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Topological effects manifest in a variety of lattice geometries. While square lattices, due to their simplicity, have been used for models supporting nontrivial topology, several exotic topological phenomena such as Dirac points, Weyl points, and Haldane phases are most commonly supported by non-square lattices. Examples of prototypical non-square lattices include the honeycomb lattice of graphene and 2D materials, and the Kagome lattice, both of which break fundamental symmetries and can exhibit quantized transport, especially when long-range hoppings and gauge fields are incorporated. The challenge of controllably realizing such long-range hoppings and gauge fields has motivated a large body of research focused on harnessing lattices encoded in synthetic dimensions. Photons in particular have many internal degrees of freedom and hence show promise for implementing these synthetic dimensions; however, most photonic synthetic dimensions have hitherto created 1D or 2D square lattices. Here we show that non-square lattice Hamiltonians such as the Haldane model and its variations can be implemented using Floquet synthetic dimensions. Our construction uses dynamically modulated ring resonators and provides the capacity for directk-space engineering of lattice Hamiltonians. Thisk-space construction lifts constraints on the orthogonality of lattice vectors that make square geometries simpler to implement in lattice-space constructions and instead transfers the complexity to the engineering of tailored, complex Floquet drive signals. We simulate topological signatures of the Haldane and the brick-wall Haldane model and observe them to be robust in the presence of external optical drive and photon loss, and discuss unique characteristics of their topological transport when implemented on these Floquet lattices. Our proposal demonstrates the potential of driven-dissipative Floquet synthetic dimensions as a new architecture fork-space Hamiltonian simulation of high-dimensional lattice geometries, supported by scalable photonic integration, that lifts the constraints of several existing platforms for topological photonics and synthetic dimensions.
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Lattice Green's Functions
Computes lattice Green's functions in two dimensions for square lattices. The method is a set of recurrence relations implemented in high-precision arithmetic. The square lattice results were needed for a particular research project, which is why this was developed, but similar recursion relations are available for three-dimensional lattices and the same method should work.
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- Award ID(s):
- 1810196
- PAR ID:
- 10454971
- Publisher / Repository:
- Texas Data Repository
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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