Sufficient conditions for localized vibrational modes in one- and two-dimensional discrete lattices

TL;DR

The paper establishes that arbitrarily weak defects that reduce the net mass in 1D and 2D discrete monatomic lattices with nearest-neighbor coupling generate localized phonon modes. It employs a variational min–max framework with dimension-specific trial functions—$v^*_n = \alpha^{|n|}$ in 1D and $v^*_{n,m} = (-1)^{n+m} e^{-(n^2+m^2+1)^\alpha}$ in 2D—to show an eigenfrequency exits the bulk spectrum, i.e., $\omega^2 > \omega_{\max}^2$, ensuring localization. The analysis relies on invariance of the essential spectrum under localized perturbations and uses Rayleigh quotients to bound eigenvalues outside the continuous spectrum. The results extend prior continuous-wave localization and provide a generalizable framework for more complex lattices, interactions, and defect geometries, suggesting broad applicability to discrete phonon systems and related wave problems.

Abstract

This paper presents a rigorous proof that arbitrarily weak perturbations produce localized vibrational (phonon) modes in one- and two-dimensional discrete lattices, inspired by analogous results for the Schr{ö}dinger and Maxwell equations, and complementing previous explicit solutions for specific perturbations (e.g., decreasing a single mass). In particular, we study monatomic crystals with nearest-neighbor harmonic interactions, corresponding to square lattices of masses and springs, and prove that arbitrary localized perturbations that decrease the net mass lead to localized vibrating modes. The proof employs a straightforward variational method that should be extensible to other discrete lattices, interactions, and perturbations.

Figures (3)

• Figure 1: Schematic monotomic lattices with masses $M$, spring constants $J$ (harmonic nearest-neighbor interactions), and period $a$. Localized perturbations: "light" ($< M$, red) and "heavy" ($> M$, blue) defect masses. (a) 1d lattice. (b) 2d square lattice.
• Figure 2: Plot of the well-known dispersion relation Ashcroft for the unperturbed 1d monatomic lattice. When the system is perturbed (e.g., decreasing some masses), new discrete frequencies can arise outside of this continuous spectrum, corresponding to localized states.
• Figure 3: Some 1d examples of perturbations and their corresponding localized modes. (a) Single light-mass perturbation ($M_{\textnormal{light}} = 0.5M$); (b) Two light-mass, one heavy-mass perturbation ($M_{\textnormal{light}} = 0.5M$, $M_{\textnormal{heavy}} = 2M$); (c) single light-mass weak perturbation ($M_{\textnormal{light}} = 0.99M$). Eigenvectors were computed numerically by a Lanczos method Trefethen97KrylovKit from a sparse-matrix representation of $\hat{T}$ truncated to a finite supercell of 1000 masses (with Dirichlet boundaries $u_0 = u_{1001}=0$), much larger than the localization length of the bound modes.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2