Journal reference: Phys. Rev. B 97, 035161 (2018) We propose a framework for the connection between local symmetries of discrete Hamiltonians and the design of compact localized states. Such compact localized states are used for the creation of tunable, local symmetry-induced bound states in an energy continuum and flat energy bands for periodically repeated local symmetries in one-and two-dimensional lattices. The framework is based on very recent theorems in graph theory which are here employed to obtain a block partitioning of the Hamiltonian induced by the symmetry of a given system under local site permutations. The diagonalization of the Hamiltonian is thereby reduced to finding the eigenspectra of smaller matrices, with eigenvectors automatically divided into compact localized and extended states. We distinguish between local symmetry operations which commute with the Hamiltonian, and those which do not commute due to an asymmetric coupling to the surrounding sites. While valuable as a computational tool for versatile discrete systems with locally symmetric structures, the approach provides in particular a unified, intuitive, and efficient route to the flexible design of compact localized states at desired energies.

We study the scattering resonances of one-dimensional deterministic aperiodic chains of electric dipoles using the vectorial Green's matrix method, which accounts for both short-and long-range electromagnetic interactions in open scattering systems. We discover the existence of edge-localized scattering states within fractal energy gaps with characteristic topological band structures. Notably, we report and characterize edge-

We propose a real-space approach explaining and controlling the occurrence of edge-localized gap states between the spectral quasibands of binary tight binding chains with deterministic aperiodic long-range order. The framework is applied to the Fibonacci, Thue-Morse and Rudin-Shapiro chains, representing different structural classes. Our approach is based on an analysis of the eigenstates at weak inter-site coupling, where they are shown to generically localize on locally reflection-symmetric substructures which we call local resonators. A perturbation theoretical treatment demonstrates the local symmetries of the eigenstates. Depending on the degree of spatial complexity of the chain, the proposed local resonator picture can be used to predict the occurrence of gap-edge states even for stronger couplings. Moreover, we connect the localization behavior of a given eigenstate to its energy, thus providing a quantitative connection between the real-space structure of the chain and its eigenvalue spectrum. This allows for a deeper understanding, based on local symmetries, of how the energy spectra of binary chains are formed. The insights gained allow for a systematic analysis of aperiodic binary chains and offers a pathway to control structurally induced edge states.

We propose modulation protocols designed to generate, store and transfer compact localized states in a quantum network. Induced by parameter tuning or local reflection symmetries, such states vanish outside selected domains of the complete system and are therefore ideal for information storage. Their creation and transfer is here achieved either via amplitude phase flips or via optimal temporal control of inter-site couplings. We apply the concept to a decorated, locally symmetric Lieb lattice where one sublattice is dimerized, and also demonstrate it for more complex setups. The approach allows for a flexible storage and transfer of states along independent paths in lattices supporting flat energetic bands. The generic network and protocols proposed can be utilized in various physical setups such as atomic or molecular spin lattices, photonic waveguide arrays, and acoustic setups.

Local symmetries are spatial symmetries present in a subdomain of a complex system. By using and extending a framework of so-called non-local currents that has been established recently, we show that one can gain knowledge about the structure of eigenstates in locally symmetric setups through a Kirchhoff-type law for the non-local currents. The framework is applicable to all discrete planar Schrödinger setups, including those with non-uniform connectivity. Conditions for spatially constant non-local currents are derived and we explore two types of locally symmetric subsystems in detail, closed-loops and one-dimensional open ended chains. We find these systems to support locally similar or even locally symmetric eigenstates.Local symmetries, i.e. symmetries that are only present in a spatial part of a given system, are ubiquitous in nature, a popular example being quasicrystals [1][2][3][4][5]. Due to the long-range order of quasicrystals, one can always find structures of equal structure which can be described by local symmetries. Other examples are large molecules [6,7] and, in general, systems where the global symmetry is broken due to defects. Beyond this, a second class of systems are those which are specifically designed in such a way that they possess local symmetries. Examples therefore are photonic multilayers [8][9][10][11] or photonic waveguide arrays [12][13][14][15].Despite their widespread presence in both natural and artificial physical systems, a systematic and in-depth treatment of the influence of local symmetries on a system's behaviour is still missing. A reason for this might lie in the tools used. In quantum systems, for example, the treatment of symmetries is based on the determination of the Hamiltonian group, i.e. the set of operators commuting with the Hamiltonian of the considered system. These operators usually refer to global symmetry transforms such as translation or reflection. The corresponding Hamiltonian eigenstates are also eigenstates of the irreducible representations of the symmetry operators, leading to states with definite parity (reflection) [16] or Bloch-states (discrete translation) [17]. For operatorsΣ L describing local symmetries valid only in a limited spatial domain, however, we have [Ĥ,Σ L ] = 0 in general. Does this mean that local symmetries do not affect the eigenstates of the Hamiltonian? Or is it possible to gain additional knowledge about the structure of eigenstates in locally symmetric systems using other means?Recently a framework for the treatment of local symmetries in one-dimensional discrete setups has been established [18], motivated by corresponding results for discrete local symmetries in continuous one-dimensional systems [19][20][21][22]. The very spirit of this framework is to use local symmetries to define new quantities, so-called non-local currents obeying suitably defined continuity equations. For eigenstates in one-dimensional discrete systems, the non-local currents have two interesting properties: Firstly, for a given site the sum of a sour...