New Duality Relations for Classical Ground States

We derive new duality relations that link the energy of configurations associated with a class of soft pair potentials to the corresponding energy of the dual (Fourier-transformed) potential. We apply them by showing how information about the classical ground states of short-ranged potentials can be used to draw new conclusions about the nature of the ground states of long-ranged potentials and vice versa. They also lead to bounds on the T=0 system energies in density intervals of phase coexistence, the identification of a one-dimensional system that exhibits an infinite number of ``phase transitions," and a conjecture regarding the ground states of purely repulsive monotonic potentials.Comment: 11 pages, 2 figures. Slightly revised version that corrects typos. This article will be appearing in Physical Review Letters in a slightly shortened for

Quaternionic Root Systems and Subgroups of the Aut(F4)Aut(F_{4})

Cayley-Dickson doubling procedure is used to construct the root systems of some celebrated Lie algebras in terms of the integer elements of the division algebras of real numbers, complex numbers, quaternions and octonions. Starting with the roots and weights of SU(2) expressed as the real numbers one can construct the root systems of the Lie algebras of SO(4),SP(2)= SO(5),SO(8),SO(9),F_{4} and E_{8} in terms of the discrete elements of the division algebras. The roots themselves display the group structures besides the octonionic roots of E_{8} which form a closed octonion algebra. The automorphism group Aut(F_{4}) of the Dynkin diagram of F_{4} of order 2304, the largest crystallographic group in 4-dimensional Euclidean space, is realized as the direct product of two binary octahedral group of quaternions preserving the quaternionic root system of F_{4}.The Weyl groups of many Lie algebras, such as, G_{2},SO(7),SO(8),SO(9),SU(3)XSU(3) and SP(3)X SU(2) have been constructed as the subgroups of Aut(F_{4}). We have also classified the other non-parabolic subgroups of Aut(F_{4}) which are not Weyl groups. Two subgroups of orders192 with different conjugacy classes occur as maximal subgroups in the finite subgroups of the Lie group $G_{2}$ of orders 12096 and 1344 and proves to be useful in their constructions. The triality of SO(8) manifesting itself as the cyclic symmetry of the quaternionic imaginary units e_{1},e_{2},e_{3} is used to show that SO(7) and SO(9) can be embedded triply symmetric way in SO(8) and F_{4} respectively

Toward the Jamming Threshold of Sphere Packings: Tunneled Crystals

We have discovered a new family of three-dimensional crystal sphere packings that are strictly jammed (i.e., mechanically stable) and yet possess an anomalously low density. This family constitutes an uncountably infinite number of crystal packings that are subpackings of the densest crystal packings and are characterized by a high concentration of self-avoiding "tunnels" (chains of vacancies) that permeate the structures. The fundamental geometric characteristics of these tunneled crystals command interest in their own right and are described here in some detail. These include the lattice vectors (that specify the packing configurations), coordination structure, Voronoi cells, and density fluctuations. The tunneled crystals are not only candidate structures for achieving the jamming threshold (lowest-density rigid packing), but may have substantially broader significance for condensed matter physics and materials science.Comment: 19 pages, 5 figure

Quintics with Finite Simple Symmetries

We construct all quintic invariants in five variables with simple Non-Abelian finite symmetry groups. These define Calabi-Yau three-folds which are left invariant by the action of A_5, A_6 or PSL_2(11).Comment: 18 pages, typos corrected, matches published versio

Estimates of the optimal density and kissing number of sphere packings in high dimensions

The problem of finding the asymptotic behavior of the maximal density of sphere packings in high Euclidean dimensions is one of the most fascinating and challenging problems in discrete geometry. One century ago, Minkowski obtained a rigorous lower bound that is controlled asymptotically by $1/2^d$, where $d$ is the Euclidean space dimension. An indication of the difficulty of the problem can be garnered from the fact that exponential improvement of Minkowski's bound has proved to be elusive, even though existing upper bounds suggest that such improvement should be possible. Using a statistical-mechanical procedure to optimize the density associated with a "test" pair correlation function and a conjecture concerning the existence of disordered sphere packings [S. Torquato and F. H. Stillinger, Experimental Math. {\bf 15}, 307 (2006)], the putative exponential improvement was found with an asymptotic behavior controlled by $1/2^{(0.77865...)d}$. Using the same methods, we investigate whether this exponential improvement can be further improved by exploring other test pair correlation functions correponding to disordered packings. We demonstrate that there are simpler test functions that lead to the same asymptotic result. More importantly, we show that there is a wide class of test functions that lead to precisely the same exponential improvement and therefore the asymptotic form $1/2^{(0.77865...)d}$ is much more general than previously surmised.Comment: 23 pages, 4 figures, submitted to Phys. Rev.

Direct observation of irradiation-induced nanocavity shrinkage in Si

Nanocavities in Si substrates, formed by conventional H implantation and thermal annealing, are shown to evolve in size during subsequent Si irradiation. Both ex situ and in situ analytical techniques were used to demonstrate that the mean nanocavity diameter decreases as a function of Si irradiation dose in both the crystalline and amorphous phases. Potential mechanisms for this irradiation-induced nanocavity evolution are discussed. In the crystalline phase, the observed decrease in diameter is attributed to the gettering of interstitials. When the matrix surrounding the cavities is amorphized, cavity shrinkage may be mediated by one of two processes: nanocavities can supply vacancies into the amorphous phase and/or the amorphous phase may flow plastically into the nanocavities. Both processes yield the necessary decrease in density of the amorphous phase relative to crystalline material

Template-based searches for gravitational waves: efficient lattice covering of flat parameter spaces

The construction of optimal template banks for matched-filtering searches is an example of the sphere covering problem. For parameter spaces with constant-coefficient metrics a (near-) optimal template bank is achieved by the A_n* lattice, which is the best lattice-covering in dimensions n <= 5, and is close to the best covering known for dimensions n <= 16. Generally this provides a substantially more efficient covering than the simpler hyper-cubic lattice. We present an algorithm for generating lattice template banks for constant-coefficient metrics and we illustrate its implementation by generating A_n* template banks in n=2,3,4 dimensions.Comment: 10 pages, submitted to CQG for proceedings of GWDAW1

Tail resonances of FPU q-breathers and their impact on the pathway to equipartition

Upon initial excitation of a few normal modes the energy distribution among all modes of a nonlinear atomic chain (the Fermi-Pasta-Ulam model) exhibits exponential localization on large time scales. At the same time resonant anomalies (peaks) are observed in its weakly excited tail for long times preceding equipartition. We observe a similar resonant tail structure also for exact time-periodic Lyapunov orbits, coined q-breathers due to their exponential localization in modal space. We give a simple explanation for this structure in terms of superharmonic resonances. The resonance analysis agrees very well with numerical results and has predictive power. We extend a previously developed perturbation method, based essentially on a Poincare-Lindstedt scheme, in order to account for these resonances, and in order to treat more general model cases, including truncated Toda potentials. Our results give qualitative and semiquantitative account for the superharmonic resonances of q-breathers and natural packets

From Operator Algebras to Superconformal Field Theory

We make a review on the recent progress in the operator algebraic approach to (super)conformal field theory. We discuss representation theory, classification results, full and boundary conformal field theories, relations to supervertex operator algebras and Moonshine, connections to subfactor theory and noncommutative geometry