1,998 research outputs found
A short introduction to Fibonacci anyon models
We discuss how to construct models of interacting anyons by generalizing
quantum spin Hamiltonians to anyonic degrees of freedom. The simplest
interactions energetically favor pairs of anyons to fuse into the trivial
("identity") channel, similar to the quantum Heisenberg model favoring pairs of
spins to form spin singlets. We present an introduction to the theory of anyons
and discuss in detail how basis sets and matrix representations of the
interaction terms can be obtained, using non-Abelian Fibonacci anyons as
example. Besides discussing the "golden chain", a one-dimensional system of
anyons with nearest neighbor interactions, we also present the derivation of
more complicated interaction terms, such as three-anyon interactions in the
spirit of the Majumdar-Ghosh spin chain, longer range interactions and two-leg
ladders. We also discuss generalizations to anyons with general non-Abelian
su(2)_k statistics. The k to infinity limit of the latter yields ordinary SU(2)
spin chains
Why some heaps support constant-amortized-time decrease-key operations, and others do not
A lower bound is presented which shows that a class of heap algorithms in the
pointer model with only heap pointers must spend Omega(log log n / log log log
n) amortized time on the decrease-key operation (given O(log n) amortized-time
extract-min). Intuitively, this bound shows the key to having O(1)-time
decrease-key is the ability to sort O(log n) items in O(log n) time; Fibonacci
heaps [M.L. Fredman and R. E. Tarjan. J. ACM 34(3):596-615 (1987)] do this
through the use of bucket sort. Our lower bound also holds no matter how much
data is augmented; this is in contrast to the lower bound of Fredman [J. ACM
46(4):473-501 (1999)] who showed a tradeoff between the number of augmented
bits and the amortized cost of decrease-key. A new heap data structure, the
sort heap, is presented. This heap is a simplification of the heap of Elmasry
[SODA 2009: 471-476] and shares with it a O(log log n) amortized-time
decrease-key, but with a straightforward implementation such that our lower
bound holds. Thus a natural model is presented for a pointer-based heap such
that the amortized runtime of a self-adjusting structure and amortized lower
asymptotic bounds for decrease-key differ by but a O(log log log n) factor
Heisenberg antiferromagnet on Cayley trees: low-energy spectrum and even/odd site imbalance
To understand the role of local sublattice imbalance in low-energy spectra of
s=1/2 quantum antiferromagnets, we study the s=1/2 quantum nearest neighbor
Heisenberg antiferromagnet on the coordination 3 Cayley tree. We perform
many-body calculations using an implementation of the density matrix
renormalization group (DMRG) technique for generic tree graphs. We discover
that the bond-centered Cayley tree has a quasidegenerate set of a low-lying
tower of states and an "anomalous" singlet-triplet finite-size gap scaling. For
understanding the construction of the first excited state from the many-body
ground state, we consider a wave function ansatz given by the single-mode
approximation, which yields a high overlap with the DMRG wave function.
Observing the ground-state entanglement spectrum leads us to a picture of the
low-energy degrees of freedom being "giant spins" arising out of sublattice
imbalance, which helps us analytically understand the scaling of the
finite-size spin gap. The Schwinger-boson mean-field theory has been
generalized to nonuniform lattices, and ground states have been found which are
spatially inhomogeneous in the mean-field parameters.Comment: 19 pages, 12 figures, 6 tables. Changes made to manuscript after
referee suggestions: parts reorganized, clarified discussion on Fibonacci
tree, typos correcte
Task-based Augmented Contour Trees with Fibonacci Heaps
This paper presents a new algorithm for the fast, shared memory, multi-core
computation of augmented contour trees on triangulations. In contrast to most
existing parallel algorithms our technique computes augmented trees, enabling
the full extent of contour tree based applications including data segmentation.
Our approach completely revisits the traditional, sequential contour tree
algorithm to re-formulate all the steps of the computation as a set of
independent local tasks. This includes a new computation procedure based on
Fibonacci heaps for the join and split trees, two intermediate data structures
used to compute the contour tree, whose constructions are efficiently carried
out concurrently thanks to the dynamic scheduling of task parallelism. We also
introduce a new parallel algorithm for the combination of these two trees into
the output global contour tree. Overall, this results in superior time
performance in practice, both in sequential and in parallel thanks to the
OpenMP task runtime. We report performance numbers that compare our approach to
reference sequential and multi-threaded implementations for the computation of
augmented merge and contour trees. These experiments demonstrate the run-time
efficiency of our approach and its scalability on common workstations. We
demonstrate the utility of our approach in data segmentation applications
Hyperbolic tilings and formal language theory
In this paper, we try to give the appropriate class of languages to which
belong various objects associated with tessellations in the hyperbolic plane.Comment: In Proceedings MCU 2013, arXiv:1309.104
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