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We develop a formalism relating nonlocal current continuity to spatialsymmetries of subparts in discrete Schr\"odinger systems. Breaking of suchlocal symmetries hereby generates sources or sinks for the associated nonlocalcurrents. The framework is applied to locally inversion-(time-) andtranslation-(time-) symmetric one-dimensional photonic waveguide arrays withHermitian or non-Hermitian effective tight-binding Hamiltonians. For stationarystates the nonlocal currents become translationally invariant within symmetricdomains, exposing different types of local symmetry. They are further employedto derive a mapping between wave amplitudes of symmetry-related sites,generalizing also the global Bloch and parity mapping to local symmetry indiscrete systems. In scattering setups, perfectly transmitting states arecharacterized by aligned invariant currents in attached symmetry domains, whosevanishing signifies a correspondingly symmetric density. For periodicallydriven arrays, the invariance of the nonlocal currents is retained on periodaverage for quasi-energy eigenstates. The proposed theory of symmetry-inducedcontinuity and local invariants may contribute to the understanding of wavestructure and response in systems with localized spatial order.

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