Topological ordering

Bonchev, D. (1995). Topological Order in Molecules. 1. Molecular Branching Revisited. J.Mol. 

Figure 6. Left pane Location of the discrete degrees of freedom responsible for the dynamics of the Josephson junction array shown in Fig. 5. The spin degrees of freedom describing the state of the elementary rhombi are located on the bonds of the triangular lattice (shown in thick lines) while the constraints are defined on the sites of this lattice. The dashed line indicates the boundary condition imposed by a physical circuitry shown in Fig. 5 a. Contours 7 and 7 are used in the construction of topological order parameter and excitations. Right pane The lattice with K = 3 openings, the ground state of Josephson junction array on this lattice is 2fc = 8 fold degenerate.

Two-dimensional systems possess a unique topological ordering - not found in either three- or one-dimensional systems. 

The contribution from electrons at the origin of the ID is, for most of the simulated topologies, order of magnitudes less than the contribution coming from inside the ID. For (mj, Hi -7) and topologies, the k = 0 contribution is... 

Bonchey D. (1995) Topological order in molecules. 1. Molecular branching reyisited. J. Mol. Struct. (Thcochem), 336, 137-156. 

Davis summarized the concepts about HLB, PIT, and Windsor s ternary phase diagrams for the case of microemulsions and reported topologically ordered models connected with the Helfrich membrane bending energy. Because the curvature of surfactant lamellas plays a major role in determining the patterns of phase behavior in microemulsions, it is important to reveal how the optimal microemulsion state is affected by the surface forces determining the curvature... 

Local topological order in 2D systems can be characterized by the fraction of c-coordinated particles, /. Figures 27-29 show for c = 4-8 as a function of p for the time-averaged 3584-particle WCA system, /g, the fraction of six-coordinated particles, serves as a sort of order parameter, being nearly unity in the time-averaged WCA solid (/g = 0.9961 ... 

In [6.41-43] we find reports of the experimental investigations of partial distribution functions of atoms in amorphous alloys. The strong LO is discovered in the alloys of the metal-metalloid and metal-metal types. The compositional order is closely (but not always unambiguously) connected with the topological order, so that the presence of the former testifies to the existence of the definite local topological order. 

This remarkable topological order of the FIEM is not only theoretically appealing, but also didactically useful, because this order reveals well-defined universal logical structures in the immense wealth of individual facts. 

Lewenstein, M., Polar molecules in topological order. Nature Phys., 2, 309, 2006. 

J. Haines, C. Levelul, A. Isambert, P. H ert, S. Koharad, D. Keen, T. Hammouda, and D. Andrault, Topologically ordered amorphous silica obtained from the collapsed siliceous zeolite, silicalite-1-F a step toward perfect glasses, J. Am. Chem. Soc. 131, 12333-12338 (2009). 

Note that in Figure C.3 there is only one partial profile S (Q) since, in pure silica glass, there is only one type of topological order of the constituting Si04 building blocks. 

Relatively, efficient microemulsion formulations, typically those with less than 20 wt% surfactant, differ significantly from non-ideal solutions in that they contain a distinct type of microstructure, i.e. topologically ordered oil and water domains coated by surfactant. The microstructures within microemulsions typically range from 3 to 100 nm in size, and fluctuate rapidly in time. As the oil-to-water ratio in microemulsion phases is increased, many experimental studies indicate that a continuous progression of microstructures is observed. At low concentrations of oil, droplets of oil, coated with surfactant, swim in a continuous water domain (an oil-in-water microemulsion). As the concentration of oil is increased, the spherical droplets form oblong and globular structures. At intermediate ratios of oil to water (near 50/50), a so-called bicontinuous structure of oil and water domains is observed (12). As a... 

Since the elastic constants are calculated using measurements of density and acoustic wave velocity, the individual contributions of MCN dependence of these quantities on these elastic transitions should be considered. As evident from the previous section, the density itself changes with the MCN in a manner that reflects the topological order in Ge-As-Se ternary glasses. However, the fluctuation of the density is less than 2% compared with 50%... 

Both theoretical and experimental approaches have been developed to estimate and measure the composition dependence of the optical bandgap of glasses. Lucovsky found local extrema of the average bandgap for the chemically stoichiometric As2Se3 and GeSe2 compositions, but their relation to the topological order was not explored [102]. 

This gravitational topological adiabatic quantum algorithm, constitutes an example of a quantum adiabatic speedup that relies on the topological order that it is inherent to the space-time in loop quantum gravity. 

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