Topological Disorder in Spin Models on Hierarchical Lattices

The renormalization flow of the hierarchical Anderson model at weak disorder

We study the flow of the renormalized model parameters obtained from a sequence of simple transformations of the 1D Anderson model with long-range hierarchical hopping. Combining numerical results with a perturbative approach for the flow equations, we identify three qualitatively different regimes at weak disorder. For a sufficiently fast decay of the hopping energy, the Cauchy distribution is the only stable fixed-point of the flow equations, whereas for sufficiently slowly decaying hopping energy the renormalized parameters flow to a delta peak fixed-point distribution. In an intermediate r…

The effect of disorder on the hierarchical modularity in complex systems

We consider a system hierarchically modular, if besides its hierarchical structure it shows a sequence of scale separations from the point of view of some functionality or property. Starting from regular, deterministic objects like the Vicsek snowflake or the deterministic scale free network by Ravasz et al. we first characterize the hierarchical modularity by the periodicity of some properties on a logarithmic scale indicating separation of scales. Then we introduce randomness by keeping the scale freeness and other important characteristics of the objects and monitor the changes in the modul…

无序二维拓扑绝缘体中局部拓扑序参量的涨落与关联

Fluctuations and Correlations of Local Topological Order Parameters in Disordered Two-dimensional Topological Insulators

本文论证空间分辨拓扑标记可作为局部序参量，通过超胞数值模拟研究Haldane与Kane-Mele模型中无序驱动的拓扑相变，发现拓扑关联函数具有普适临界指数且能区分不同相变。

Spread of wave packets in disordered hierarchical lattices

We consider the spreading of the wave packet in the generalized Rosenzweig-Porter random matrix ensemble in the region of non-ergodic extended states $1<γ<2$. We show that despite non-trivial fractal dimensions $0 < D_{q}=2-γ<1$ characterize wave function statistics in this region, the wave packet spreading $\langle r^{2} \rangle \propto t^β$ is governed by the "diffusion" exponent $β=1$ outside the ballistic regime $t>τ\sim 1$ and $\langle r^{2}\rangle \propto t^{2}$ in the ballistic regime for $t<τ\sim 1$. This demonstrates that the multifractality exhibits itself only in {\it local} quantit…

Two-Dimensional Diffusion in the Presence of Topological Disorder

How topological defects affect the dynamics of particles hopping between lattice sites of a distorted, two-dimensional crystal is addressed. Perturbation theory and numerical simulations show that weak, short-ranged topological disorder leads to a finite reduction of the diffusion coefficient. Renormalization group theory and numerical simulations suggest that longer-ranged disorder, such as that from randomly placed dislocations or random disclinations with no net disclinicity, leads to subdiffusion at long times.

存在无序和长程耦合时的拓扑相

Topological phases in the presence of disorder and longer-range couplings

研究扩展Kitaev链中无序与长程耦合对一维拓扑超导相图的共同影响，推导Majorana零模条件并分析边缘模局域化行为。

一维机械准晶体中由无序驱动的拓扑相变

Disorder driven topological phase transitions in 1D mechanical quasicrystals

该研究通过分析准周期环境中的无序影响，揭示了一维机械准晶体中由无序驱动的拓扑相变及拓扑安德森绝缘体的出现。

Local disorder, topological ground state degeneracy and entanglement entropy, and discrete anyons

In this comprehensive study of Kitaev's abelian models defined on a graph embedded on a closed orientable surface, we provide complete proofs of the topological ground state degeneracy, the absence of local order parameters, compute the entanglement entropy exactly and characterise the elementary anyonic excitations. The homology and cohomolgy groups of the cell complex play a central role and allow for a rigorous understanding of the relations between the above characterisations of topological order.