[Paper Review] Discrete Bethe--Sommerfeld Conjecture for Triangular, Square, and Hexagonal Lattices

Jake Fillman, Rui Han|arXiv (Cornell University)|Jun 6, 2018

Spectral Theory in Mathematical Physics24 references3 citations

TL;DR

This paper proves the discrete Bethe-Sommerfeld conjecture for triangular, hexagonal, and square lattices with next-nearest-neighbor interactions by analyzing small periodic potentials. Using a perturb-and-count technique based on Floquet theory, it identifies exceptional energies where gaps may open and provides sharp arithmetic criteria on periods for gap formation, showing that at most finitely many gaps open perturbatively, with explicit examples achieving maximal gap counts and scaling laws for gap lengths as the coupling constant vanishes.

ABSTRACT

We study discrete Schrödinger operators on the graphs corresponding to the triangular lattice, the hexagonal lattice, and the square lattice with next-nearest neighbor interactions. For each of these lattice geometries, we analyze the behavior of small periodic potentials. In particular, we provide sharp bounds on the number of gaps that may perturbatively open, we describe sharp arithmetic criteria on the periods that ensure that no gaps open, and we characterize those energies at which gaps may open in the perturbative regime. In all three cases, we provide examples that open the maximal number of gaps and estimate the scaling behavior of the gap lengths as the coupling constant goes to zero.

Motivation & Objective

• To extend the discrete Bethe-Sommerfeld conjecture to non-square lattices, including triangular, hexagonal, and square lattices with next-nearest-neighbor interactions.
• To determine the exceptional energies at which gaps may open under small periodic perturbations in these lattice geometries.
• To establish sharp arithmetic conditions on the periods of the potential that determine whether gaps open perturbatively.
• To construct explicit potentials that realize the maximal number of perturbatively opened gaps and estimate the scaling of gap lengths with the coupling constant.
• To resolve geometrically dependent band-touching phenomena in free Laplacians on these lattices, particularly for rational fluxes.

Proposed method

• Adapts the perturb-and-count technique from Han–Jitomirskaya (2017) to non-square lattices, using Floquet theory for periodic Schrödinger operators on periodic graphs.
• Employs a contradiction-based argument: assumes two bands overlap trivially at energy E, then shows inconsistent eigenvalue counting under perturbations along different directions.
• For exceptional energies, constructs two or more perturbation directions with differing quadratic terms in the eigenvalue expansion to detect gap formation.
• Uses Taylor expansions of Floquet eigenvalues up to λ⁹ order to bound the determinant of the perturbed Hamiltonian and prove gap existence or absence.
• Applies the AM-GM inequality and careful estimation of trigonometric and polynomial terms (Y_j(s)) to lower-bound the determinant and ensure positivity in the spectral gap region.
• For upper bounds on gap length, evaluates the determinant at specific Floquet parameters (e.g., (π,0)) to show sign changes, proving the gap contains energies 1±λ/4.

Experimental results

Research questions

• RQ1 For the triangular, hexagonal, and EHM lattices, at which exceptional energies can gaps open under small periodic perturbations?
• RQ2 What arithmetic conditions on the periods of the potential determine whether a gap opens perturbatively at these exceptional energies?
• RQ3 What is the maximal number of gaps that can be opened by small periodic potentials on these lattices, and can this maximum be achieved?
• RQ4 How do the lengths of perturbatively opened gaps scale with the coupling constant λ as λ→0?
• RQ5 What is the mechanism behind band-touching in free Laplacians on these lattices, particularly for rational fluxes?

Key findings

• The discrete Bethe-Sommerfeld conjecture holds for the triangular, hexagonal, and EHM lattices: only finitely many gaps open under small periodic potentials.
• For all three lattices, gaps can only open at specific exceptional energies, determined by the lattice geometry and symmetry.
• Sharp arithmetic criteria are derived: for the triangular lattice, gaps open only if the period satisfies certain Diophantine conditions; similar criteria are given for the hexagonal and EHM lattices.
• Explicit potentials are constructed that open the maximal number of gaps allowed by the theory, confirming the sharpness of the bounds.
• The length of the perturbatively opened gaps scales as λ⁶ for the EHM lattice, with upper and lower bounds established via determinant estimates.
• For the EHM lattice, a novel construction shows that approximately 2/3 of degenerate eigenvalues move upward under a specific perturbation direction, resolving a symmetry obstruction in the perturb-and-count method.