The Anderson transition in random graphs has raised great interest, partly because of its analogy with the many-body localization (MBL) transition. Unlike the latter, many results for random graphs are now well established, in particular the existence and precise value of a critical disorder separating a localized from an ergodic delocalized phase. However, the renormalization group flow and the nature of the transition are not well understood. In turn, recent works on the MBL transition have made the remarkable prediction that the flow is of Kosterlitz-Thouless type. Here we show that the Anderson transition on graphs displays the same type of flow. Our work attests to the importance of rare branches along which wave functions have a much larger localization length $\xi_\parallel$ than the one in the transverse direction, $\xi_\perp$. Importantly, these two lengths have different critical behaviors: $\xi_\parallel$ diverges with a critical exponent $\nu_\parallel=1$, while $\xi_\perp$ reaches a finite universal value ${\xi_\perp^c}$ at the transition point $W_c$. Indeed, $\xi_\perp^{-1} \approx {\xi_\perp^c}^{-1} + \xi^{-1}$, with $\xi \sim (W-W_c)^{-\nu_\perp}$ associated with a new critical exponent $\nu_\perp = 1/2$, where $\exp( \xi)$ controls finite-size effects. The delocalized phase inherits the strongly non-ergodic properties of the critical regime at short scales, but is ergodic at large scales, with a unique critical exponent $\nu=1/2$. This shows a very strong analogy with the MBL transition: the behavior of $\xi_\perp$ is identical to that recently predicted for the typical localization length of MBL in a phenomenological renormalization group flow. We demonstrate these important properties for a smallworld complex network model and show the universality of our results by considering different network parameters and different key observables of Anderson localization.

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随机图中安德森跃迁的关键性质：两参数标度理论、Kosterlitz-Thouless 型流和多体定位

随机图中的 Anderson 跃迁引起了极大的兴趣，部分原因是它与多体定位 (MBL) 跃迁类似。与后者不同，随机图的许多结果现在已经很好地建立起来，特别是区分局部和遍历离域相的临界无序的存在和精确值。然而，重整化群流和转变的性质还不是很清楚。反过来，最近关于 MBL 过渡的工作已经做出了显着的预测，即流动是 Kosterlitz-Thouless 类型的。在这里，我们展示了图表上的 Anderson 转换显示了相同类型的流。我们的工作证明了稀有分支的重要性，在这些分支上，波函数的定位长度 $\xi_\parallel$ 远大于横向方向的定位长度 $\xi_\perp$。重要的，这两个长度具有不同的临界行为：$\xi_\parallel$ 以临界指数 $\nu_\parallel=1$ 发散，而 $\xi_\perp$ 达到有限的通用值 ${\xi_\perp^c}$在过渡点 $W_c$。确实，$\xi_\perp^{-1} \approx {\xi_\perp^c}^{-1} + \xi^{-1}$，与 $\xi \sim (W-W_c)^{ -\nu_\perp}$ 与一个新的临界指数 $\nu_\perp = 1/2$ 相关联，其中 $\exp( \xi)$ 控制有限尺寸效应。离域相在小尺度上继承了临界状态的强非遍历特性，但在大尺度上是遍历的，具有唯一的临界指数 $\nu=1/2$。这与 MBL 转换非常相似：$\xi_\perp$ 的行为与最近预测的现象学重整化群流中 MBL 的典型定位长度的行为相同。