Fractal sets of dual topological quantum numbers

Topological Order, Quantum Codes and Quantum Computation on Fractal Geometries

We investigate topological order on fractal geometries embedded in $n$ dimensions. In particular, we diagnose the existence of the topological order through the lens of quantum information and geometry, i.e., via its equivalence to a quantum error-correcting code with a macroscopic code distance or the presence of macroscopic systoles in systolic geometry. We first prove a no-go theorem that $\mathbb{Z}_N$ topological order cannot survive on any fractal embedded in 2D. For fractal lattice models embedded in 3D or higher spatial dimensions, $\mathbb{Z}_N$ topological order survives if the bound…

具有自相似或分形结构介质的散射

Topological properties and a multiplicative Bloch-Floquet-Zak transform for scattering by self-similar or fractal media

该研究构建了自相似或分形介质波散射的拓扑框架，通过边界积分模型与Riesz投影实现严格拓扑约化，得到正则单形构型的非零环绕数公式等结果。

分形节点能带结构

Fractal Nodal Band Structures

本文报告了在厄米与非厄米拓扑能带结构中分形特征值简并与费米面的出现，揭示了非整数维度下的拓扑现象，对黑洞物理及超导等费米面不稳定性驱动现象具有深远意义。

Butterflies and topological quantum numbers

The Hofstadter model illustrates the notion of topological quantum numbers and how they account for the quantization of the Hall conductance. It gives rise to colorful fractal diagrams of butterflies where the colors represent the topological quantum numbers.

On the dual topological quantum numbers filling factors

We consider recent experimental results [W. Pan {\it et al}, Phys. Rev. Lett. {\bf 90}, 016801 (2003)] for occurrence of the fractional quantum Hall effect-FQHE under the perspective of our formulation in terms of {\it fractons}. These objects carry rational or irrational values of spin and satisfy a {\it fractal distribution function} associated with a {\it fractal von Neumann entropy}. According to our approach the {\it FQHE occurs in pairs of dual topological quantum numbers fillings factors} and this geometrical character comes from the {\it connection betwenn the fractal parameter or Haus…

The Hausdorff dimension of fractal sets and fractional quantum Hall effect

We consider Farey series of rational numbers in terms of {\it fractal sets} labeled by the Hausdorff dimension with values defined in the interval 1$ $$ < $$ $$h$$ $$ <$$ $$ 2$ and associated with fractal curves. Our results come from the observation that the fractional quantum Hall effect-FQHE occurs in pairs of {\it dual topological quantum numbers}, the filling factors. These quantum numbers obey some properties of the Farey series and so we obtain that {\it the universality classes of the quantum Hall transitions are classified in terms of $h$}. The connection between Number Theory and Phy…