4,391 research outputs found
Generalized Fibonacci cubes
AbstractGeneralized Fibonacci cube Qd(f) is introduced as the graph obtained from the d-cube Qd by removing all vertices that contain a given binary string f as a substring. In this notation, the Fibonacci cube Γd is Qd(11). The question whether Qd(f) is an isometric subgraph of Qd is studied. Embeddable and non-embeddable infinite series are given. The question is completely solved for strings f of length at most five and for strings consisting of at most three blocks. Several properties of the generalized Fibonacci cubes are deduced. Fibonacci cubes are, besides the trivial cases Qd(10) and Qd(01), the only generalized Fibonacci cubes that are median closed subgraphs of the corresponding hypercubes. For admissible strings f, the f-dimension of a graph is introduced. Several problems and conjectures are also listed
Maximal hypercubes in Fibonacci and Lucas cubes
The Fibonacci cube is the subgraph of the hypercube induced by the
binary strings that contain no two consecutive 1's. The Lucas cube
is obtained from by removing vertices that start and end with 1. We
characterize maximal induced hypercubes in and and
deduce for any the number of maximal -dimensional hypercubes in
these graphs
Infinite Randomness Phases and Entanglement Entropy of the Disordered Golden Chain
Topological insulators supporting non-abelian anyonic excitations are at the
center of attention as candidates for topological quantum computation. In this
paper, we analyze the ground-state properties of disordered non-abelian anyonic
chains. The resemblance of fusion rules of non-abelian anyons and real space
decimation strongly suggests that disordered chains of such anyons generically
exhibit infinite-randomness phases. Concentrating on the disordered golden
chain model with nearest-neighbor coupling, we show that Fibonacci anyons with
the fusion rule exhibit two
infinite-randomness phases: a random-singlet phase when all bonds prefer the
trivial fusion channel, and a mixed phase which occurs whenever a finite
density of bonds prefers the fusion channel. Real space RG analysis
shows that the random-singlet fixed point is unstable to the mixed fixed point.
By analyzing the entanglement entropy of the mixed phase, we find its effective
central charge, and find that it increases along the RG flow from the random
singlet point, thus ruling out a c-theorem for the effective central charge.Comment: 16 page
Strong-disorder renormalization for interacting non-Abelian anyon systems in two dimensions
We consider the effect of quenched spatial disorder on systems of
interacting, pinned non-Abelian anyons as might arise in disordered Hall
samples at filling fractions \nu=5/2 or \nu=12/5. In one spatial dimension,
such disordered anyon models have previously been shown to exhibit a hierarchy
of infinite randomness phases. Here, we address systems in two spatial
dimensions and report on the behavior of Ising and Fibonacci anyons under the
numerical strong-disorder renormalization group (SDRG). In order to manage the
topology-dependent interactions generated during the flow, we introduce a
planar approximation to the SDRG treatment. We characterize this planar
approximation by studying the flow of disordered hard-core bosons and the
transverse field Ising model, where it successfully reproduces the known
infinite randomness critical point with exponent \psi ~ 0.43. Our main
conclusion for disordered anyon models in two spatial dimensions is that
systems of Ising anyons as well as systems of Fibonacci anyons do not realize
infinite randomness phases, but flow back to weaker disorder under the
numerical SDRG treatment.Comment: 12 pages, 12 figures, 1 tabl
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