Spectral properties of periodic systems cut at an angle

Abstract

We consider a semi-periodic two-dimensional Schrödinger operator which is cut at an angle. When the cut is commensurate with the periodic lattice, the spectrum of the operator has the band-gap Bloch structure. We prove that in the incommensurable case, there are no gaps: the gaps of the bulk operator are filled with edge spectrum.

Discrete honeycombs, rational edges, and edge states

“…where 𝑁 𝑘 , 𝑁 𝐸 , 𝐸 lim are specified for each simulation. In all figures, the dark areas correspond to the essential spectrum; see Figures 1,2,8,9,12,15,16,17. Outside the dark areas, in particular for (𝑘, 𝐸) ∈ 𝒦 (see (7.1)), we seek edge state curves by studying where (𝑘, 𝐸) ↦ log |Δ(𝑘, 𝐸)| takes on very large negative values. In Figure 12, we consider 4 different edges, three of zigzag-type and one of armchair-type, with numerical parameters and 𝑁 𝑘 = 𝑁 𝐸 = 1000 and specified values of 𝐸 lim .…”

“…(a) Are there states which are bounded and oscillatory parallel to an irrational edge and which decay into the bulk? Related to this question is the article [17], which demonstrates that the edge spectrum for a rationally terminated continuum periodic Schroedinger operator (with Dirichlet boundary conditions) has a band‐gap spectrum, while for an irrational termination the gaps are filled with “edge spectrum”. (b) Can one realize an edge state for irrational termination as the limit of a sequence of edge state wave‐packets (superpositions of edge states) of rationally terminated structures?…”

Discrete honeycombs, rational edges and edge states

“…(a) Are there states which are bounded and oscillatory parallel to an irrational edge and which decay into the bulk? Related to this question is the article [15], which demonstrates that the edge spectrum for a rationally terminated continuum periodic Schroedinger operator (with Dirichlet boundary conditions) has a bandgap spectrum, while for an irrational termination the gaps are filled with "edge spectrum". (b) Can one realize an edge state for irrational termination as the limit of a sequence of edge state wave-packets (superpositions of edge states) of rationally terminated structures?…”