is the very narrow valleypolarization transition peak at D ¼ 0, which is a strongindicator of higher sample quality. Figure 7(d) comparesRxxðDÞ traces taken at the ν ¼ 2=3 minimum in devices002, 011, and 703. Both 002 and 011 are fabricated with theCHF3=O2 etching step highlighted in red in Fig. 6 while703 is fabricated without this step. The full width at halfmaximum of the D ¼ 0 peak δD is, respectively, 0.38,0.24, and 1.43 mV=nm in devices 002, 011, and 703.While it remains unclear to us how the extra etching stepreduces δD, a small disorder broadening of the valleyZeeman splitting is crucial to the observations reported inthis work.We use the top and bottom graphite gates to tune thecarrier density n and the perpendicular electricFIG. 7. Device characterization. (a) Rxx as a function of VBG at fixed VTG’s as indicated in the plot. Tracking the peaks allows us todetermine the gating relation of VBG and VTG, which is plotted in panel (b). The global minimum occurs atD ¼ 0, which corresponds toVBG0 ¼ 0.137 V and VTG0 ¼ −0.155 V. (c) Hall resistance RxyðnÞ at B ¼ 0.05 T, T ¼ 50 mK, andD ¼ 0 mV=nm. The shaded regiongives an estimated disorder-induced density broadening of δn ¼ 7 × 109 cm−2. (a)–(c) Device 002. (d) Rxx (D) sweeps obtained at theν ¼ 2=3minimum in three devices as labeled in the plot. The FWHM δD ¼ 0.38, 0.24, and 1.43 mV=nm, respectively, for devices 002,011, and 703. Traces are shifted horizontally to overlap the peaks at D ¼ 0.TABLE II. Parameters used in the RIE processes.GasPressure(mTorr)Temperature(°C)RF forwardpower (W)Flow(sccm)O2 20 25 14 25CHF3 þ O2 75 25 40 CHF3: 40O2: 4VALLEY ISOSPIN CONTROLLED FRACTIONAL QUANTUM HALL … PHYS. REV. X 12, 031019 (2022)031019-9displacement fieldD following established practices [28,30].Figure 7(a) plots the longitudinal resistanceRxx as a functionof the bottom gate voltage VBG at fixed top gate voltage VTGin device 002. Tracking the charge neutrality points (CNP)(Rxx peaks) yields the gating relation VBG ¼ −0.81 ×VTG þ 0.012 as well as (VBG0,VTG0) corresponding to theD ¼ 0 point marked in Fig. 7(b).Figure 7(c) plots the Hall resistance Rxy vs n at B ¼0.05 T in device 002, from which we estimate a disorder-induced density broadening of δn ¼ 7 × 109 cm−2. Notethat δn is typical of our dual-graphite-gated devices andcomparable to what is reported in the literature [11,12]. InFig. 8, we show that device 002 exhibits a bulk disorderenergy scale of about 10 K, similar to other high-qualityBLG devices reported in the literature [52]. However,transport in the quantum Hall regime is carried by edgestates and as a result sensitively depends upon disorderclose to the sample edges. Our devices that were fabricatedusing the process illustrated in Fig. 6 show unprecedentedquality in this measurement, as indicated by the narrowpeaks that devices 002 and 011 exhibit at the FQH stateν ¼ 2=3 in Fig. 7(d).APPENDIX B: SUPPORTING MEASUREMENTSON DEVICE 0021. Measurement of the bare valleyZeeman g-factor gvWe define the valley Zeeman energy E�v ¼ �1=2gvD,where � corresponds to the two valley indices and gv isthe bare valley g-factor. The valley Zeeman splittingbetween the “þ” and the “−” levels is Ev ¼ gvD. Forthe N ¼ 0 orbital, Ev ¼ U, where U is the potentialdifference between the top and bottom graphene layers.FIG. 8. The D-field dependence of the band gap in BLG.(a) Charge neutrality point resistance RCNP as a function of D atdifferent temperatures. (b) T-dependent RCNP obtained fromtraces in panel (a) and plotted on an Arrhenius plot for selectedD fields as labeled. Solid lines are fits to expð−Δ=2kBTÞ, fromwhich we extract Δ. (c) Extracted ΔðDÞ for both positive andnegative D fields. We fit data above 20 mV=nm with a linecorresponding to ΔðDÞ ¼ 1.43D − 16 ½K� (green dashed line).Disorder leads to the reduction of Δ, which manifests as anegative offset of 16 K. The formation of electron-hole puddles atthe CNP gives rise to a finite energy scale of 5 K at D ¼ 0. Bothpoint to bulk disorder on the scale of about 10 K, similar to otherstate-of-the-art BLG devices. (d) Data from device 002 here and23L in Ref. [30]. The red line corresponds to the fit obtained inpanel (c).FIG. 9. The Rxx and Rxy vs filling factor ν near 5=2. (a) Deep Rxx minimum and a quantized Rxy plateau observed in device 015.