The peculiar nature of electron scattering in graphene is among many exciting theoretical predictions for the physical properties of this material. To investigate electron scattering properties in a graphene plane, we have created a gate-tunable potential barrier within a single-layer graphene sheet. We report measurements of electrical transport across this structure as the tunable barrier potential is swept through a range of heights. When the barrier is sufficiently strong to form a bipolar junction (n-p-n or p-n-p) within the graphene sheet, the resistance across the barrier sharply increases. We compare these results to predictions for both diffusive and ballistic transport, as the barrier rises on a length scale comparable to the mean free path. Finally, we show how a magnetic field modifies transport across the barrier.

It was recently pointed out that topological liquid phases arising in the fractional quantum Hall effect (FQHE) are not required to be rotationally invariant, as most variational wavefunctions proposed to date have been. Instead, they possess a geometric degree of freedom corresponding to a shear deformation that acts like an intrinsic metric. We apply this idea to a system with an anisotropic band mass, as is intrinsically the case in many-valley semiconductors such as AlAs and Si, or in isotropic systems like GaAs in the presence of a tilted magnetic field, which breaks the rotational invariance. We perform exact diagonalization calculations with periodic boundary conditions (torus geometry) for various filling fractions in the lowest, first and second Landau levels. In the lowest Landau level, we demonstrate that FQHE states generally survive the breakdown of rotational invariance by moderate values of the band mass anisotropy. At 1/3 filling, we generate a variational family of Laughlin wavefunctions parametrized by the metric degree of freedom. We show that the intrinsic metric of the Laughlin state adjusts as the band mass anisotropy or the dielectric tensor are varied, while the phase remains robust. In the n = 1 Landau level, mass anisotropy drives transitions between incompressible liquids and compressible states with charge density wave ordering. In n ≥ 2 Landau levels, mass anisotropy selects and enhances stripe ordering with compatible wave vectors at partial 1/3 and 1/2 fillings.

We construct model wave functions for the collective modes of fractional quantum Hall systems. The wave functions are expressed in terms of symmetric polynomials characterized by a root partition that defines a "squeezed" basis, and show excellent agreement with exact diagonalization results for finite systems. In the long wavelength limit, we prove that the model wave functions are identical to those predicted by the single-mode approximation, leading to intriguing interpretations of the collective modes from the perspective of the ground-state guiding-center metric.

We generalize the notion of Haldane pseudopotentials to anisotropic fractional quantum Hall (FQH) systems that are physically realized, e.g., in tilted magnetic field experiments or anisotropic band structures. This formalism allows us to expand any translation-invariant interaction over a complete basis, and directly reveals the intrinsic metric of incompressible FQH fluids. We show that purely anisotropic pseudopotentials give rise to new types of bound states for small particle clusters in the infinite plane, and can be used as a diagnostic of FQH nematic order. We also demonstrate that generalized pseudopotentials quantify the anisotropic contribution to the effective interaction potential, which can be particularly large in models of fractional Chern insulators. DOI: 10.1103/PhysRevLett.118.146403 The fractional quantum Hall (FQH) system is host to a wide variety of topological phases of matter [1]. This complexity belies the deceivingly simple microscopic Hamiltonian containing only the effective Coulomb interaction projected to a single Landau level (LL) [2]. The understanding of different topological states was greatly facilitated by the concept of pseudopotentials (PPs) introduced by Haldane [3,4]. This formalism allows one to expand any rotation-invariant interaction over the complete basis of the PPs, which are projection operators onto two-particle states with a given value of relative angular momentum. Furthermore, a combination of a small number of PPs naturally defines parent Hamiltonians for some FQH model states, such as the Laughlin states [5,6]. The method has also been generalized to many-body PPs [7,8], which form the parent Hamiltonians of the non-Abelian FQH states [9,10]. In many cases, the ground state of these model Hamiltonians is believed to be adiabatically connected to the actual ground state of the experimental system. Thus, the relatively simple (and to some degree analytically tractable) model wave functions and Hamiltonians give much insight into the nature of the experimentally realized FQH states.Recently, interest in the FQH effect has been renewed due to emerging connections between topological order, geometry, and broken symmetry. An early precursor of these ideas was the realization that rotational invariance is not necessary for the FQH effect [4]. This lead to the conclusion that FQH states possess new "geometrical" degrees of freedom [11], uncovering a more complete description of their low-energy properties [12][13][14]. The notion of geometry has also inspired the construction of a more general class of Laughlin states with the nonEuclidean metric [15], which was shown to be physically relevant in situations where the band mass or dielectric tensor is anisotropic [16][17][18], or in the tilted magnetic field [19]. On the other hand, an intriguing coexistence of topological order with broken symmetry [20,21], leading to the nematic FQH effect, has also been proposed [22,23]. The nematic order is expected to arise due to spontaneous symmetry breaking, as suggested ...