Quadratic pseudospectrum for identifying localized states

Abstract

We examine the utility of the quadratic pseudospectrum in photonics and condensed matter. Specifically, the quadratic pseudospectrum represents a method for approaching systems with incompatible observables, as it both minimizes the "eigen-error" in the joint approximate spectrum of the incompatible observables and does not increase the system's computational complexity. Moreover, we derive an important estimate relating the Clifford and quadratic pseudospectra. Finally, we prove that the quadratic pseudospectrum is local, and derive the bounds on the errors that are incurred by truncating the system in the vicinity of where the pseudospectrum is being calculated.

20 pages, 9 figures

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• Tutorial: Classifying Photonic Topology Using the Spectral Localizer and Numerical $K$-Theory
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• Real-space topological invariant for time-quasiperiodic Majoranas
• Versatile Control of Nonlinear Topological States in Non-Hermitian Systems
• Even spheres as joint spectra of matrix models
• Local and energy-resolved topological invariants for Floquet systems
• Gapless superconductivity and its real-space topology in quasicrystals
• Topological photonic crystal fibre
• Laser-induced topological phases in monolayer amorphous carbon
• Quantitative measure of topological protection in Floquet systems through the spectral localizer
• Semiconductor Meta-Graphene and Valleytronics
• Dimensional crossover of class D real-space topological invariants