This research proposal establishes a comprehensive framework for studying topological phases induced by quasiperiodic and fractal lattice modulations. It synthesizes recent theoretical advances and proposes a rigorous methodology bridging theoretical predictions with experimental realizations across photonic, cold atom, and electronic platforms. The framework encompasses unified tight-binding Hamiltonians, real-space topological invariant calculations using Chern markers and Bott indices, and numerical protocols for spectral analysis. Key contributions include identifying research gaps in higher-order topological phases on fractal geometries and providing a roadmap for topological quantum device applications.
Key findings
Development of a unified tight-binding Hamiltonian framework for describing topological states in aperiodic lattices
Real-space topological invariant calculations using Chern markers and Bott indices enable characterization without translational symmetry
Identification of critical research gaps including higher-order topological phases in fractal geometries and non-Hermitian effects
Comprehensive experimental implementation strategies for photonic waveguide arrays, cold atom optical lattices, and electronic circuit networks
Establishment of a roadmap advancing the field from fundamental understanding to practical topological quantum devices
Limitations & open questions
Framework primarily addresses single-particle physics with limited treatment of many-body interaction effects
Experimental realization of fractal topological states faces significant technical challenges in fabrication and measurement
Non-Hermitian generalizations and three-dimensional extensions require further theoretical development