Invariant currents and scattering off locally symmetric potential landscapes

TL;DR

This paper introduces invariant nonlocal currents in one-dimensional wave scattering with local symmetries, generalizing classical theorems and aiding analysis of complex aperiodic systems.

Contribution

It develops a framework of spatially invariant currents for locally symmetric potentials, extending Bloch and parity theorems to broken symmetries and enabling systematic scattering analysis.

Findings

Invariant currents characterize wave propagation in locally symmetric potentials.

Zero invariants indicate restored global symmetry.

Transfer matrix expressed via invariants aids in analyzing transmission spectra.

Abstract

We study the effect of discrete symmetry breaking in inhomogeneous scattering media within the framework of generic wave propagation. Our focus is on one-dimensional scattering potentials exhibiting local symmetries. We find a class of spatially invariant nonlocal currents, emerging when the corresponding generalized potential exhibits symmetries in arbitrary spatial domains. These invariants characterize the wave propagation and provide a spatial mapping of the wave function between any symmetry related domains. This generalizes the Bloch and parity theorems for broken reflection and translational symmetries, respectively. Their nonvanishing values indicate the symmetry breaking, whereas a zero value denotes the restoration of the global symmetry where the well-known forms of the two theorems are recovered. These invariants allow for a systematic treatment of systems with any local symmetry combination, providing a tool for the investigation of the scattering properties of aperiodic but locally symmetric systems. To this aim we express the transfer matrix of a locally symmetric potential unit via the corresponding invariants and derive quantities characterizing the complete scattering device which serve as key elements for the investigation of transmission spectra and particularly of perfect transmission resonances.