Nonlocal discrete continuity and invariant currents in locally symmetric effective Schr\"odinger arrays

TL;DR

This paper introduces a formalism linking nonlocal current continuity to local symmetries in discrete Schrödinger systems, revealing how symmetry breaking affects currents and enabling new insights into wave behavior in symmetric arrays.

Contribution

It develops a novel framework connecting nonlocal currents with local symmetries, generalizing global mappings, and applies it to photonic arrays with both Hermitian and non-Hermitian Hamiltonians.

Findings

Nonlocal currents are invariant within symmetric domains for stationary states.

Symmetry-related wave amplitudes can be mapped using the developed formalism.

Perfect transmission states correspond to aligned invariant currents in symmetry domains.

Abstract

We develop a formalism relating nonlocal current continuity to spatial symmetries of subparts in discrete Schr\"odinger systems. Breaking of such local symmetries hereby generates sources or sinks for the associated nonlocal currents. The framework is applied to locally inversion-(time-) and translation-(time-) symmetric one-dimensional photonic waveguide arrays with Hermitian or non-Hermitian effective tight-binding Hamiltonians. For stationary states the nonlocal currents become translationally invariant within symmetric domains, exposing different types of local symmetry. They are further employed to derive a mapping between wave amplitudes of symmetry-related sites, generalizing also the global Bloch and parity mapping to local symmetry in discrete systems. In scattering setups, perfectly transmitting states are characterized by aligned invariant currents in attached symmetry domains, whose vanishing signifies a correspondingly symmetric density. For periodically driven arrays, the invariance of the nonlocal currents is retained on period average for quasi-energy eigenstates. The proposed theory of symmetry-induced continuity and local invariants may contribute to the understanding of wave structure and response in systems with localized spatial order.