215 research outputs found
Resonating singlet valence plaquettes
We consider the simplest generalizations of the valence bond physics of SU(2)
singlets to SU(N) singlets that comprise objects with N sites -- these are
SU(N) singlet plaquettes with N=3 and N=4 in three spatial dimensions.
Specifically, we search for a quantum mechanical liquid of such objects -- a
resonating singlet valence plaquette phase that generalizes the celebrated
resonating valence bond phase for SU(2) spins. We extend the Rokhsar-Kivelson
construction of the quantum dimer model to the simplest SU(4) model for valence
plaquette dynamics on a cubic lattice. The phase diagram of the resulting
quantum plaquette model is analyzed both analytically and numerically. We find
that the ground state is solid everywhere, including at the Rokhsar-Kivelson
point where the ground state is an equal amplitude sum. By contrast, the equal
amplitude sum of SU(3) singlet triangular plaquettes on the face centered cubic
lattice is liquid and thus a candidate for describing a resonating single
valence plaquette phase, given a suitably defined local Hamiltonian.Comment: 12 pages, 15 figures, minor changes, references added, Phys Rev B
versio
Gray Codes and Symmetric Chains
We consider the problem of constructing a cyclic listing of all bitstrings of length 2n+1 with Hamming weights in the interval [n+1-l,n+l], where 1 = 12
On the central levels problem
The \emph{central levels problem} asserts that the subgraph of the -dimensional hypercube induced by all bitstrings with at least many 1s and at most many 1s, i.e., the vertices in the middle levels, has a Hamilton cycle for any and .
This problem was raised independently by Buck and Wiedemann, Savage, Gregor and {\v{S}}krekovski, and by Shen and Williams, and it is a common generalization of the well-known \emph{middle levels problem}, namely the case , and classical binary Gray codes, namely the case .
In this paper we present a general constructive solution of the central levels problem.
Our results also imply the existence of optimal cycles through any sequence of consecutive levels in the -dimensional hypercube for any and .
Moreover, extending an earlier construction by Streib and Trotter, we construct a Hamilton cycle through the -dimensional hypercube, , that contains the symmetric chain decomposition constructed by Greene and Kleitman in the 1970s, and we provide a loopless algorithm for computing the corresponding Gray code
Fastest mixing Markov chain on graphs with symmetries
We show how to exploit symmetries of a graph to efficiently compute the
fastest mixing Markov chain on the graph (i.e., find the transition
probabilities on the edges to minimize the second-largest eigenvalue modulus of
the transition probability matrix). Exploiting symmetry can lead to significant
reduction in both the number of variables and the size of matrices in the
corresponding semidefinite program, thus enable numerical solution of
large-scale instances that are otherwise computationally infeasible. We obtain
analytic or semi-analytic results for particular classes of graphs, such as
edge-transitive and distance-transitive graphs. We describe two general
approaches for symmetry exploitation, based on orbit theory and
block-diagonalization, respectively. We also establish the connection between
these two approaches.Comment: 39 pages, 15 figure
Gray codes and symmetric chains
We consider the problem of constructing a cyclic listing of all bitstrings of length~ with Hamming weights in the interval , where , by flipping a single bit in each step.
This is a far-ranging generalization of the well-known middle two levels problem (the case~).
We provide a solution for the case~ and solve a relaxed version of the problem for general values of~, by constructing cycle factors for those instances.
Our proof uses symmetric chain decompositions of the hypercube, a concept known from the theory of posets, and we present several new constructions of such decompositions.
In particular, we construct four pairwise edge-disjoint symmetric chain decompositions of the -dimensional hypercube for any~
On 1-factorizations of Bipartite Kneser Graphs
It is a challenging open problem to construct an explicit 1-factorization of
the bipartite Kneser graph , which contains as vertices all -element
and -element subsets of and an edge between any
two vertices when one is a subset of the other. In this paper, we propose a new
framework for designing such 1-factorizations, by which we solve a nontrivial
case where and is an odd prime power. We also revisit two classic
constructions for the case --- the \emph{lexical factorization} and
\emph{modular factorization}. We provide their simplified definitions and study
their inner structures. As a result, an optimal algorithm is designed for
computing the lexical factorizations. (An analogous algorithm for the modular
factorization is trivial.)Comment: We design the first explicit 1-factorization of H(2,q), where q is a
odd prime powe
Parametric binary dissection
Binary dissection is widely used to partition non-uniform domains over parallel computers. This algorithm does not consider the perimeter, surface area, or aspect ratio of the regions being generated and can yield decompositions that have poor communication to computation ratio. Parametric Binary Dissection (PBD) is a new algorithm in which each cut is chosen to minimize load + lambda x(shape). In a 2 (or 3) dimensional problem, load is the amount of computation to be performed in a subregion and shape could refer to the perimeter (respectively surface) of that subregion. Shape is a measure of communication overhead and the parameter permits us to trade off load imbalance against communication overhead. When A is zero, the algorithm reduces to plain binary dissection. This algorithm can be used to partition graphs embedded in 2 or 3-d. Load is the number of nodes in a subregion, shape the number of edges that leave that subregion, and lambda the ratio of time to communicate over an edge to the time to compute at a node. An algorithm is presented that finds the depth d parametric dissection of an embedded graph with n vertices and e edges in O(max(n log n, de)) time, which is an improvement over the O(dn log n) time of plain binary dissection. Parallel versions of this algorithm are also presented; the best of these requires O((n/p) log(sup 3)p) time on a p processor hypercube, assuming graphs of bounded degree. How PBD is applied to 3-d unstructured meshes and yields partitions that are better than those obtained by plain dissection is described. Its application to the color image quantization problem is also discussed, in which samples in a high-resolution color space are mapped onto a lower resolution space in a way that minimizes the color error
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