Graphene and its multilayers have attracted considerable interest because their fourfold spin and valley degeneracy enables a rich variety of broken-symmetry states arising from electron-electron interactions, and raises the prospect of controlled phase transitions among them. Here we report local electronic compressibility measurements of ultraclean suspended graphene that reveal a multitude of fractional quantum Hall states surrounding filling factors ν=-1/2 and -1/4. Several of these states exhibit phase transitions that indicate abrupt changes in the underlying order, and we observe many additional oscillations in compressibility as ν approaches -1/2, suggesting further changes in spin and/or valley polarization. We use a simple model based on crossing Landau levels of composite fermions with different internal degrees of freedom to explain many qualitative features of the experimental data. Our results add to the diverse array of many-body states observed in graphene and demonstrate substantial control over their order parameters.

The nature of fractional quantum Hall (FQH) states is determined by the interplay between the Coulomb interaction and the symmetries of the system. The unique combination of spin, valley, and orbital degeneracies in bilayer graphene is predicted to produce novel and tunable FQH ground states. Here we present local electronic compressibility measurements of the FQH effect in the lowest Landau level of bilayer graphene. We observe incompressible FQH states at filling factors ν = 2p + 2/3 with hints of additional states appearing at ν = 2p + 3/5, where p = −2, −1, 0, and 1. This sequence of states breaks particle-hole symmetry and instead obeys a ν → ν + 2 symmetry, which highlights the importance of the orbital degeneracy for many-body states in bilayer graphene.One Sentence Summary: Bilayer graphene exhibits a unique sequence of electron-hole asymmetric fractional quantum Hall states because of its twofold orbital degeneracy. The charge carriers in bilayer graphene obey an electron-hole symmetric dispersion at zero magnetic field. Application of a perpendicular magnetic field B breaks this dispersion into energy bands known as Landau levels (LLs). In addition to the standard spin and valley degeneracy found in monolayer graphene, the N = 0 and N = 1 orbital states in bilayer graphene are also degenerate and occur at zero energy (1). This results in a sequence of single-particle quantum Hall states at ν = 4M e 2 /h, where M is a nonzero integer (2).When the disorder is sufficiently low, the eightfold degeneracy of the lowest LL is lifted by electron-electron interactions, which results in quantum Hall states at all integer filling factors (3,4). The nature of these broken-symmetry states has been studied extensively both experimentally and theoretically, with particular attention given to the insulating ν = 0 state. Multiple groups have been able to induce transitions between different spin and valley orders of the ground states using external electric and magnetic fields, which indicates the extensive tunability of many-body states in bilayer graphene (6-9). The interplay between externally applied fields and intrinsic electron-electron interactions, both of which break the degeneracies of bilayer graphene, produces a rich phase diagram not found in any other system.Knowledge of the ground state at integer filling factors is especially important for investigating the physics of partially filled LLs, where in exceptionally clean samples, the charge carriers condense into fractional quantum Hall (FQH) states. The above-mentioned degrees of freedom as well as the strong screening of the Coulomb interaction in bilayer graphene are expected to result in an interesting sequence of FQH states in the lowest LL (10-15).Indeed, partial breaking of the SU(4) symmetry in monolayer graphene has already resulted in sequences of FQH states with multiple missing fractions (17-22).Experimental observation of FQH states, however, has proven to be difficult in bilayer 2 graphene. Hints of a ν = 1/3 state were first report...

An indirect electronic transition from the L point valence band to the T point valence band has been previously observed in Bi nanowires oriented along the ͓0112͔ crystalline direction ͑used by Black et al. and by Reppert et al.͒ but not in ͓1120͔-oriented nanowires ͑used by Cornelius et al.͒ or in bulk bismuth. Here we measure the Bi nanowire samples from each of these prior experimental studies on the same Fourier transform infrared apparatus, confirming that the differences are indeed physical and are not associated with the experimental setup. We develop an analytical model for the threshold energy of the indirect L to T point valence-band transition that takes as parameters the nanowire diameter and crystalline orientation. Our model shows good agreement with experimental results, and demonstrates that the nonparabolic nature of the L point bands is essential for calculating the energy of this transition. Finally, we propose a mechanism based on symmetry lowering to explain why this indirect transition is observed for ͓0112͔-oriented but not for ͓1120͔-oriented nanowires.

In the preceding Comment, the authors raise several questions regarding the properties of bismuth nanowire samples studied in the experimental section of Phys. Rev. B 79, 165117 ͑2009͒ and the interpretation of the resulting infrared spectra. We address their concerns by discussing each sample in detail and providing additional relevant information.