Triggered by the exchange of ideas between computer science and theoretical physics, several disordered systems with complex energy landscapes can now be analyzed numerically exact through computer simulations [1] by using fast combinatorial optimization algorithms. E.g., the ground state problem for the planar 2d random bond Ising model (RBIM) can be mapped to an auxiliary minimum-weight perfe...

From the point of view of coupled systems developed by Stewart, Golubitsky and Pivato, lattice differential equations consist of choosing a phase space R for each point in a lattice, and a system of differential equations on each of these spaces R such that the whole system is translation invariant. The architecture of a lattice differential equation specifies the sites that are coupled to each...

A planar self-avoiding walk (SAW) is a nearest neighbor random walk path in the square lattice with no self-intersection. A planar self-avoiding polygon (SAP) is a loop with no self-intersection. In this paper we present conjectures for the scaling limit of the uniform measures on these objects. The conjectures are based on recent results on the stochastic Loewner evolution and nondisconnecting...

At certain tilt alignments between a MeV proton beam and a planar channeling direction, a single interface lattice rotation within a crystal can result in a lower rate of dechanneling than at planar alignment in a perfect crystal. Such planar channeling enhancement arises when the beam passes through a layer thickness which is a half-multiple of the oscillation wavelength and then encounters a ...

We study low temperature properties of a spinless interacting Fermi gas in the trimerized kagomé lattice. The case of two fermions per trimer is described by a quantum spin 1/2 model on the triangular lattice with couplings depending on the bond directions. Using exact diagonalizations we show that the system exhibits nonstandard properties of a quantum spin-liquid crystal, combining a planar a...

It is shown that it is possible to obtain complete planar phononic bandgaps in square and hexagonal (honeycomb) lattice phononic crystals formed by etching a periodic array of circular holes in a thin silicon plate (or membrane). Also, better bandgap properties are obtained using the hexagonal lattice structure; and, with practical structure sizes, it is possible-to-obtain a gap-to-midgap ratio...

We present experimental evidence for strong angular dispersion in a planar photonic crystal (PC) structure by properly engineering the modes in the second PC band. We show that by using the second photonic band of a square lattice PC, angular dispersion of 4 degrees /nm can be achieved. We also show that major challenges in designing practical PC devices using second band modes can be addressed...

In abstract algebra courses, teachers are often confronted with the task of drawing subgroup lattices. For purposes of instruction, it is usually desirable that these lattices be planar graphs (with no crossings). We present a characterization of abelian groups with this property. We also resolve the following problem in the abelian case: if the subgroup lattice is required to be drawn hierarch...

We report laser micro-fabrication of photonic crystal structures in silica glass and photopolymerizing resins, and present a detailed analysis of their structural and optical properties. Photonic crystals with 2D triangular lattice and 3D diamond lattice in glass are found to exhibit moderately pronounced photonic pseudogaps. Formation of planar microcavity is demonstrated in log-pile 3D photon...