Mathematical Foundations of the Non-Hermitian Skin Effect

Area of Science:

• Physics
• Condensed Matter Physics
• Wave Phenomena

Background:

• The skin effect describes the tendency of waves to concentrate near the surface of a conductor.
• Non-Hermitian systems exhibit unique properties not found in their Hermitian counterparts.
• Subwavelength resonators offer unique electromagnetic and acoustic properties.

Purpose of the Study:

• To investigate the skin effect in a finite one-dimensional system of subwavelength resonators.
• To analyze the influence of a non-Hermitian imaginary gauge potential on eigenmode condensation.
• To establish a theoretical framework for understanding spectral properties in such systems.

Main Methods:

• Utilizing Toeplitz matrix theory to analyze the system's spectral properties.
• Introducing a generalized complex Brillouin zone for infinite periodic structures.
• Comparing systems with imaginary gauge potentials to those with complex material parameters.

Main Results:

• Proved the condensation of bulk eigenmodes at one edge of the finite resonator system.
• Computed spectral bands of the associated infinitely periodic structure using the complex Brillouin zone.
• Demonstrated that the spectral bands of the infinite system are the limit of finite systems.
• Showcased fundamental distinctions between non-Hermitian systems with imaginary gauge potentials and those with complex material parameters.

Conclusions:

• The skin effect in this non-Hermitian system leads to edge localization of bulk modes.
• The complex Brillouin zone provides a powerful tool for analyzing spectral properties of finite and infinite non-Hermitian systems.
• Non-Hermiticity introduced via imaginary gauge potentials results in distinct physics compared to complex material parameters.