Cayley树上的Anderson跃迁作为各种概率分布的行波临界点
Anderson transition on the Cayley tree as a traveling wave critical point for various probability distributions
摘要：
对于Cayley树上的Anderson局域化，我们研究了各种观测量作为无序强度$W$和世代数$N$的函数的统计特性。我们首先考虑Landauer传输$T_N$。在局部化阶段，其对数遵循行波形式$\ln T_N \simeq \bar{\ln T_N} + \ln t^*$，其中（i）无序平均值线性移动$\bar{\ln (T_N)} \simeq - \frac{N}{ξ_{loc}}$，并且局部化长度发散为具有$ν_{loc}=1$的$ξ_{loc} \sim (W-W_c)^{-ν_{loc}}$（ii）对于具有$0<β(W) \leq 1/2$的大$t^*$，变量$t^*$是具有幂律尾部$P^*(t^*) \sim 1/(t^*)^{1+β(W)}$的固定随机变量，使得$T_N$的所有整数矩由罕见事件支配。在非定域阶段，透射率$T_N$仍然是一个有限的随机变量$N \to \infty$，我们测量了近临界的本质奇异性$\bar{\ln (T)} \sim - | W_c-W |^{-κ_T}$与$κ_T \sim 0.25$。然后，我们考虑归一化本征态的统计特性，特别是熵和逆参与比（I.P.R.）。在定域相，典型的熵发散为$(W-W_c)^{- ν_S}$与$ν_S \sim 1.5$，而它在$N$在离域相线性增长。最后是I.P.R.我们解释如何密切相关的变量传播作为行波在离域阶段。总之，定域相和离域相都是以某些概率分布的行波传播为特征的，Anderson定域/离域转变对应于一个行波/非行波临界点.此外，我们的研究结果指向存在几个指数$ν$在临界状态。
英文摘要：
For Anderson localization on the Cayley tree, we study the statistics of various observables as a function of the disorder strength $W$ and the number $N$ of generations. We first consider the Landauer transmission $T_N$. In the localized phase, its logarithm follows the traveling wave form $\ln T_N \simeq \bar{\ln T_N} + \ln t^*$ where (i) the disorder-averaged value moves linearly $\bar{\ln (T_N)} \simeq - \frac{N}{ξ_{loc}}$ and the localization length diverges as $ξ_{loc} \sim (W-W_c)^{-ν_{loc}}$ with $ν_{loc}=1$ (ii) the variable $t^*$ is a fixed random variable with a power-law tail $P^*(t^*) \sim 1/(t^*)^{1+β(W)}$ for large $t^*$ with $0<β(W) \leq 1/2$, so that all integer moments of $T_N$ are governed by rare events. In the delocalized phase, the transmission $T_N$ remains a finite random variable as $N \to \infty$, and we measure near criticality the essential singularity $\bar{\ln (T)} \sim - | W_c-W |^{-κ_T}$ with $κ_T \sim 0.25$. We then consider the statistical properties of normalized eigenstates, in particular the entropy and the Inverse Participation Ratios (I.P.R.). In the localized phase, the typical entropy diverges as $(W-W_c)^{- ν_S}$ with $ν_S \sim 1.5$, whereas it grows linearly in $N$ in the delocalized phase. Finally for the I.P.R., we explain how closely related variables propagate as traveling waves in the delocalized phase. In conclusion, both the localized phase and the delocalized phase are characterized by the traveling wave propagation of some probability distributions, and the Anderson localization/delocalization transition then corresponds to a traveling/non-traveling critical point. Moreover, our results point towards the existence of several exponents $ν$ at criticality.
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