Topology in the Sierpiński-Hofstadter problem

Abstract

Using the Sierpiński carpet and gasket, we investigate whether fractal lattices embedded in two-dimensional space can support topological phases when subjected to a homogeneous external magnetic field. To this end, we study the localization property of eigenstates, the Chern number, and the evolution of energy level statistics when disorder is introduced. Combining these theoretical tools, we identify regions in the phase diagram of both the carpet and the gasket, for which the systems exhibit properties normally associated to gapless topological phases with a mobility edge.

arXiv comment9 pages, 8 figures

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• Observation of nonlinear fractal higher-order topological insulator
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• Universal quantum transport and impurity band super metallicity in self-similar graphene carpets
• Quantum transport in flat bands and super-metallicity
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• Non-Hermitian Quantum Fractals
• Laughlin topology on fractal lattices without area law entanglement
• Corner and Edge States in Topological Sierpinski Carpet Systems
• Anomalous quantum transport in fractal lattices
• Loop current states and their stability in small fractal lattices of Bose-Einstein condensates
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• Josephson effect in a fractal geometry
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• Homogeneous Electron Liquid in Arbitrary Dimensions: Exchange and Correlation Using the Singwi-Tosi-Land-Sjölander Approach
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• Projected branes as platforms for crystalline, superconducting, and higher-order topological phases