Abstract

The physical observation that it is energetically favorable for interacting particles to arrange in a periodic fashion remains a largely open problem, referred to as the crystallization conjecture. In specific models, e.g., coming from atomic pair potentials, periodic configurations can be shown to be stable under, both, global ℓ2 or local perturbations. The situation changes when one studies aperiodic media. The specific class of aperiodic media we are interested in arise from taking two 2D periodic crystals and stacking them parallel at a relative twist. We study different notions of stability for such systems. The goal of our analysis is to provide phonons in such systems with meaning. In periodic media, phonons are generalized eigenvectors for a stability operator acting on ℓ2, coming from a mechanical energy. Using ideas rigorously established for the 1D Frenkel-Kontorova model, we assume that we can parametrize minimizing lattice deformations w.r.t. local perturbations via continuous stacking-periodic functions, see [9], for which we previously derived a continuous energy density functional [18]. Such (continuous) energy densities are analytically and computationally much better accessible compared to discrete energy functionals. In order to pass to an ℓ2-based energy functional, we also study the offset energy w.r.t. given lattice deformations, under ℓ1-perturbations. Our findings show that, in the case of an undeformed bilayer heterostructure, while the energy density can be shown to be stable under the assumption of stability of individual layers, the offset energy fails to be stable in the case of twisted bilayer graphene. We then establish conditions for stability and instability of the offset energy w.r.t. the relaxed lattice. Finally, we show that, in the case of incommensurate bilayer homostructures, i.e., two equal layers, if we choose minimizing deformations according to the global energy density above, the offset energy is stable in the limit of zero twist angle. Consequently, in this case, one can then define phonons as generalized eigenvectors w.r.t. the stability operator associated with the offset energy.

Abstract

The physical observation that it is energetically favorable for interacting particles to arrange in a periodic fashion remains a largely open problem, referred to as the crystallization conjecture. In specific models, e.g., coming from atomic pair potentials, periodic configurations can be shown to be stable under, both, global ℓ2 or local perturbations. The situation changes when one studies aperiodic media. The specific class of aperiodic media we are interested in arise from taking two 2D periodic crystals and stacking them parallel at a relative twist. We study different notions of stability for such systems. The goal of our analysis is to provide phonons in such systems with meaning. In periodic media, phonons are generalized eigenvectors for a stability operator acting on ℓ2, coming from a mechanical energy. Using ideas rigorously established for the 1D Frenkel-Kontorova model, we assume that we can parametrize minimizing lattice deformations w.r.t. local perturbations via continuous stacking-periodic functions, see [9], for which we previously derived a continuous energy density functional [18]. Such (continuous) energy densities are analytically and computationally much better accessible compared to discrete energy functionals. In order to pass to an ℓ2-based energy functional, we also study the offset energy w.r.t. given lattice deformations, under ℓ1-perturbations. Our findings show that, in the case of an undeformed bilayer heterostructure, while the energy density can be shown to be stable under the assumption of stability of individual layers, the offset energy fails to be stable in the case of twisted bilayer graphene. We then establish conditions for stability and instability of the offset energy w.r.t. the relaxed lattice. Finally, we show that, in the case of incommensurate bilayer homostructures, i.e., two equal layers, if we choose minimizing deformations according to the global energy density above, the offset energy is stable in the limit of zero twist angle. Consequently, in this case, one can then define phonons as generalized eigenvectors w.r.t. the stability operator associated with the offset energy.

Abstract

The physical observation that it is energetically favorable for interacting particles to arrange in a periodic fashion remains a largely open problem, referred to as the crystallization conjecture. In specific models, e.g., coming from atomic pair potentials, periodic configurations can be shown to be stable under, both, global ℓ2 or local perturbations. The situation changes when one studies aperiodic media. The specific class of aperiodic media we are interested in arise from taking two 2D periodic crystals and stacking them parallel at a relative twist. We study different notions of stability for such systems. The goal of our analysis is to provide phonons in such systems with meaning. In periodic media, phonons are generalized eigenvectors for a stability operator acting on ℓ2, coming from a mechanical energy. Using ideas rigorously established for the 1D Frenkel-Kontorova model, we assume that we can parametrize minimizing lattice deformations w.r.t. local perturbations via continuous stacking-periodic functions, see [9], for which we previously derived a continuous energy density functional [18]. Such (continuous) energy densities are analytically and computationally much better accessible compared to discrete energy functionals. In order to pass to an ℓ2-based energy functional, we also study the offset energy w.r.t. given lattice deformations, under ℓ1-perturbations. Our findings show that, in the case of an undeformed bilayer heterostructure, while the energy density can be shown to be stable under the assumption of stability of individual layers, the offset energy fails to be stable in the case of twisted bilayer graphene. We then establish conditions for stability and instability of the offset energy w.r.t. the relaxed lattice. Finally, we show that, in the case of incommensurate bilayer homostructures, i.e., two equal layers, if we choose minimizing deformations according to the global energy density above, the offset energy is stable in the limit of zero twist angle. Consequently, in this case, one can then define phonons as generalized eigenvectors w.r.t. the stability operator associated with the offset energy.

Abstract

The physical observation that it is energetically favorable for interacting particles to arrange in a periodic fashion remains a largely open problem, referred to as the crystallization conjecture. In specific models, e.g., coming from atomic pair potentials, periodic configurations can be shown to be stable under, both, global ℓ2 or local perturbations. The situation changes when one studies aperiodic media. The specific class of aperiodic media we are interested in arise from taking two 2D periodic crystals and stacking them parallel at a relative twist. We study different notions of stability for such systems. The goal of our analysis is to provide phonons in such systems with meaning. In periodic media, phonons are generalized eigenvectors for a stability operator acting on ℓ2, coming from a mechanical energy. Using ideas rigorously established for the 1D Frenkel-Kontorova model, we assume that we can parametrize minimizing lattice deformations w.r.t. local perturbations via continuous stacking-periodic functions, see [9], for which we previously derived a continuous energy density functional [18]. Such (continuous) energy densities are analytically and computationally much better accessible compared to discrete energy functionals. In order to pass to an ℓ2-based energy functional, we also study the offset energy w.r.t. given lattice deformations, under ℓ1-perturbations. Our findings show that, in the case of an undeformed bilayer heterostructure, while the energy density can be shown to be stable under the assumption of stability of individual layers, the offset energy fails to be stable in the case of twisted bilayer graphene. We then establish conditions for stability and instability of the offset energy w.r.t. the relaxed lattice. Finally, we show that, in the case of incommensurate bilayer homostructures, i.e., two equal layers, if we choose minimizing deformations according to the global energy density above, the offset energy is stable in the limit of zero twist angle. Consequently, in this case, one can then define phonons as generalized eigenvectors w.r.t. the stability operator associated with the offset energy.