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

Abstract

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 onedimensional 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. arXiv:1607.06577v2 [quant-ph]

Local symmetry theory of resonator structures for the real-space control of edge states in binary aperiodic chains

“…In this work we have given an explanation for this finding at high contrast, and we believe that the study of local symmetries in complex setups is a very promising field with rich perspectives and potential applications. The recently established framework of local symmetries 52,53,56,57,92,93 provides dedicated tools for this purpose, and extensions of it are of immediate relevance. In this line, our work may enable the local-symmetry assisted design of novel optical devices that support desired quasiband structures and strongly localized edge states at prescribed energies, offering exciting opportunities to control light-matter coupling in complex aperiodic environments.…”

“…Overall, the perturbation theoretical treatment demonstrates the crucial role of local reflection symmetries in the eigenstate localization profiles of binary aperiodic chains. A promising direction of research would thus be to treat this class of systems within the recently developed theoretical framework of local symmetries 52,53,56 .…”

Designing pretty good state transfer via isospectral reductions

“…3 (a1), which is invariant under the permutation of sites 2 and 4. Given the high number of local symmetries in latently symmetric setups, it would be interesting to investigate such systems under the recently established framework of local symmetries [20][21][22][23][24][25], which provides dedicated tools for the analysis of such setups.…”

“…To extract P ± (ξ , λ), we test whether p ± (ξ , λ), given by the respective numerators of Eqs. (24) and (25), have leading order coefficients +1 and that p ± (ξ , λ)/q ± (ξ , λ) are irreducible fractions. The latter is indeed the case, but the leading order coefficients are −1.…”

Compact localized states and flat bands from local symmetry partitioning

“…on the invariance of a subset of matrix elements under a site permutation. In general, local symmetries of the underlying Hamiltonian are indirectly encoded into its eigenstates, as has been demonstrated recently in various contexts [30][31][32][33][34][35][36][37][38]. However, not every locally symmetric system features CLSs, and a systematic framework linking a theory of local symmetries to the formation and control of both CLSs and the resulting flat bands is still missing.…”