一维随机近邻跳跃链的超越普适性
Beyond universal behavior in the one-dimensional chain with random nearest neighbor hopping
摘要：
我们研究了一维近邻紧束缚模型的电子与独立分布的随机跳跃和没有现场电位（即非对角无序与粒子-空穴对称性，导致亚晶格对称性，为每个实现）。对于跳跃的非奇异分布，已知该模型在带中心$E = 0$附近表现出态密度$ρ(E) \sim 1/|E \ln^3|E||$和局域化长度$ξ(E) \sim |\ln|E||$的普遍奇异行为。(This奇异行为也适用于随机XY和海森堡自旋链;它首先由戴森针对特定的随机谐振子链获得）。同时，$E = 0$的状态在大距离$\sim \exp [ -\sqrt{r/r_0} ]$处显示出普遍的亚指数衰减。在这项研究中，我们考虑奇异的，但规范化的，跳跃的分布，其行为在小$t$的形式是$\sim 1/ [t \ln^{λ+1}(1/t) ]$，其特征在于由一个单一的，连续可调的参数$λ> 0$。我们发现，使用分析和数值方法相结合，而普遍的结果适用于$λ> 2$，它不再持有的间隔$0 < λ< 2$。特别是，我们发现，状态的奇异性密度的形式是增强（相对于戴森结果）在一个连续的方式依赖于非通用参数$λ$ ;同时，本地化长度显示出较少的发散形式在低能量下，并停止发散低于$λ= 1$。对于$λ< 2$，$E = 0$态在大距离处的衰减也偏离了普遍的结果，并且具有$\sim \exp [-(r/r_0)^{1/λ}]$的形式，其衰减速度比$λ< 1$的指数衰减速度更快。
英文摘要：
We study the one-dimensional nearest neighbor tight binding model of electrons with independently distributed random hopping and no on-site potential (i.e. off-diagonal disorder with particle-hole symmetry, leading to sub-lattice symmetry, for each realization). For non-singular distributions of the hopping, it is known that the model exhibits a universal, singular behavior of the density of states $ρ(E) \sim 1/|E \ln^3|E||$ and of the localization length $ξ(E) \sim |\ln|E||$, near the band center $E = 0$. (This singular behavior is also applicable to random XY and Heisenberg spin chains; it was first obtained by Dyson for a specific random harmonic oscillator chain). Simultaneously, the state at $E = 0$ shows a universal, sub-exponential decay at large distances $\sim \exp [ -\sqrt{r/r_0} ]$. In this study, we consider singular, but normalizable, distributions of hopping, whose behavior at small $t$ is of the form $\sim 1/ [t \ln^{λ+1}(1/t) ]$, characterized by a single, continuously tunable parameter $λ> 0$. We find, using a combination of analytic and numerical methods, that while the universal result applies for $λ> 2$, it no longer holds in the interval $0 < λ< 2$. In particular, we find that the form of the density of states singularity is enhanced (relative to the Dyson result) in a continuous manner depending on the non-universal parameter $λ$; simultaneously, the localization length shows a less divergent form at low energies, and ceases to diverge below $λ= 1$. For $λ< 2$, the fall-off of the $E = 0$ state at large distances also deviates from the universal result, and is of the form $\sim \exp [-(r/r_0)^{1/λ}]$, which decays faster than an exponential for $λ< 1$.
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