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Fractal geometries can host topological phases of matter, revealing protected boundary and corner states without magnetic fields. Isospectral reduction simplifies fractal structures, enabling new topological material designs.

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Area of Science:

  • Condensed Matter Physics
  • Materials Science
  • Theoretical Physics

Background:

  • Fractal geometries exhibit self-similarity and noninteger dimensions, offering unique properties for exploring exotic states of matter.
  • Topological phases of matter are typically realized with specific driving mechanisms like magnetic fields or spin-orbit coupling.

Purpose of the Study:

  • To introduce a theoretical framework for identifying topological phases in fractal geometries.
  • To demonstrate that fractal structures can support topological phases without conventional driving mechanisms.

Main Methods:

  • Leveraging isospectral reduction to simplify complex fractal structures.
  • Analyzing the emergence of topologically protected boundary and corner states within these simplified models.

Main Results:

  • Fractal geometries can intrinsically host topological phases, characterized by protected boundary and corner states.
  • The proposed framework, based on isospectral reduction, is broadly applicable to various fractal systems.
  • Topological phases may naturally exist in naturally occurring fractal materials.

Conclusions:

  • Fractal-based topological materials can be designed and explored, expanding the field of topological matter.
  • This work provides a new perspective on the interplay between fractal geometry and topological physics.
  • Opens new avenues for theoretical and experimental research into topology in complex systems.