Generalization to d-dimensions of a fermionic path integral for exact enumeration of polygons on hypercubic lattices.

The generating function for polygons on the square lattice has been known for many decades and is closely related to the path integral formulation of a free fermion model. On the cubic and hypercubic lattices the generating function is still unknown and the problem remains open. It has been conjectured that the three-dimensional (3D) and higher dimensional problems are not solvable-or, to be more precise, that there are no differentiably finite (D-finite) solutions. In this context, very recently a Berezin integral of an exponentiated Grassmann action was found for the polygon generating function on the cubic lattice, making explicit the connection between 3D polygons and a model of interacting fermions. Here we address the problem of how to generalize the 3D result to higher dimensions. We derive a Grassmann representation in terms of a Berezin integral for the generating function of polygons on d-dimensional hypercubic lattices. On the one hand, this new result admittedly brings us no closer to the problem of finding an explicit analytic expression for the desired generating function for polygons. On the other hand, however, the significant advance reported here precisely quantifies the remarkable mathematical difficulty of finding the explicit generating function. Indeed, the non-quadratic functional form of the Grassmann action that we derive here provides a very clear picture of the formidable mathematical obstruction that would need to be overcome. Specifically, in d dimensions, the Grassmann action contains terms of degree 2 ( d - 1 ) , so the model describes interacting rather than free fermions. It is an open problem whether or not these models of interacting fermions can in principle be free fermionized through some still undiscovered algebraic method, but it is widely believed that this goal is mathematically impossible.

Intermediate statistics: Addressing the thermoelectric properties of solids.

We study the thermodynamics of a crystalline solid by applying intermediate statistics obtained by deforming known solid state models using the mathematics of q analogs. We apply the resulting q deformation to both the Einstein and Debye models and study the deformed thermal and electrical conductivities and the deformed Debye specific heat. We find that the q deformation acts in two different ways-but not necessarily as independent mechanisms. First, it acts as an effective factor of disorder or impurity, modifying the characteristics of a crystalline structure, which are phenomena described by q bosons. Second, it also manifests intermediate statistics, namely, the B anyons (or B-type systems). For the latter case we have identified the Schottky effect, normally associated with high-T_{c} superconductors in the presence of rare-earth-ion impurities. We also find that it increases the specific heat of the solids beyond the Dulong-Petit limit at high temperature. Such an effect is usually related to anharmonicity of interatomic interactions. Alternatively, since in the q-boson's case the statistics are in principle maintained, the effect of the deformation acts more slowly due to a small change in the crystal lattice. On the other hand, B anyons that belong to modified statistics are more sensitive to the deformation. The results reported here may be verified experimentally, for instance, in experimental samples by inserting impurities, or changes in pressure or temperature if one assumes these tuning quantities are related with the q-deformation parameter.

Lévy walkers inside spherical shells with absorbing boundaries: Towards settling the optimal Lévy walk strategy for random searches.

The Lévy flight foraging hypothesis states that organisms must have evolved adaptations to exploit Lévy walk search strategies. Indeed, it is widely accepted that inverse square Lévy walks optimize the search efficiency in foraging with unrestricted revisits (also known as nondestructive foraging). However, a mathematically rigorous demonstration of this for dimensions D≥2 is still lacking. Here we study the very closely related problem of a Lévy walker inside annuli or spherical shells with absorbing boundaries. In the limit that corresponds to the foraging with unrestricted revisits, we show that inverse square Lévy walks optimize the search. This constitutes the strongest formal result to date supporting the optimality of inverse square Lévy walks search strategies.

Investigating size effects in graphene-BN hybrid monolayers: a combined density functional theory-molecular dynamics study.

We combine Density Functional Theory (DFT) and classical Molecular Dynamics (MD) simulations to study graphene-boron nitride (BN) hybrid monolayers spanning a wide range of sizes (from 2 nm to 100 nm). Our simulations show that the elastic properties depend on the fraction of BN contained in the monolayer, with Young's modulus values decreasing as the BN concentration increases. Furthermore, our calculations reveal that the mechanical properties are weakly anisotropic. We also analyze the evolution of the stress distribution during our MD simulations. Curiously, we find that stress does not concentrate on the graphene-BN interface, even though fracture always starts in this region. Hence, we find that fracture is caused by the lower strength of C-N and C-B bonds, rather than by high local stress values. Still, in spite of the fact that the weaker bonds in the interface region become a lower fraction of the total as size increases, we find that the mechanical properties of the hybrid monolayers do not depend on the size of the structure, for constant graphene/BN concentrations. Our results indicate that the mechanical properties of the hybrid monolayers are independent of scale, so long as the graphene sheet and the h-BN nanodomain decrease or increase proportionately.

