Nonlocal discrete continuity and invariant currents in locally symmetric effective Schrödinger arrays

• Published in: Annals of physics Vol. 385; pp. 623 - 649
• Main Authors: Morfonios, C.V., Kalozoumis, P.A., Diakonos, F.K., Schmelcher, P.
• Format: Journal Article
• Language: English
• Published: United States Elsevier Inc 01.10.2017
• Subjects: [formula omitted] symmetry; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CURRENT DENSITY; Current-density continuity; Discrete local symmetry; Discrete Schrödinger equation; MAPPING; ONE-DIMENSIONAL CALCULATIONS; PERIODICITY; Photonic waveguide array; SCHROEDINGER EQUATION; SYMMETRY; Current-density continuity; Discrete local symmetry; Discrete Schrödinger equation; Photonic waveguide array; PT symmetry
• ISSN: 0003-4916; 1096-035X
• DOI: 10.1016/j.aop.2017.07.019

We develop a formalism relating nonlocal current continuity to spatial symmetries of subparts in discrete Schrödinger 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. •Nonlocal current continuity related to subsystem symmetries in discrete Schrödinger arrays.•Application to 1D (non-) Hermitian bound and scattering tight-binding setups.•Local inversion/translation symmetries described by invariant nonlocal currents.•Generalized parity/Bloch mapping between local-symmetry-related sites via currents.•Time-averaged invariant currents of Floquet states in periodically driven arrays.