Restricted and Unrestricted Coverings of Complete Bipartite Graphs with Hexagons

A minimal covering of a graph G with isomorphic copies of graph H is a set {H1, H2, H3, ... , Hn} where Hi is isomorphic to H, the vertex set of Hi is a subset of G, the edge set of G is a subset of the union of Hi\u27s, and the cardinality of the union of Hi\u27s minus G is minimum. Some studies have been made of covering the complete graph in which case an added condition of the edge set of Hi is the subset of the edge set of G for all i which implies no additional restrictions. However, if G is not the complete graph, then this condition may have implications. We will give necessary and sufficient conditions for minimal coverings of complete bipartite graph with 6-cycles, which we call minimal unrestricted coverings. We also give necessary and sufficient conditions for minimal coverings of the complete bipartite graph with 6-cycles with the added condition the edge set of Hi is a subset of G for all i, and call these minimal restricted coverings

Enumeration of Matchings: Problems and Progress

This document is built around a list of thirty-two problems in enumeration of matchings, the first twenty of which were presented in a lecture at MSRI in the fall of 1996. I begin with a capsule history of the topic of enumeration of matchings. The twenty original problems, with commentary, comprise the bulk of the article. I give an account of the progress that has been made on these problems as of this writing, and include pointers to both the printed and on-line literature; roughly half of the original twenty problems were solved by participants in the MSRI Workshop on Combinatorics, their students, and others, between 1996 and 1999. The article concludes with a dozen new open problems. (Note: This article supersedes math.CO/9801060 and math.CO/9801061.)Comment: 1+37 pages; to appear in "New Perspectives in Geometric Combinatorics" (ed. by Billera, Bjorner, Green, Simeon, and Stanley), Mathematical Science Research Institute publication #37, Cambridge University Press, 199

Quantum dimer model with Z_2 liquid ground-state: interpolation between cylinder and disk topologies and toy model for a topological quantum-bit

We consider a quantum dimer model (QDM) on the kagome lattice which was introduced recently [Phys. Rev. Lett. 89, 137202 (2002)]. It realizes a Z_2 liquid phase and its spectrum was obtained exactly. It displays a topological degeneracy when the lattice has a non-trivial geometry (cylinder, torus, etc). We discuss and solve two extensions of the model where perturbations along lines are introduced: first the introduction of a potential energy term repelling (or attracting) the dimers along a line is added, second a perturbation allowing to create, move or destroy monomers. For each of these perturbations we show that there exists a critical value above which, in the thermodynamic limit, the degeneracy of the ground-state is lifted from 2 (on a cylinder) to 1. In both cases the exact value of the gap between the first two levels is obtained by a mapping to an Ising chain in transverse field. This model provides an example of solvable Hamiltonian for a topological quantum bit where the two perturbations act as a diagonal and a transverse operator in the two-dimensional subspace. We discuss how crossing the transitions may be used in the manipulation of the quantum bit to optimize simultaneously the frequency of operation and the losses due to decoherence.Comment: 11 pages, 7 (.eps) figures. Improved discussion of the destruction of the topological degeneracy and other minor corrections. Version to appear in Phys. Rev.

Decompositions, Packings, and Coverings of Complete Directed Gaphs with a 3-Circuit and a Pendent Arc.

In the study of Graph theory, there are eight orientations of the complete graph on three vertices with a pendant edge, K3 ∪ {e}. Two of these are the 3-circuit with a pendant arc and the other six are transitive triples with a pendant arc. Necessary and sufficient conditions are given for decompositions, packings, and coverings of the complete digraph with the two 3-circuit with a pendant arc orientations

Competing states in the SU(3) Heisenberg model on the honeycomb lattice: Plaquette valence-bond crystal versus dimerized color-ordered state

Conflicting predictions have been made for the ground state of the SU(3) Heisenberg model on the honeycomb lattice: Tensor network simulations found a plaquette order [Zhao et al, Phys. Rev. B 85, 134416 (2012)], where singlets are formed on hexagons, while linear flavor-wave theory (LFWT) suggested a dimerized, color ordered state [Lee and Yang, Phys. Rev. B 85, 100402 (2012)]. In this work we show that the former state is the true ground state by a systematic study with infinite projected-entangled pair states (iPEPS), for which the accuracy can be systematically controlled by the so-called bond dimension $D$. Both competing states can be reproduced with iPEPS by using different unit cell sizes. For small $D$ the dimer state has a lower variational energy than the plaquette state, however, for large $D$ it is the latter which becomes energetically favorable. The plaquette formation is also confirmed by exact diagonalizations and variational Monte Carlo studies, according to which both the dimerized and plaquette states are non-chiral flux states.Comment: 11 pages, 12 figures, small changes, added more reference

Packings and Coverings of Complete Graphs with a Hole with the 4-Cycle with a Pendant Edge

In this thesis, we consider packings and coverings of various complete graphs with the 4-cycle with a pendant edge. We consider both restricted and unrestricted coverings. Necessary and sufficient conditions are given for such structures for (1) complete graphs Kv, (2) complete bipartite graphs Km,n, and (3) complete graphs with a hole K(v,w)

Strongly correlated fermions on a kagome lattice

We study a model of strongly correlated spinless fermions on a kagome lattice at 1/3 filling, with interactions described by an extended Hubbard Hamiltonian. An effective Hamiltonian in the desired strong correlation regime is derived, from which the spectral functions are calculated by means of exact diagonalization techniques. We present our numerical results with a view to discussion of possible signatures of confinement/deconfinement of fractional charges.Comment: 10 pages, 10 figure

Geometrically frustrated quantum magnets

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2004.Includes bibliographical references (p. 215-219).(cont.) more general lessons on frustrated quantum magnetism. At the end, we demonstrate some new mathematical tools on two other frustrated two-dimensional systems, and summarize our conclusions, with an outlook to remaining open problems.In this thesis we attempt to reach a physical understanding and theoretical description of some of the greatest challenges in the field of frustrated quantum magnetism, mainly the Kagome lattice antiferromagnets. After an introductory review of concepts, we closely examine the Kagome lattice quantum Heisenberg and Ising models. We apply several new techniques based on lattice gauge theories, duality mappings and field theory in order to explore phase diagrams of these models. Our approach provides a microscopic picture of the mysterious phenomena observed numerically and experimentally in the Kagome Heisenberg antiferromagnets. Namely, we argue that the spinless excitations, thought to be gapless in absence of any symmetry breaking in this system, are in fact gapped, but at an extremely small emergent energy scale. This scenario is realized in an unconventional valence-bond ordered phase, with a very large unit-cell and complex structure. We also discuss properties of a spin liquid that could be realized in the Kagome antiferromagnet, and argue that its elementary excitations are clearly gapped and extremely massive or even localized. We demonstrate that the Kagome lattice quantum Ising models are an excellent platform for learning about effects of quantum fluctuations on classically degenerate ground-states. We consider several ways in which spins can acquire quantum dynamics, including transverse field, XXZ exchange and ring-exchange perturbations. Using two different setups of compact U(1) gauge theory we find circumstances in which many characteristic quantum phases occur: disordered phase, topologically ordered spin liquid, valence-bond crystal, and a phase with coexistence of magnetic and valence-bond order. From this variety of results we attemptby Predrag NikoliÄ.Ph.D