2025 Theses Doctoral

Band structure degeneracies and edge states in square lattice media and their deformations

Chaban, Jonah

We study the spectral properties of conservative linear wave systems modeling two-dimensional periodic "bulk" media, as well as "edge" media formed by the junction of two distinct bulk media. In both scenarios, we consider the cases of (1) periodic bulk media possessing the symmetries of a square lattice, and (2) linear deformations of such media. The dissertation is divided in two parts:

In part one, we begin with a spatially infinite bulk medium modeled by the Schrödinger operator H = -Δ + V, where the potential V is real-valued, periodic with respect to a square lattice, and invariant under the symmetry group of the square. The band structure of H is known to contain quadratic band degeneracy points [43, 44]; see also [8]. We prove that, under typical small linear deformations of V, each quadratic band degeneracy splits into a pair of nearby (tilted, elliptical) conical degeneracies, or Dirac points. Further, we show that both types of degeneracies are lifted upon the introduction of appropriate symmetry-breaking perturbations, and discuss the topology of the now non-degenerate bands.

In part two, we consider a class of Schrödinger operators, modeling either an undeformed or deformed square lattice bulk medium, perturbed by a spatially non-compact line defect commensurate with the underlying periodic bulk medium. For both cases (1) and (2), we construct edge states, which propagate parallel to the interface but are localized transverse to it. The edge states bifurcate from the band structure degeneracies associated with the unperturbed bulk medium; the bifurcation is controlled by effective (homogenized) edge Hamiltonians derived via multiple-scale analysis. In case (1), the bifurcation is governed by a matrix Schrödinger operator; in case (2), it is governed by a pair of Dirac operators. We present analytical results as well as numerical simulations, all consistent with the bulk-edge correspondence principle of topological physics.