Degeneracy doubling and sublattice polarization in strain-induced pseudo-Landau levels

The degeneracy and spatial support of pseudo-Landau levels (pLLs) in strained honeycomb lattices systematically depends on the geometry -- for instance, in hexagonal and rectangular flakes the 0th pLL displays a twofold increased degeneracy, while the characteristic sublattice polarization of the 0th pLL is only fully realized in a zigzag-terminated triangle. These features are dictated by algebraic constraints in the atomistic theory, and signify a departure from the standard picture in which all qualitative differences between pLLs and Landau levels induced by a magnetic field trace back to the valley-antisymmetry of the pseudomagnetic field.Comment: 5 pages, 2 figure

New algebraic relationships between tight binding models

In this thesis, we present a new perspective on tight binding models. Utilising the rich algebraic toolkit provided by a combination of graph and matrix theory allows us to explore tight binding systems related through polynomial relationships. By utilising ring operations of weighted digraphs through intermediate König digraph representations, we establish a polynomial algebra over finite and infinite periodic graphs, analogous to polynomial operations on adjacency matrices. Exploring the microscopic and macroscopic behaviour of polynomials in a graph-theoretic setting, we reveal elegant relationships between the symmetrical, topological, and spectral properties of a parent graph G and its family of child graphs p(G). Drawing a correspondence between graphs and tight binding models, we investigate deep-rooted connections between different quantum systems, providing a fresh angle from which to view established tight binding models. Finally, we visit topological chains, demonstrate how their properties relate to more trivial underlying chains through effective “square root” operations, and provide new insights into their spectral characteristics

Topological tight-binding models from nontrivial square roots

We describe a versatile mechanism that provides tight-binding models with an enriched, topologically nontrivial band structure. The mechanism is algebraic in nature, and leads to tight-binding models that can be interpreted as a nontrivial square root of a parent lattice Hamiltonian—in analogy to the passage from a Klein-Gordon equation to a Dirac equation. In the tight-binding setting, the square-root operation admits to induce spectral symmetries at the expense of broken crystal symmetries. As we illustrate in detail for a simple one-dimensional example, the emergent and inherited spectral symmetries equip the energy gaps with independent topological quantum numbers that control the formation of topologically protected states. We also describe an implementation of this system in silicon photonic structures, outline applications in higher dimensions, and provide a general argument for the origin and nature of the emergent symmetries, which are typically nonsymmorphic

Flux-induced midgap states between strain-engineered flat bands

Half-integer quantized flux vortices appear in honeycomb lattices when the signs of an odd number of couplings around a plaquette are inverted. We show that states trapped at these vortices can be isolated by applying inhomogeneous strain to the system. A vortex then results in localized midgap states lying between the strain-induced pseudo-Landau levels, with 2n+1midgap states appearing between the nth and the (n+1)th level. These states are well-defined spectrally isolated and spatially localized excitations that could be realized in electronic and photonic systems based on graphenelike honeycomb lattices. In the context of Kitaev's honeycomb model of interacting spins, the mechanism improves the localization of non-Abelian anyons in the spin-liquid phase, and reduces their mutual interactions. The described states also serve as a testbed for fundamental physics in the emerging low-energy theory, as the correct energies and degeneracies of the excitations are only replicated if one accounts for the effective hyperbolic geometry induced by the strain. We further illuminate this by considering the effects of an additional external magnetic field, resulting in a characteristic spatial dependence that directly maps out the inhomogeneous metric of the emerging hyperbolic space

Universal sign control of coupling in tight-binding lattices

We present a method of locally inverting the sign of the coupling term in tight-binding systems, by means of inserting a judiciously designed ancillary site and eigenmode matching of the resulting vertex triplet. Our technique can be universally applied to all lattice configurations, as long as the individual sites can be detuned. We experimentally verify this method in laser-written photonic lattices and confirm both the magnitude and the sign of the coupling by interferometric measurements. Based on these findings, we demonstrate how such universal sign-flipped coupling links can be embedded into extended lattice structures to impose a Z2-gauge transformation. This opens a new avenue for investigations on topological effects arising from magnetic fields with aperiodic flux patterns or in disordered systems

Experimental Realization of Multiple Topological Edge States in a 1D Photonic Lattice

Topological photonic systems offer light transport that is robust against defects and disorder, promising a new generation of chip-scale photonic devices and facilitating energy-efficient on-chip information routing and processing. However, present quasi one dimensional (1D) designs, such as the Su–Schrieffer–Heeger and Rice–Mele models, support only a limited number of nontrivial phases due to restrictions on dispersion band engineering. Here, a flexible topological photonic lattice on a silicon photonic platform is experimentally demonstrated that realizes multiple topologically nontrivial dispersion bands. By suitably setting the couplings between the 1D waveguides, different lattices can exhibit the transition between multiple different topological phases and allow the independent realization of the corresponding edge states. Heterodyne measurements clearly reveal the ultrafast transport dynamics of the edge states in different phases at a femtosecond scale, validating the designed topological features. The study equips topological models with enriched edge dynamics and considerably expands the scope to engineer unique topological features into photonic, acoustic, and atomic systems

Flux-induced midgap states between strain-engineered flat bands

Half-integer quantized flux vortices appear in honeycomb lattices when the signs of an odd number of couplings around a plaquette are inverted. We show that states trapped at these vortices can be isolated by applying inhomogeneous strain to the system. A vortex then results in localized midgap states lying between the strain-induced pseudo-Landau levels, with 2n+1 midgap states appearing between the nth and the (n+1)th level. These states are well-defined spectrally isolated and spatially localized excitations that could be realized in electronic and photonic systems based on graphenelike honeycomb lattices. In the context of Kitaev's honeycomb model of interacting spins, the mechanism improves the localization of non-Abelian anyons in the spin-liquid phase, and reduces their mutual interactions. The described states also serve as a testbed for fundamental physics in the emerging low-energy theory, as the correct energies and degeneracies of the excitations are only replicated if one accounts for the effective hyperbolic geometry induced by the strain. We further illuminate this by considering the effects of an additional external magnetic field, resulting in a characteristic spatial dependence that directly maps out the inhomogeneous metric of the emerging hyperbolic space. © 2023 American Physical Society.11Nsciescopu

Edge-state dynamics in a one-dimensional topological photonic lattice of multiple quantum numbers

© OSA 2018. By manipulating the couplings in a flexible topological waveguide lattice with multiple nontrivial dispersion bands, we demonstrate the independent control of edge states associated with different quantum numbers through ultrafast heterodyne measurements