Spectral Band Degeneracies of $\frac{\pi}{2}$-Rotationally Invariant Periodic Schrödinger Operators

The dynamics of waves in periodic media is determined by the band structure of the underlying periodic Hamiltonian. Symmetries of the Hamiltonian can give rise to novel properties of the band structure. Here we consider a class of periodic Schrödinger operators, $H_V=-\Delta+V$, where $V$ is periodic with respect to the lattice of translates $\Lambda=\mathbb{Z}^2$. The potential is also assumed to be real-valued, sufficiently regular, and such that, with respect to some origin of coordinates, inversion symmetric (even) and invariant under $\pi/2$ rotation. We present general conditions ensuring that the band structure of $H_V$ contains dispersion surfaces which touch at multiplicity two eigenvalues at the vertices (high-symmetry quasi momenta) of the Brillouin zone. Locally, the band structure consists of two intersecting dispersion surfaces described by a normal form which is $\pi/2$-rotationally invariant, and to leading order homogeneous of degree two. Furthermore, the effective dynamics of wave-packets, which are spectrally concentrated near high-symmetry quasi momenta, is given by a system of coupled Schrödinger equations with indefinite effective mass tensor. For small amplitude potentials, $\varepsilon V$ with $\varepsilon$ small or weak coupling, certain distinguished Fourier coefficients of the potential control which of the low-lying dispersion surfaces (first four) of $H^\varepsilon=H_{_{\varepsilon V}}$ intersect and have the above local behavior. The existence of quadratically touching dispersion surfaces with the above properties persists for all real $\varepsilon$, without restriction on the size of $\varepsilon$, except for $\varepsilon$ in a discrete set. Our results apply to periodic superpositions of spatially localized “atomic potentials” centered on the square ($\mathbb{Z}^2$) and Lieb lattices. We show, in particular, that the well-known conical plus flat-band structure of the three dispersion surfaces of the Lieb lattice tight-binding model does not persist in the corresponding Schrödinger operator with finite depth potential wells. Finally, we corroborate our analytical results with extensive numerical simulations. The present results are the $\mathbb{Z}^2$-analogue of results obtained for conical degenerate points (Dirac points) in the band structure for honeycomb structures.