Band Unfolding via the Quadratic Pseudospectrum

TL;DR

This paper introduces a novel band unfolding method using a pseudospectral approach to analyze electronic structures in systems lacking perfect periodicity, applicable to aperiodic, disordered, and finite materials.

Contribution

It generalizes traditional band theory to non-periodic systems by employing a pseudospectral framework that identifies approximate eigenstates and isolates bulk properties.

Findings

Successfully applied to a Fibonacci chain, revealing dispersive features and spectral gaps.

Provides a systematic way to distinguish bulk states from boundary-localized states.

Enables momentum-resolved analysis in systems where conventional methods fail.

Abstract

Band theory provides the foundation for understanding electronic structure in crystalline materials, but its reliance on exact translational symmetry limits its applicability to systems with defects, disorder, incommensurate modulations, or large unit cells. Here, we introduce a band unfolding framework that directly generalizes traditional band theory to systems where exact periodicity is absent, and which remains well-defined for both aperiodic and finite systems. To do so, we employ a pseudospectral approach to identify approximate joint eigenvectors of a system's Hamiltonian and translation operators, thereby yielding an unfolded band structure whose features are directly connected to the manifestation of approximate extended states simultaneously localized in energy and crystalline momentum. To reveal bulk-only spectral phenomena in finite systems, we further show that this pseudospectral framework naturally accommodates additional operators that suppress contributions from boundary-localized states, enabling the systematic isolation of intrinsic bulk behavior. We benchmark the scheme on several representative systems in one and two dimensions, including a Fibonacci chain, where our approach is able to both reveal a dispersive envelope while preserving the underlying hierarchy of spectral gaps. Looking forward, this pseudospectral approach may yield a broad framework for predicting momentum-resolved material responses in aperiodic, disordered, and finite systems where conventional band-theoretic methods are not applicable.