We present a comprehensive numerical study of a microscopic model of the fractional quantum Hall system at filling fraction =5/ 2, based on the disk geometry. Our model includes Coulomb interaction and a semirealistic confining potential. We also mix in a three-body interaction in some cases to help elucidate the physics. We obtain a phase diagram, discuss the conditions under which the ground state can be described by the Moore-Read state, and study its competition with neighboring stripe phases. We also study quasihole excitations and edge excitations in the Moore-Read-like state. From the evolution of the edge spectrum, we obtain the velocities of the charge and neutral edge modes, which turn out to be very different. This separation of velocities is a source of decoherence for a non-Abelian quasihole and/or quasiparticle ͑with charge Ϯe / 4͒ when propagating at the edge; using numbers obtained from a specific set of parameters, we estimate the decoherence length to be around 4 m. This sets an upper bound for the separation of the two point contacts in a double point-contact interferometer, designed to detect the non-Abelian nature of such quasiparticles. We also find a state that is a potential candidate for the recently proposed anti-Pfaffian state. We find the speculated anti-Pfaffian state is favored in weak confinement ͑smooth edge͒, while the Moore-Read Pfaffian state is favored in strong confinement ͑sharp edge͒.

We present comprehensive results on the edge-mode velocities in a quantum Hall droplet with realistic interaction and confinement at various filling fractions. We demonstrate that the charge-mode velocity scales roughly with the valence Landau level filling fraction and the Coulomb energy in the corresponding Landau level. At Landau level filling fraction =5/ 2, the stark difference between the bosonic charge-mode velocity and the fermionic neutral-mode velocity can manifest itself in the thermal smearing of the non-Abelian quasiparticle interference. We estimate the dependence of the coherence temperature on the confining potential strength, which may be tunable experimentally to enhance the non-Abelian state.

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 ...