Invariant currents and scattering off locally symmetric potential landscapes

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.

29 pages, 6 figures

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• PT-symmetry breaking and laser-absorber modes in optical scattering systems
• Slow sound propagation in lossy locally resonant periodic structures
• Invariants of broken discrete symmetries
• Local symmetries in one-dimensional quantum scattering
• Disordered one-dimensional photonic structures composed by more than two materials with the same optical thickness
• Systematic pathway to $\mathcal{P}\mathcal{T}$ symmetry breaking in scattering systems

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• Generalized continuity equations from two-field Schrödinger Lagrangians
• Super-Lagrangian and variational principle for generalized continuity equations
• Non-local divergence-free currents for the account of symmetries in two-dimensional wave scattering
• Duality of bounded and scattering wave systems with local symmetries
• Robustness of perfect transmission resonances to asymmetric perturbation
• Compact localized states of open scattering media