key: cord-0574216-tpjg1vyx
authors: Huang, Ke; Fu, Hailong; Hickey, Danielle Reifsnyder; Alem, Nasim; Lin, Xi; Watanabe, Kenji; Taniguchi, Takashi; Zhu, Jun
title: Valley Isospin Controlled Fractional Quantum Hall States in Bilayer Graphene
date: 2021-05-14
journal: nan
DOI: nan
sha: ead44607c3cc6b88f9f94a7f4e8cca18c3835e34
doc_id: 574216
cord_uid: tpjg1vyx

Electron spin and pseudospin degrees of freedom play a critical role in many-body phenomena through exchange interactions, the understanding and control of which enable the construction of states with complex topological orders and exotic excitations. In this work, we demonstrate fine control of the valley isospin in high-quality bilayer graphene devices and its profound impact in realizing fractional quantum Hall effect with different ground state orders. We present evidence for a new even-denominator fractional quantum Hall state in bilayer graphene, its spontaneous valley polarization in the limit of zero valley Zeeman energy, and the breaking of particle-hole symmetry. These observations support the Moore-Read anti-Pfaffian order. Our experiments establish valley isospin in bilayer graphene to be a powerful experimental knob and open the door to engineering non-Abelian states and quantum information processes in a quantum Hall platform.

(FQH) effect. The FQH regime hosts many-body ground states with complex wave functions, non-trivial topology, and unconventional quantum exchange statistics that are potentially useful for topological quantum computing (1-9). The electron spin plays an essential role in constructing and understanding these interaction-driven phenomena (1, 2, 6). However, methods used to probe spin, e.g. magnetic field or light, are often difficult to miniaturize and control locally.

New materials and structures that possess spin-like electronic degrees of freedom, such as the valley isospin (VIS) in Bernal-stacked bilayer graphene (BLG), enable alternative pathways to pursue the rich physics and application potentials of correlated phenomena in 2D systems (10) (11) (12) (13) (14) (15) . In the E = 0 Landau level (LL) of BLG, the VIS is nearly synonymous with the top/bottom layer number and the valley Zeeman splitting Ev can be continuously tuned by a perpendicular electric field. This tuning is independent of the magnetic field B, which controls the strength of the Coulomb interactions. It is realized through gating, which gives VIS in BLG a distinct device advantage.

In this work, we have made ultra-high-quality BLG devices that enable fine control of the valley Zeeman splitting Ev and exploited FQH states with different ground state VIS polarizations. At filling factor = 5/2, an unprecedented even-denominator FQH state emerges and exhibits spontaneous valley polarization in the limit of vanishing Ev, thus validating an outstanding prediction of the Moore-Read wave function. We observe clear evidence of particlehole asymmetry at three even-denominator FQH states = 3/2, 5/2 and 7/2, with the 3/2 and 7/2 states supporting the Pfaffian wave function and the 5/2 state supporting the anti-Pfaffian. We construct a comprehensive experimental phase diagram of the ground state valley polarization for FQH states between 0 < < 2. These measurements establish the valley isospin as an SU(2) spin-like entity and pave the pathway to future efforts exploiting its utility in addressing fundamental questions of correlated electronic states and quantum information processing.

Our dual-graphite-gated, h-BN encapsulated Hall bar devices are made using dry van der Waals transfer and side contact techniques (15, 16) . Figure 1A shows an optical micrograph of device 002 while device fabrication, operation and characteristics are described in Sections 1 and 2 of the Supplementary Materials (SM). The carrier density n and displacement field D are respectively given by , and /2 , with the gating efficiencies = 7.3 10 11 V -1 cm -2 and = 5.9 10 11 V -1 cm -2 in device 002. We focus on device 002 in the main text while results obtained on device 011 are given in Section 8 of the SM. Figure 1B shows a false color map of the longitudinal resistance Rxx (D, ) in the filling factor range 1 < < 3 at B = 18 T. Integer and fractional quantum Hall states appear as dark lines. They occupy the | 0⟩ and | 1⟩ LLs of the BLG, the wavefunction of which are illustrated in Fig. 1C (12) . Figure 1D gives an energy level diagram of this regime. The valley Zeeman splitting Ev increases with increasing D following Ev = gvD, where gv = 1.43 K/(mV/nm) is the bare "valley Zeeman g-factor" we experimentally obtained ( Fig. S4 and Ref (14) ). The | 0⟩ and | 1⟩ levels become degenerate at D = D * , where Ev = E10 (12, 14, 17, 18) . This level crossing leads to the closing of gaps for states occupying these two LLs, resulting in an Rxx increase in our measurements. We mark the D * transitions in Fig. 1B using four red dashed lines. In regimes of D > D * , previous studies (17, 18) have identified two even-denominator FQH states at = 3/2 and 7/2, which also occur in our devices (See Fig. S5 for data on 7/2 and Fig. S8 for a Ddependence study of the gap energy at = 3/2). In this work, we focus on the regime of D < D * , where disorder has obscured previous studies.

