Branch-selective phonon localization through correlated sublattice disorder

Multiband lattices possess an internal degree of freedom absent from one-band Anderson models. At each Bloch wavenumber the acoustic and optical eigenvectors distribute their motion differently over the sublattices of the unit cell, and manufactured phononic lattices acquire tolerance-induced randomness separately on each sublattice, with tolerances that can be correlated within a cell. This paper develops a branch-resolved sublattice-participation formulation of weak-disorder localization for a grounded diatomic chain, building on the Lyapunov-exponent theory of correlated multicomponent media. To leading order in the disorder strength and inside an isolated passband, the localization exponent of each branch equals the disorder covariance projected through the squared Bloch participation weights of that branch and divided by the squared group velocity. The intra-cell cross-correlation enters this law through the backscattering phase between the two sublattices. An exact finite-chain eigensolution, branch-resolved random eigenvalue statistics and the weak-disorder law are derived and compared with transfer-matrix and Monte Carlo computation, which confirms the prefactor of the law across both band interiors. Moving a fixed disorder strength from one sublattice to the other reverses which branch localizes more strongly, so disorder placement becomes a design variable. Finite-chain eigenvalue statistics and localization weight the same correlation with different phase factors. Anticorrelated disorder therefore reduces eigenvalue scatter across a whole band, and it reduces localization near the zone centre and enhances it near the zone edge. The ensemble-mean response carries the same projection as a decoherence signature that depends on disorder placement. Disorder covariance in sublattice space thereby becomes a quantitative control for multiband phononic media, with direct extensions to more sublattices and to two dimensions.