Macroscopic approximation of tight-binding models near spectral degeneracies and validity for wave packet propagation

This paper concerns the derivation and validity of macroscopic descriptions of wave packets supported in the vicinity of degenerate points $(K,E)$ in the dispersion relation of tight-binding models accounting for macroscopic variations. We show that such wave packets are well approximated over long times by macroscopic models with varying orders of accuracy. Our main applications are in the analysis of single- and multilayer graphene tight-binding Hamiltonians modeling macroscopic variations such as those generated by shear or twist. Numerical simulations illustrate the theoretical findings.

💡 Research Summary

This paper develops a rigorous multiscale framework for approximating the dynamics of wave packets that are spectrally localized near degenerate points ((K,E)) of tight‑binding (TB) Hamiltonians by continuous macroscopic partial differential equations (PDEs). The authors begin by representing a TB Hamiltonian as a Weyl pseudo‑differential operator (\operatorname{Op}^{\mathrm{W}}(a(x,\xi))) whose symbol (a) is a finite sum of shift operators and smooth matrix‑valued multipliers. Near a degeneracy the symbol is rescaled with a small parameter (\delta) (slow spatial variation) and expanded in a Taylor series: \