Algebraic Theory of Crystal Vibrations: Localization Properties of Wave Functions in Two-Dimensional Lattices

Abstract

Abstract:The localization properties of the wave functions of vibrations in two-dimensional (2D) crystals are studied numerically for square and hexagonal lattices within the framework of an algebraic model. The wave functions of 2D lattices have remarkable localization properties, especially at the van Hove singularities (vHs). Finite-size sheets with a hexagonal lattice (graphene-like materials), in addition, exhibit at zero energy a localization of the wave functions at zigzag edges, so-called edge states. The striped structure of the wave functions at a vHs is particularly noteworthy. We have investigated its stability and that of the edge states with respect to perturbations in the lattice structure, and the effect of the boundary shape on the localization properties. We find that the stripes disappear instantaneously at the vHs in a square lattice when turning on the perturbation, whereas they broaden but persist at the vHss in a hexagonal lattice. For one of them, they eventually merge into edge states with increasing coupling, which, in contrast to the zero-energy edge states, are localized at armchair edges. The results are corroborated based on participation ratios, obtained under various conditions.

Excited-state quantum phase transitions in systems with two degrees of freedom. III. Interacting boson systems

“…If the f = 2 collective dynamics of such a system is described by means of a finite dynamical algebra, there might be a close link to the present work enabled by a suitable bosonic representation of the algebra [54]. Second, a similar study as presented here can be performed in the context of extended lattice systems with collective excitations, like two-dimensional crystals [23][24][25]. ACKNOWLEDGMENTS M.M.…”

“…[38]. The cited work applied the simplified IBM Hamiltonian (25) along several symmetry-connecting paths in parameters η and χ. It turned out that energy dependencies of expectation values of the d-boson numbern d in individual excited states with L = 0 exhibit some special structures near the ESQPT borderlines, considered in Ref.…”

Excited-state quantum phase transitions and their manifestations in an extended Dicke model

“…As examples we mention van Hove singularity in two-dimensional lattice systems [19,20,21] and quantum monodromy in some molecules [22,23]. Recently, the esqpts have been related to Jacobi-type shape transitions in nuclei within the framework of the algebraic Interacting Boson Model [24].…”

Van Hove singularities and excited-state quantum phase transitions in graphene-like microwave billiards

“…Most interestingly, at both van Hove singularities, the eigenstates form stripes (panel f shows a state at the lower vHs, cf. [14]). Similar to the edge states, they are oriented parallel to the zig-zag direction, the vHs-stripes however appear in the bulk.…”