Mechanical and electronic properties of boron nitride nanosheets with graphene domains under strain.

Hybrid structures comprised of graphene domains embedded in larger hexagonal boron nitride (h-BN) nanosheets were first synthesized in 2013. However, the existing theoretical investigations on them have only considered relaxed structures. In this work, we use Density Functional Theory (DFT) and Molecular Dynamics (MD) simulations to investigate the mechanical and electronic properties of this type of nanosheet under strain. Our results reveal that the Young's modulus of the hybrid sheets depends only on the relative concentration of graphene and h-BN in the structure, showing little dependence on the shape of the domain or the size of the structure for a given concentration. Regarding the tensile strength, we obtained higher values using triangular graphene domains. We find that the studied systems can withstand large strain values (between 15% and 22%) before fracture, which always begins at the weaker C-B bonds located at the interface between the two materials. Concerning the electronic properties, we find that by combining composition and strain, we can produce hybrid sheets with band gaps spanning an extensive range of values (between 1.0 eV and 3.5 eV). Our results also show that the band gap depends more on the composition than on the external strain, particularly for structures with low carbon concentration. The combination of atomic-scale thickness, high ultimate strain, and adjustable band gap suggests applications of h-BN nanosheets with graphene domains in wearable electronics.

Electronic and optical properties of B x C y N z hybrid α-graphynes.

Hybrid two-dimensional (2D) materials composed of carbon, boron, and nitrogen constitute a hot topic of research, as their flexible composition allows for tunable properties. However, while graphene-like hybrid lattices have been well characterized, systematic investigations are lacking for various 2D materials. Hence, in the present contribution, we employ first-principles calculations to investigate the structural, electronic and optical properties of what we call B x C y N z hybrid α-graphynes. We considered eleven structures with stoichiometry BC2N and varied atomic arrangements. We calculated the formation energy for each arrangement, and determined that it is low (high) when the number of boron-carbon and nitrogen-carbon bonds is low (high). We found that the formation energy of many our structures compared favorably with a previous literature proposal. Regarding the electronic properties, we found that the investigated structures are semiconducting, with band gaps ranging from 0.02 to 2.00 eV. Moreover, we determined that most of the B x C y N z hybrid α-graphynes proposed here strongly absorb infrared light, and so could potentially find applications in optoelectronic devices such as heat sensors and infrared filters.

B x C y N z hybrid graphenylene: stability and electronic properties.

Interest in hybrid monolayers with arrangements that differ from that of the honeycomb lattice has been growing. However, systematic investigations on the properties of these structures are still lacking. In this work, we combined density functional theory (DFT) and molecular dynamics (MD) simulations to study the stability and electronic properties of nanosheets composed of B, C, and N atoms arranged in the pattern of the carbon allotrope graphenylene. We considered twenty structures with varied atomic arrangements and stoichiometries, which we call B x C y N z hybrid graphenylenes. We calculated the formation energy for each arrangement, and found that it decreases as the number of B-C and N-C bonds decreases. We also found that the structure with minimum energy has stoichiometry B2CN and an atomic arrangement with BN and C stripes connected along the zigzag direction. Regarding the electronic properties, we found that all investigated structures are semiconductors, with band gaps ranging from 0.14 to 1.65 eV. Finally, we found that the optimized hybrid lattices presented pores of varied sizes and shapes. This diversity in pore geometry suggests that these structures might be particularly suited for molecular sieve applications.

Density of states of helically symmetric boron carbon nitride nanotubes.

Motivated by the existence of helical wrapping patterns in composite nanotube systems, in this work we study the effects of the helical incorporation of carbon atoms in boron nitride nanotubes. We consider the substitutional carbon atoms distributed in stripes forming helical patterns along the nanotube axis. The density of states and energy band gap were calculated adopting Green function formalism by using the Rubio-Sancho technique in order to solve the matrix Dyson equation. We report the effects of the helical atomic distribution of carbon atoms on the behaviour of the density of states and the energy band gap. In particular, we show that the electronic energy band gap displays a non-monotonical dependence on the helical pattern, oscillating as a function of the helical angle θ.