In this small-D regime, states in the range of 2 < < 4 occupy the | 1⟩ LL levels.

Remarkably, a strong Rxx minimum develops at = 5/2 in our devices. In Fig. S6 of the SM, we show that down to B = 9 T, the Rxx minimum persists and is accompanied by the onset of a plateau in Rxy. Thus, the state at = 5/2 is an even-denominator FQH state. While the state is spin polarized by virtue of the large magnetic field (14, 18) , by continuously tuning the valley Zeeman splitting Ev to zero, we can probe a key property of the Moore-Read Pfaffian/anti-Pfaffian, that is, its predicted spontaneous spin/isospin polarization (19) (20) (21) . In spin-based experiments, reaching Ez = 0 has not been possible and investigations of this issue remain ongoing (6, (22) (23) (24) (25) (26) .

We measure the 5/2 state gap / using thermally activated transport and obtain its D-field dependence. Figure 2A At = 0, the 5/2 state changes its ground state valley polarization, exhibiting a peak in Rxx with a full-width-at-half-maximum of 0.7 mV/nm; this corresponds to an energy broadening of approximately 1 K in Ev. Similar Rxx peaks also accompany other VIS transitions occurring at both integer and fractional fillings, suggesting a common origin. Inhomogeneous broadening of the D-field caused by potential fluctuations provides a natural explanation. In the vicinity of = 0, domains of opposite valley polarizations form a bulk network that supports conduction through domain wall excitations (16, 27, 28) , and the T-dependence of Rxx is no longer intrinsic to the physics of the 5/2 state. The data and analysis of Rxx (T) and / in the disorder-dominated regime are given in Section 6 of the SM.

The particle-hole (p-h) symmetry of a half-filled LL is an open question of keen interest in the quantum Hall community (2, (29) (30) (31) (32) (33) . While the experimental situation is less clear in GaAs (32, 33) , the p-h symmetry appears to be clearly broken in our devices. In Fig. 3A -C, we plot Rxx ( ) traces taken near the three even-denominator FQH states at = 3/2, 5/2 and 7/2. Near = 3/2 and 7/2, we observe only the Levin-Halperin daughter states of the Pfaffian (34), i.e. = 7/13 and 8/17. Near = 5/2, they are replaced by the daughter states of the anti-Pfaffian, i.e. = 6/13 and 9/17. A Pfaffian order is also found for the 3/2 state in device 011 (Fig. S12 ) and in Ref. (17) . Calculations capable of explaining the p-h asymmetry of all three half-fillings will shed much theoretical light on this fundamental question. Furthermore, we note the strong appearance of unconventional FQH states at partial fillings of = 2/5, 3/5, 3/7 and 4/7 in our ultra-highquality devices, the observation of which offers the possibility to explore the proposed non-Abelian orders and topological phase transitions (4, 35) .

The understanding and control of the spin/isospin configuration of a FQH state is a fundamental and key step towards the generation of parafermions (2, 7, 11, 36, 37) . The ease to tune Ev and continuously in a single device enables us to systematically study the ground state VIS polarization of FQH states. Figures 4A and B show maps of Rxx (D, ) near D = 0 for 4/3 < < 5/3 and 1/3 < < 2/3 respectively. In Fig. 4A , we observe numerous gap closing points reminiscent of spin/pseudospin transitions observed in other 2D systems (2, 11, 36, 37) . In a 2flux composite fermion (CF) model, the fractional filling near 3/2 maps to an integer filling of * Λ levels through Extending similar measurement and analysis to higher magnetic field (18 -31 T), we plot in Fig. 4D Fig. 4D represent exact diagonalization calculations performed for spin polarization transitions in a zero thickness 2D system (38) , which is applicable to any SU (2) isospins. The qualitative agreement between our data and theory supports the SU(2) character of the VIS in BLG. Quantitatively, our results of α crit are more symmetric around 3/2 and are approximately 4 -5 times smaller than theory. Extrapolation to 3/2 yields / * ~0 .50 √ , in comparison to the theoretical value of 0.13 √ in graphene (38) . Measurements on device 011 yield nearly identical transitions ( Figure S13 of the SM), indicating that the underlying physics is insensitive to sample details. We attribute the small α crit to the effect of LL mixing, the inclusion of which is necessary to accurately capture the energetics of correlated phenomena in BLG.

