27,405 research outputs found

    Flip Distance to some Plane Configurations

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    We study an old geometric optimization problem in the plane. Given a perfect matching M on a set of n points in the plane, we can transform it to a non-crossing perfect matching by a finite sequence of flip operations. The flip operation removes two crossing edges from M and adds two non-crossing edges. Let f(M) and F(M) denote the minimum and maximum lengths of a flip sequence on M, respectively. It has been proved by Bonnet and Miltzow (2016) that f(M)=O(n^2) and by van Leeuwen and Schoone (1980) that F(M)=O(n^3). We prove that f(M)=O(n Delta) where Delta is the spread of the point set, which is defined as the ratio between the longest and the shortest pairwise distances. This improves the previous bound for point sets with sublinear spread. For a matching M on n points in convex position we prove that f(M)=n/2-1 and F(M)={{n/2} choose 2}; these bounds are tight. Any bound on F(*) carries over to the bichromatic setting, while this is not necessarily true for f(*). Let M\u27 be a bichromatic matching. The best known upper bound for f(M\u27) is the same as for F(M\u27), which is essentially O(n^3). We prove that f(M\u27)<=slant n-2 for points in convex position, and f(M\u27)= O(n^2) for semi-collinear points. The flip operation can also be defined on spanning trees. For a spanning tree T on a convex point set we show that f(T)=O(n log n)

    Phase diagram of an extended quantum dimer model on the hexagonal lattice

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    We introduce a quantum dimer model on the hexagonal lattice that, in addition to the standard three-dimer kinetic and potential terms, includes a competing potential part counting dimer-free hexagons. The zero-temperature phase diagram is studied by means of quantum Monte Carlo simulations, supplemented by variational arguments. It reveals some new crystalline phases and a cascade of transitions with rapidly changing flux (tilt in the height language). We analyze perturbatively the vicinity of the Rokhsar-Kivelson point, showing that this model has the microscopic ingredients needed for the "devil's staircase" scenario [E. Fradkin et al., Phys. Rev. B 69, 224415 (2004)], and is therefore expected to produce fractal variations of the ground-state flux.Comment: Published version. 5 pages + 8 (Supplemental Material), 31 references, 10 color figure

    Atomistic simulations of kinks in 1/2a<111> screw dislocations in bcc tantalum

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    Two types of equilibrium core structures (denoted symmetric and asymmetric) for 1/2a screw dislocations in bcc metals have been found in atomistic simulations. In asymmetric (or polarized) cores, the central three atoms simultaneously translate along the Burgers vector direction. This collective displacement of core atoms is called polarization. In contrast, symmetric (nonpolarized) cores have zero core polarization. To examine the possible role of dislocation core in kink-pair formation process, we studied the multiplicity, structural features, and formation energies of 1/3a kinks in 1/2a screw dislocations with different core structures. To do this we used a family of embedded atom model potentials for tantalum (Ta) all of which reproduce bulk properties (density, cohesive energy, and elastic constants) from quantum mechanics calculations but differ in the resulting polarization of 1/2a screw dislocations. For dislocations with asymmetric core, there are two energy equivalent core configurations [with positive (P) and negative (N) polarization], leading to 2 types of (polarization) flips, 8 kinds of isolated kinks, and 16 combinations of kink pairs. We find there are only two elementary kinks, while the others are composites of elementary kinks and flips. In contrast, for screw dislocations with symmetric core, there are only two types of isolated kinks and one kind of kink pair. We find that the equilibrium dislocation core structure of 1/2a screw dislocations is an important factor in determining the kink-pair formation energy

    Flip dynamics in octagonal rhombus tiling sets

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    We investigate the properties of classical single flip dynamics in sets of two-dimensional random rhombus tilings. Single flips are local moves involving 3 tiles which sample the tiling sets {\em via} Monte Carlo Markov chains. We determine the ergodic times of these dynamical systems (at infinite temperature): they grow with the system size NTN_T like Cst.NT2lnNTCst. N_T^2 \ln N_T; these dynamics are rapidly mixing. We use an inherent symmetry of tiling sets and a powerful tool from probability theory, the coupling technique. We also point out the interesting occurrence of Gumbel distributions.Comment: 5 Revtex pages, 4 figures; definitive versio

    Geometry fluctuations and Casimir effect in a quantum antiferromagnet

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    We show the presence of a Casimir type force between domain walls in a two dimensional Heisenberg antiferromagnet subject to geometrical fluctuations. The type of fluctuations that we consider, called phason flips, are well known in quasicrystals, but less so in periodic structures. As the classical ground state energy of the antiferromagnet is unaffected by this type of fluctuation, energy changes are purely of quantum origin. We calculate the effective interaction between two parallel domain walls, defining a slab of thickness d, in such an antiferromagnet within linear spin wave theory. The interaction is anisotropic, and for a particular orientation of the slab we find that it decays as 1/d, thus, more slowly than the electromagnetic Casimir effect in the same geometry.Comment: 5 pages, 5 figures, minor modifications, accepted for publication in EPJ
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