(b) Signatures of the 5=2 state still present at B ¼ 9 T and T ¼ 40 mK in device 002.KE HUANG et al. PHYS. REV. X 12, 031019 (2022)031019-10Note that UðDÞ increases linearly with the applied Dfield and equates the band gap in bilayer graphene ΔðDÞ atzero magnetic field when D is not too large [53].We measure ΔðDÞ at B ¼ 0 to determine the bare valleyZeeman g-factor gv using ΔðDÞ ¼ gvD. Figures 8(a)–8(c)show our measurements and analysis of ΔðDÞ, followingapproaches described in the supplementary informationof Ref. [30] and extending the data of Ref. [30] to the small-D regime. Our analyses yield gv ¼ 1.43 K=mVnm−1, inexcellent agreement with previous results [30].2. FQH states occupying the N = 0 and 1Landau levelsFigure 9 shows concomitant measurements of Rxy andRxx near ν ¼ 5=2 in two devices. Device 015 exhibits awell-quantized plateau in Rxy at 18 T while device 002exhibits a developing Rxy plateau down to 9 T, bothconfirming the 5=2 state to be a FQH state. Figure 10shows examples of FQH states occupying the N ¼ 0 LL.They are well described by two-flux composite fermions.Figure 11 provides additional evidence to support thebreaking of the particle-hole symmetry near even-denom-inator FQH states ν ¼ 3=2 and −5=2 in bilayer graphene.The Levin-Halperin states manifest as Rxx minima at theexpected filling factors. They are robust in thermalcycling and appear reproducibly at the same fillingfactors in different D fields [Fig. 11(a)] and B fields[Fig. 11(b)]. We are able to resolve the small differencesbetween the two sets of filling factors, e.g., between 6=13and 8=17, here and in the data shown in Fig. 3 of themain text.FIG. 10. FQH states on the N ¼ 0 LL. (a) Same false color map of Rxx (D, ν) as in Fig. 1(c) in the main text. (b) Trace of Rxx (ν) for2 < ν < 3 following the white dashed line in panel (a). (c) Trace of Rxx (ν) for 1 < ν < 2 following the orange dashed line in panel (a).The two-flux composite fermion FQH states corresponding to ν̃ ¼ ν − ½ν� ¼ p=ð2p� 1Þ, where p is an integer, are labeled in panels (b)and (c) and provide an excellent description of the data. Here, B ¼ 18 T, T ¼ 20 mK.FIG. 11. Additional data from device 002 showing the particle-hole asymmetry near the even-denominator FQH states in bilayergraphene. (a) Rxx (ν) at selected D fields in the vicinity of ν ¼3=2 showing the appearance of Levin-Halperin states at ν̃ ¼ 8=17and 7=13. (b) Rxx (ν) at different B and D fields in the vicinity ofν ¼ −5=2. The appearance of the weak but reproducible ν̃ ¼6=13 state suggests an anti-Pfaffian state, although the ν̃ ¼ 9=17is missing. Here, T ¼ 20 mK.VALLEY ISOSPIN CONTROLLED FRACTIONAL QUANTUM HALL … PHYS. REV. X 12, 031019 (2022)031019-113. Valley isospin polarization transitionsof the FQH states near ν = 3=2Figure 12 shows the measured RxxðDÞ traces fromwhich we extract the data points plotted in Fig. 4(d) ofthe main text. In Fig. 13, we plot transitions to a fully VISpolarized ground state at B ¼ 18 T for filling factors 4=3,7=5, 10=7, and 13=9 (open red triangles) and 5=3, 8=5,11=7, and 14=9 (open black squares). The data pointsare read from Fig. 4(a) of the main text and fit to the freeCF modelEcritv ¼ ðν� − 1ÞðℏeBeff=m�3=2Þ; ðB1Þfrom which we obtain effective polarization mass ofm�3=2 ¼ 2.6me and 1.9me, respectively, using data fromthe regimes of ν < 3=2 and ν > 3=2.APPENDIX C: DATA FROM DEVICE 011This section presents measurements from device 011.Device 011 was fabricated using the processes described inAppendix A and has top and bottom h-BN sheets withsimilar thickness to that of device 002. Figure 14 comparesthe valley isospin polarization transitions in devices 002FIG. 12. Valley isospin polarization transitions of the FQH states near ν ¼ 3=2. (a)–(d) Traces taken at different B fields as labeled andat different fractions using the color scheme labeled in panel (d). Here, T ¼ 350 mK. VIS transitions manifest as resistance peaks. Themiddle peaks of the red trace and the green trace mark the true D ¼ 0 locations. We interpolate or extrapolate them linearly to find theD ¼ 0 locations for the black and the blue traces. Data points in Fig. 4(d) correspond to readings of positive D’s from device 002.FIG. 13. Effective polarization mass of the CFs at ν ¼ 3=2.(a) Ecritv vs (ν� − 1) for ν ¼ 4=3, 7=5, 10=7, 13=9 (ν� ¼ 2, 3, 4, 5,respectively, black symbols) and for ν ¼ 5=3, 8=5, 11=7, 14=9(ν� ¼ 1, 2, 3, 4, respectively, red symbols). The solid lines are fitsto Eq. (B1). The slopes yield m�3=2 ¼ 2.6me for ν < 3=2 andm�3=2 ¼ 1.9me for ν > 3=2.FIG. 14. Comparison of VIS polarization transitions in devices002 (black traces) and 011 (red traces) at ν ¼ 4=3 [panel (a)] and7=5 [panel (b)]. Here, B ¼ 18 T and T ¼ 20 mK. Device 011exhibits very narrow VIS transitions shortly after the device wasfabricated [see Fig. 7(d)]. Its quality degraded over many monthsof storage before measurements shown here were taken. Tracesplotted here are three-terminal measurements, so the contactresistance is included. Nonetheless, nearly identical VIS polari-zation transitions are found in these two devices, indicating thatthe underlying physics is insensitive to sample details andrelatively robust against disorder.KE HUANG et al. PHYS. REV. X 12, 031019 (2022)031019-12and 011 at two fractional states. 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