In summary, we observe a spontaneously valley-polarized even-denominator FQH state in Bernal-stacked bilayer graphene, which confirms an outstanding prediction of the Moore-Read postulate. The even-denominator states are particle-hole asymmetric, with the asymmetry consistent with a Pfaffian order at filling factors 3/2 and 7/2 and the anti-Pfaffian at 5/2. The valley isospin behaves like an SU(2) spin with excellent experimental maneuverability. We demonstrate the control of the ground state valley polarization of a large family of FQH states and envision the use of valley isospin in probing other correlated electronic phenomena and constructing quantum information devices. 

Perspectives in Quantum Hall Effects: Novel Quantum Liquids in Low-Dimensional Semiconductor Structures

Fractional Quantum Hall Effects

Nonabelions in the fractional quantum hall effect

Beyond paired quantum Hall states: Parafermions and incompressible states in the first excited Landau level

Non-Abelian anyons and topological quantum computation

The quantum Hall effect at 5/2 filling factor

Universal Topological Quantum Computation from a Superconductor-Abelian Quantum Hall Heterostructure

Direct observation of anyonic braiding statistics

Fractional statistics in anyon collisions

New fractional quantum Hall state in double-layer two-dimensional electron systems

Density dependence of valley polarization energy for composite fermions

Direct measurement of discrete valley and orbital quantum numbers in bilayer graphene

Excitonic superfluid phase in double bilayer graphene

Effective Landau Level Diagram of Bilayer Graphene

A valley valve and electron beam splitter

Metallic Phase and Temperature Dependence of the ν = 0 Quantum Hall State in Bilayer Graphene

Tunable interacting composite fermion phases in a half-filled bilayer-graphene Landau level

Even-denominator fractional quantum Hall states in bilayer graphene

Transition from quantum Hall to compressible states in the second Landau level: New light on the ν = 5/2 enigma

Spin order in paired quantum Hall states

Spin polarization of the ν = 5/2 quantum Hall state

Contrasting Behavior of the 5/2 and 7/3 Fractional Quantum Hall Effect in a Tilted Field

Tilt-Induced Anisotropic to Isotropic Phase Transition at ν = 5/2

Enhancement of the ν = 5/2 Fractional Quantum Hall State in a Small In-Plane Magnetic Field

Unraveling the Spin Polarization of the ν = 5/2 Fractional Quantum Hall State

Complete spin phase diagram of the fractional quantum Hall liquid

Resistance Spikes and Domain Wall Loops in Ising Quantum Hall Ferromagnets

Tunable fractional quantum Hall phases in bilayer graphene

Is the Composite Fermion a Dirac Particle?

Stabilization of the Particle-Hole Pfaffian Order by Landau-Level Mixing and Impurities That Break Particle-Hole Symmetry

Landau Level Mixing and the Ground State of the ν = 5/2 Quantum Hall Effect

Nonconventional Odd-Denominator Fractional Quantum Hall States in the Second Landau Level

Observation of half-integer thermal Hall conductance

Collective states of non-Abelian quasiparticles in a magnetic field

Non-Abelian fractional quantum Hall state at 3/7-filled Landau level

Fractional Quantum Hall Effect around ν = 3/2: Composite Fermions with a Spin

Fractional Quantum Hall Phase Transitions and Four-Flux States in Graphene

Fractional quantum Hall effect in graphene: Quantitative comparison between theory and experiment

2DCC-MIP) under NSF cooperative agreement DMR-1539916. This work utilized resources provided by the NSF-MRSEC-sponsored Materials Characterization Lab at Penn State. X. L acknowledges the support of Beijing Natural Science Foundation (Grant No. JQ18002) and the National Key Research and Development Program of China

acknowledge support from the Elemental Strategy Initiative conducted by the MEXT

We thank Jainendra Jain, Ajit C. Balram 

The authors declare no competing financial interests.

Data shown in this paper are available at