Integrable lattices and their sublattices II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices

An integrable self-adjoint 7-point scheme on the triangular lattice and an integrable self-adjoint scheme on the honeycomb lattice are studied using the sublattice approach. The star-triangle relation between these systems is introduced, and the Darboux transformations for both linear problems from the Moutard transformation of the B-(Moutard) quadrilateral lattice are obtained. A geometric interpretation of the Laplace transformations of the self-adjoint 7-point scheme is given and the corresponding novel integrable discrete 3D system is constructed.Comment: 15 pages, 6 figures; references added, some typos correcte

Complex-Temperature Properties of the Ising Model on 2D Heteropolygonal Lattices

Using exact results, we determine the complex-temperature phase diagrams of the 2D Ising model on three regular heteropolygonal lattices, $(3 \cdot 6 \cdot 3 \cdot 6)$ (kagom\'{e}), $(3 \cdot 12^2)$, and $(4 \cdot 8^2)$ (bathroom tile), where the notation denotes the regular $n$-sided polygons adjacent to each vertex. We also work out the exact complex-temperature singularities of the spontaneous magnetisation. A comparison with the properties on the square, triangular, and hexagonal lattices is given. In particular, we find the first case where, even for isotropic spin-spin exchange couplings, the nontrivial non-analyticities of the free energy of the Ising model lie in a two-dimensional, rather than one-dimensional, algebraic variety in the $z=e^{-2K}$ plane.Comment: 31 pages, latex, postscript figure

Geometry Optimization of Crystals by the Quasi-Independent Curvilinear Coordinate Approximation

The quasi-independent curvilinear coordinate approximation (QUICCA) method [K. N\'emeth and M. Challacombe, J. Chem. Phys. {\bf 121}, 2877, (2004)] is extended to the optimization of crystal structures. We demonstrate that QUICCA is valid under periodic boundary conditions, enabling simultaneous relaxation of the lattice and atomic coordinates, as illustrated by tight optimization of polyethylene, hexagonal boron-nitride, a (10,0) carbon-nanotube, hexagonal ice, quartz and sulfur at the $\Gamma$-point RPBE/STO-3G level of theory.Comment: Submitted to Journal of Chemical Physics on 7/7/0

Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy

It is shown that the integrable discrete Schwarzian KP (dSKP) equation which constitutes an algebraic superposition formula associated with, for instance, the Schwarzian KP hierarchy, the classical Darboux transformation and quasi-conformal mappings encapsulates nothing but a fundamental theorem of ancient Greek geometry. Thus, it is demonstrated that the connection with Menelaus' theorem and, more generally, Clifford configurations renders the dSKP equation a natural object of inversive geometry on the plane. The geometric and algebraic integrability of dSKP lattices and their reductions to lattices of Menelaus-Darboux, Schwarzian KdV, Schwarzian Boussinesq and Schramm type is discussed. The dSKP and discrete Schwarzian Boussinesq equations are shown to represent discretizations of families of quasi-conformal mappings.Comment: 26 pages, 9 figure

Darboux cyclides and webs from circles

Motivated by potential applications in architecture, we study Darboux cyclides. These algebraic surfaces of order a most 4 are a superset of Dupin cyclides and quadrics, and they carry up to six real families of circles. Revisiting the classical approach to these surfaces based on the spherical model of 3D Moebius geometry, we provide computational tools for the identification of circle families on a given cyclide and for the direct design of those. In particular, we show that certain triples of circle families may be arranged as so-called hexagonal webs, and we provide a complete classification of all possible hexagonal webs of circles on Darboux cyclides.Comment: 34 pages, 20 figure

Dimers and cluster integrable systems

We show that the dimer model on a bipartite graph on a torus gives rise to a quantum integrable system of special type - a cluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space of line bundles with connections on the graph. The sum of Hamiltonians is essentially the partition function of the dimer model. Any graph on a torus gives rise to a bipartite graph on the torus. We show that the phase space of the latter has a Lagrangian subvariety. We identify it with the space parametrizing resistor networks on the original graph.We construct several discrete quantum integrable systems.Comment: This is an updated version, 75 pages, which will appear in Ann. Sci. EN