Cohomology of matching rules

Quasiperiodic patterns described by polyhedral "atomic surfaces" and admitting matching rules are considered. It is shown that the cohomology ring of the continuous hull of such patterns is isomorphic to that of the complement of a torus $T^N$ to an arrangement $A$ of thickened affine tori of codimension two. Explicit computation of Betti numbers for several two-dimensional tilings and for the icosahedral Ammann-Kramer tiling confirms in most cases the results obtained previously by different methods. The cohomology groups of $T^N \backslash A$ have a natural structure of a right module over the group ring of the space symmetry group of the pattern and can be decomposed in a direct sum of its irreducible representations. An example of such decomposition is shown for the Ammann-Kramer tiling

Stability of Quasicrystals Composed of Soft Isotropic Particles

Quasicrystals whose building blocks are of mesoscopic rather than atomic scale have recently been discovered in several soft-matter systems. Contrary to metallurgic quasicrystals whose source of stability remains a question of great debate to this day, we argue that the stability of certain soft-matter quasicrystals can be directly explained by examining a coarse-grained free energy for a system of soft isotropic particles. We show, both theoretically and numerically, that the stability can be attributed to the existence of two natural length scales in the pair potential, combined with effective three-body interactions arising from entropy. Our newly gained understanding of the stability of soft quasicrystals allows us to point at their region of stability in the phase diagram, and thereby may help control the self-assembly of quasicrystals and a variety of other desired structures in future experimental realizations.Comment: Revised abstract, more detailed explanations, and better images of the numerical minimization of the free energ

Bethe Ansatz solution of a decagonal rectangle triangle random tiling

A random tiling of rectangles and triangles displaying a decagonal phase is solved by Bethe Ansatz. Analogously to the solutions of the dodecagonal square triangle and the octagonal rectangle triangle tiling an exact expression for the maximum of the entropy is found.Comment: 17 pages, 4 figures, some remarks added and typos correcte

Nonlocal Scalar Quantum Field Theory: Functional Integration, Basis Functions Representation and Strong Coupling Expansion

Nonlocal QFT of one-component scalar field $\varphi$ in $D$-dimensional Euclidean spacetime is considered. The generating functional (GF) of complete Green functions $\mathcal{Z}$ as a functional of external source $j$, coupling constant $g$, and spatial measure $d\mu$ is studied. An expression for GF $\mathcal{Z}$ in terms of the abstract integral over the primary field $\varphi$ is given. An expression for GF $\mathcal{Z}$ in terms of integrals over the primary field and separable Hilbert space (HS) is obtained by means of a separable expansion of the free theory inverse propagator $\hat{L}$ over the separable HS basis. The classification of functional integration measures $\mathcal{D}\left[\varphi\right]$ is formulated, according to which trivial and two nontrivial versions of GF $\mathcal{Z}$ are obtained. Nontrivial versions of GF $\mathcal{Z}$ are expressed in terms of $1$-norm and $0$-norm, respectively. The definition of the $0$-norm generator $\varPsi$ is suggested. Simple cases of sharp and smooth generators are considered. Expressions for GF $\mathcal{Z}$ in terms of integrals over the separable HS with new integrands are obtained. For polynomial theories $\varphi^{2n},\, n=2,3,4,\ldots,$ and for the nonpolynomial theory $\sinh^{4}\varphi$, integrals over the separable HS in terms of a power series over the inverse coupling constant $1/\sqrt{g}$ for both norms ($1$-norm and $0$-norm) are calculated. Critical values of model parameters when a phase transition occurs are found numerically. A generalization of the theory to the case of the uncountable integral over HS is formulated. A comparison of two GFs $\mathcal{Z}$, one in the case of uncountable HS integral and one obtained using the Parseval-Plancherel identity, is given.Comment: 26 pages, 2 figures; v2: significant additions in the text; prepared for the special issue "QCD and Hadron Structure" of the journal Particles; v3: minimal corrections; v4: paragraphs added related to Reviewer comment

Composition of Ni2+ cation solvation shell in NiCl2–methanol solution by multinuclear NMR

1H-, 2H- and 13C-NMR spectra have been used to test the Ni2+ solvation shell composition in the 1.1 molal methanol solution of NiCl2. It has been confirmed that Cl− anion takes part in the nearest environment of Ni2+ cation at all the temperatures investigated. Using 2H-NMR allowed us to detect for the first time OD-signal of methanol in the primary solvation shell of Ni2+ cation. Both 2H- and 13C-NMR spectra show that the composition of the cation solvation shell becomes more complicated at temperatures lower than 220 K