Thermoelectric properties of two-dimensional (2D) Dirac materials are calculated within linearized Boltzmann transport theory and relaxation time approximation. We find that the gapless 2D Dirac material exhibits poorer thermoelectric performance than the gapped one. This fact arises due to the cancelation effect from electron-hole contributions to the transport quantities. Opening the bandgap lifts this cancelation effect. Furthermore, there exists an optimal bandgap for maximizing figure of merit ($ZT$ ) in the gapped 2D Dirac material. The optimal bandgap ranges from $6k_{B}T$ to $18k_{B}T$ , where $k_B$ is the Boltzmann constant and $T$ is the operating temperature in kelvin. This result indicates the importance of having narrow gaps to achieve the best thermoelectrics in 2D systems. Larger maximum $ZT$ s can also be obtained by suppressing the lattice thermal conductivity. In the most ideal case where the lattice thermal conductivity is very small, the maximum $ZT$ in the gapped 2D Dirac material can be many times the $ZT$ of commercial thermoelectric materials.

We investigate properties of attractors for the scalar field in the Lorentz violating scalar-vector-tensor theory of gravity. In this framework, both the effective coupling and potential functions determine the stabilities of the fixed points. In the model, we consider the constants of the slope of the effective coupling and potential functions which lead to the quadratic effective coupling vector with the (inverse) power-law potential. For the case of a purely scalar field, there are only two stable attractor solutions in the inflationary scenario. In the presence of a barotropic fluid, the fluid dominated solution is absent. We find two scaling solutions: the kinetic scaling solution and the scalar field scaling solutions. We show the stable attractors in regions of $(\gamma ,\xi )$ parameter space and in a phase plane plot for different qualitative evolutions. From the standard nucleosynthesis, we derive the constraints for the value of the coupling parameter.

In this paper, we search for the BPS skyrmions in some BPS submodels of the generalized Skyrme model in five-dimensional spacetime using the BPS Lagrangian method. We focus on the static solutions of the Bogomolny’s equations and their corresponding energies with topological charge B > 0 is an integer. We consider two main cases based on the symmetry of the effective Lagrangian of the BPS submodels, i.e. the spherically symmetric and non-spherically symmetric cases. For the spherically symmetric case, we find two BPS submodels. The first BPS submodels consist of a potential term and a term proportional to the square of the topological current. The second BPS submodels consist of only the Skyrme term. The second BPS submodel has BPS skyrmions with the same topological charge B > 1, but with different energies, that we shall call “topological degenerate” BPS skyrmions. It also has the usual BPS skyrmions with equal energies, if the topological charge is a prime number. Another interesting feature of the BPS skyrmions, with B > 1, in this BPS submodel, is that these BPS skyrmions have non-zero pressures in the angular direction. For the non-spherically symmetric case, there is only one BPS submodel, which is similar to the first BPS submodel in the spherically symmetric case. We find that the BPS skyrmions depend on a constant k and for a particular value of k we obtain the BPS skyrmions of the first BPS submodel in the spherically symmetric case. The total static energy and the topological charge of these BPS skyrmions also depend on this constant. We also show that all the results found in this paper satisfy the full field equations of motions of the corresponding BPS submodels.