Mapping the braiding properties of the Moore-Read state Emil Prodan 1 and Frederick Duncan M. Haldane 2 $^{1}$Department of Physics, Yeshiva University, New York, NY 10016 ∗ and $^{2}$Department of Physics, Princeton University, Princeton, NJ 08544 In this paper we explore the braiding properties of the Moore-Read fractional Hall sequence, which amounts to computing the adiabatic evolution of the Hall liquid when the anyons are moved along various trajectories. In this work, the anyons are pinned to precise spatial configurations by using specific external potentials. Such external potentials break the translational symmetry and it appears that one will be forced to simulate the braidings on the entire many-body Hilbert space, an absolutely prohibitive scenario. We demonstrate how to overcome this difficulty and obtain the exact braidings for fairly large Hall systems. For this, we show that the incompressible state of a general ( k,m ) fractional Hall sequence can be viewed as the unique zero mode of a specific Hamiltonian H ( k,m $^{)}$, whose form is explicitly derived by using k-particles creation operators. The compressible Hall states corresponding to n × k anyons fixed at w$_{1}$ ,. . . , w$_{nk}$ are shown to be the zero modes of a pinning Hamiltonian H ( k,m ) w$_{1}$$_{,...,w$_{nk}$}$, which is also explicitly derived. The zero modes of H ( k,m ) w$_{1}$,...,w$_{nk}$ are shown to be contained in the space of the zero modes of H ( k,m $^{)}$. Therefore, the computation of the braidings can be done entirely within this space, which we map out for a number of Hall systems. Using this efficient computational method, we study various properties of the Moore-Read states. In particular, we give direct confirmation of their topological and non-abelian properties that were previously implied from the underlying Conformal Field Theory (CFT) structure of the Moore-Read state. PACS numbers: 71.10.-w, 71.15.-m measures the twist of the zero modes space during the adiabatic braidings. - a direct proof that the adiabatic curvature is strongly localized near the fixed anyons, thus confirming the topo- logical properties of adiabatic braids. - a direct proof that the monodromy corresponding to a braiding in which one anyon loops around another anyon is in perfect agreement with the CFT prediction. - a direct proof that the zero modes space splits ac- cording to the fusion rules of the underling Conformal Field Theory structure of the Moore-Read state. - a direct proof that the conformal blocks can be dis- tinguished by bringing two anyons together and by mea- suring the electron density for the fused anyons. The recently discovered Fractional Hall states at filling factors ν = 5 / 2, 1 and ν =12/5, 2 are thought to contain anyons obeying Non-Abelian statistics. It was argued 3 that these states are part of the Moore-Read sequence, 4 and of the Read-Rezayi sequence at k =3, 5 respectively. Both states received sustained attention from the the- oretical and experimental condensed matter community because they can provide the key to the physical real- ization of a Topological Quantum Computer (TQC). For this reason, there is a concentrated effort on both theo- retical and experimental fronts for finding direct confir- mations of the Non-Abelian and Topological properties of these Fractional Hall states. 6–14 In spite of an overwhelming indirect evidence, the non- Abelian statistics for the Moore-Read sequence was not considered resolved 20 until the relatively recent numer- ical confirmation of Ref. 21, where Monte-Carlo tech- niques were used to compute a small number of adiabatic mononodromies. The present paper reports the results of an exact diagonalization study, which allows us to take a direct and unprecedented look into the non-Abelian and topological properties of the Moore-Read sequence. We have results for systems containing 2 and 4 anyons. A model Hamiltonian (reported here for the first time) is used to pin down the anyons at specified locations and to move them adiabatically along different paths. Among our results, the reader will find the following: Let us briefly mention the relevance of our study for for TQC. The logic gates in TQC algorithms can be gen- erated by sequences of braids. The specific sequences of braids can be computed using the Solovay-Kitaev algorithm, 15 once a unitary representation of the anyons’ braid group is provided. For the Fibonacci anyons, for ex- ample, the braiding sequences implementing the elemen- tary logic gates were explicitely calculated in Refs. 16 and 17. The unitary representations of the braid group can be classified by various methods. Particularly, the Ref. 18 describes a general method for generating unitary rep- resentations of the braid group using quantum groups. These representations were connected to the braidings of the anyons in Fractional Hall Liquids via the argu- ment that braiding can be computed from an effective picture in which the anyons carry an internal quantum symmetry. This symmetry can be derived from the un- derlying CFT structure of the Fractional Hall sequence. For example, the braiding of the Fibonacci anyons in - a map of the Abelian and non-Abelian adiabatic cur- vature experienced by an itinerant anyon when the rest of the anyons are kept at fixed locations. - a map of a newly introduced Twist Density, which arXiv:1001.1930v1 [cond-mat.str-el] 12 Jan 2010 I. INTRODUCTION