随机图中Anderson转变的临界性质：双参数标度理论、Kosterlitz-Less型流和多体局域化
Critical properties of the Anderson transition in random graphs: two-parameter scaling theory, Kosterlitz-Thouless type flow and many-body localization
摘要：
随机图中的Anderson转变引起了人们极大的兴趣，部分原因是它与多体局域化（MBL）转变相似。与后者不同，随机图的许多结果现在已经很好地建立起来，特别是存在和精确值的关键障碍分离本地化的遍历离域阶段。然而，重整化群流和跃迁的性质还没有得到很好的理解。而最近关于MBL转捩的工作又作出了引人注目的预言，即流动是Kosterlitz-无边界层型的。在这里，我们表明，图上的安德森过渡显示相同类型的流。我们的工作证明了稀有分支的重要性，沿着这些分支，波函数的局部化长度$ξ_\parallel$比横向的局部化长度$ξ_\perp$大得多。重要的是，这两个长度具有不同的临界行为：$ξ_\parallel$以临界指数$ν_\parallel=1$发散，而$ξ_\perp$在转变点$W_c$处达到有限的普适值${ξ_\perp^c}$。实际上，$ξ_\perp^{-1} \approx {ξ_\perp^c}^{-1} + ξ^{-1}$，其中$ξ\sim (W-W_c)^{-ν_\perp}$与新的临界指数$ν_\perp = 1/2$相关，其中$\exp( ξ)$控制有限尺寸效应。离域相在短尺度下继承了临界区的强非遍历性质，但在大尺度下是遍历的，具有唯一的临界指数$ν=1/2$。这显示了与MBL跃迁的非常强的类比：$ξ_\perp$的行为与最近在唯象重整化群流中对MBL的典型局域化长度所预测的行为相同。我们证明了这些重要的性质的小世界复杂网络模型，并显示我们的结果的普遍性，考虑不同的网络参数和不同的关键可观的安德森本地化。
英文摘要：
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 $ξ_\parallel$ than the one in the transverse direction, $ξ_\perp$. Importantly, these two lengths have different critical behaviors: $ξ_\parallel$ diverges with a critical exponent $ν_\parallel=1$, while $ξ_\perp$ reaches a finite universal value ${ξ_\perp^c}$ at the transition point $W_c$. Indeed, $ξ_\perp^{-1} \approx {ξ_\perp^c}^{-1} + ξ^{-1}$, with $ξ\sim (W-W_c)^{-ν_\perp}$ associated with a new critical exponent $ν_\perp = 1/2$, where $\exp( ξ)$ 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 $ν=1/2$. This shows a very strong analogy with the MBL transition: the behavior of $ξ_\perp$ 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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