The traveling salesman problem, conformal invariance, and dense polymers

We propose that the statistics of the optimal tour in the planar random Euclidean traveling salesman problem is conformally invariant on large scales. This is exhibited in power-law behavior of the probabilities for the tour to zigzag repeatedly between two regions, and in subleading corrections to the length of the tour. The universality class should be the same as for dense polymers and minimal spanning trees. The conjectures for the length of the tour on a cylinder are tested numerically.Comment: 4 pages. v2: small revisions, improved argument about dimensions d>2. v3: Final version, with a correction to the form of the tour length in a domain, and a new referenc

Dense loops, supersymmetry, and Goldstone phases in two dimensions

Loop models in two dimensions can be related to O(N) models. The low-temperature dense-loops phase of such a model, or of its reformulation using a supergroup as symmetry, can have a Goldstone broken-symmetry phase for N<2. We argue that this phase is generic for -2< N <2 when crossings of loops are allowed, and distinct from the model of non-crossing dense loops first studied by Nienhuis [Phys. Rev. Lett. 49, 1062 (1982)]. Our arguments are supported by our numerical results, and by a lattice model solved exactly by Martins et al. [Phys. Rev. Lett. 81, 504 (1998)].Comment: RevTeX, 5 pages, 3 postscript figure

Exact enumeration of Hamiltonian circuits, walks, and chains in two and three dimensions

We present an algorithm for enumerating exactly the number of Hamiltonian chains on regular lattices in low dimensions. By definition, these are sets of k disjoint paths whose union visits each lattice vertex exactly once. The well-known Hamiltonian circuits and walks appear as the special cases k=0 and k=1 respectively. In two dimensions, we enumerate chains on L x L square lattices up to L=12, walks up to L=17, and circuits up to L=20. Some results for three dimensions are also given. Using our data we extract several quantities of physical interest

Dynamic rotor mode in antiferromagnetic nanoparticles

We present experimental, numerical, and theoretical evidence for a new mode of antiferromagnetic dynamics in nanoparticles. Elastic neutron scattering experiments on 8 nm particles of hematite display a loss of diffraction intensity with temperature, the intensity vanishing around 150 K. However, the signal from inelastic neutron scattering remains above that temperature, indicating a magnetic system in constant motion. In addition, the precession frequency of the inelastic magnetic signal shows an increase above 100 K. Numerical Langevin simulations of spin dynamics reproduce all measured neutron data and reveal that thermally activated spin canting gives rise to a new type of coherent magnetic precession mode. This "rotor" mode can be seen as a high-temperature version of superparamagnetism and is driven by exchange interactions between the two magnetic sublattices. The frequency of the rotor mode behaves in fair agreement with a simple analytical model, based on a high temperature approximation of the generally accepted Hamiltonian of the system. The extracted model parameters, as the magnetic interaction and the axial anisotropy, are in excellent agreement with results from Mossbauer spectroscopy

Low-loss photonic crystal fibers for transmission systems and their dispersion properties

We report on a single-mode photonic crystal fiber with attenuation and effective area at 1550 nm of 0.48 dB/km and 130 square-micron, respectively. This is, to our knowledge, the lowest loss reported for a PCF not made from VAD prepared silica and at the same time the largest effective area for a low-loss (< 1 dB/km) PCF. We briefly discuss the future applications of PCFs for data transmission and show for the first time, both numerically and experimentally, how the group velocity dispersion is related to the mode field diameterComment: 5 pages including 3 figures + 1 table. Accepted for Opt. Expres

A tree-decomposed transfer matrix for computing exact Potts model partition functions for arbitrary graphs, with applications to planar graph colourings

Combining tree decomposition and transfer matrix techniques provides a very general algorithm for computing exact partition functions of statistical models defined on arbitrary graphs. The algorithm is particularly efficient in the case of planar graphs. We illustrate it by computing the Potts model partition functions and chromatic polynomials (the number of proper vertex colourings using Q colours) for large samples of random planar graphs with up to N=100 vertices. In the latter case, our algorithm yields a sub-exponential average running time of ~ exp(1.516 sqrt(N)), a substantial improvement over the exponential running time ~ exp(0.245 N) provided by the hitherto best known algorithm. We study the statistics of chromatic roots of random planar graphs in some detail, comparing the findings with results for finite pieces of a regular lattice.Comment: 5 pages, 3 figures. Version 2 has been substantially expanded. Version 3 shows that the worst-case running time is sub-exponential in the number of vertice

The packing of two species of polygons on the square lattice

We decorate the square lattice with two species of polygons under the constraint that every lattice edge is covered by only one polygon and every vertex is visited by both types of polygons. We end up with a 24 vertex model which is known in the literature as the fully packed double loop model. In the particular case in which the fugacities of the polygons are the same, the model admits an exact solution. The solution is obtained using coordinate Bethe ansatz and provides a closed expression for the free energy. In particular we find the free energy of the four colorings model and the double Hamiltonian walk and recover the known entropy of the Ice model. When both fugacities are set equal to two the model undergoes an infinite order phase transition.Comment: 21 pages, 4 figure

Finite average lengths in critical loop models

A relation between the average length of loops and their free energy is obtained for a variety of O(n)-type models on two-dimensional lattices, by extending to finite temperatures a calculation due to Kast. We show that the (number) averaged loop length L stays finite for all non-zero fugacities n, and in particular it does not diverge upon entering the critical regime n -> 2+. Fully packed loop (FPL) models with n=2 seem to obey the simple relation L = 3 L_min, where L_min is the smallest loop length allowed by the underlying lattice. We demonstrate this analytically for the FPL model on the honeycomb lattice and for the 4-state Potts model on the square lattice, and based on numerical estimates obtained from a transfer matrix method we conjecture that this is also true for the two-flavour FPL model on the square lattice. We present in addition numerical results for the average loop length on the three critical branches (compact, dense and dilute) of the O(n) model on the honeycomb lattice, and discuss the limit n -> 0. Contact is made with the predictions for the distribution of loop lengths obtained by conformal invariance methods.Comment: 20 pages of LaTeX including 3 figure

Construction of transferable spherically-averaged electron potentials

A new scheme for constructing approximate effective electron potentials within density-functional theory is proposed. The scheme consists of calculating the effective potential for a series of reference systems, and then using these potentials to construct the potential of a general system. To make contact to the reference system the neutral-sphere radius of each atom is used. The scheme can simplify calculations with partial wave methods in the atomic-sphere or muffin-tin approximation, since potential parameters can be precalculated and then for a general system obtained through simple interpolation formulas. We have applied the scheme to construct electron potentials of phonons, surfaces, and different crystal structures of silicon and aluminum atoms, and found excellent agreement with the self-consistent effective potential. By using an approximate total electron density obtained from a superposition of atom-based densities, the energy zero of the corresponding effective potential can be found and the energy shifts in the mean potential between inequivalent atoms can therefore be directly estimated. This approach is shown to work well for surfaces and phonons of silicon.Comment: 8 pages (3 uuencoded Postscript figures appended), LaTeX, CAMP-090594-