准周期和准随机晶格中的传导：斐波那契、黎曼和安德森模型
Conduction in quasi-periodic and quasi-random lattices: Fibonacci, Riemann, and Anderson models
摘要：
本文研究了具有手征对称性的非周期非随机一维模型中非相互作用电子的基态导电性质，并与具有非确定性无序的Anderson模型进行了比较.我们考虑的第一个模型是斐波那契晶格，这是一个典型的准晶体模型;第二个是黎曼晶格，我们定义的灵感来自戴森的建议之间的可能联系的黎曼假设和一个适当定义的准晶体。我们的分析是基于Kohn的多粒子局域化张量定义的现代理论的绝缘状态。在斐波那契准晶体中，所有单粒子本征态都是临界的（即，处于遍历和局部化之间的中间），发现非相互作用的电子气在大多数电子密度下是导体，包括半填充的情况;然而，在各种特定填充$ρ$下，包括值$ρ= 1/g^n$，其中$g$是黄金分割比，$n$是任何整数，由于光谱间隙，气体变成绝缘体。在黎曼晶格中也发现了半填充的金属行为;然而，与斐波那契准晶相反，由于单粒子本征态局部化，黎曼晶格通常是绝缘体，可能在所有其他填充中。它的行为原来是类似的非对角安德森模型，虽然不同的系统规模缩放的带中心异常。科恩的本地化张量，而不是其他措施的本地化熟悉的安德森绝缘体理论（如参与比或李雅普诺夫指数）的分析的优点突出。
英文摘要：
We study the ground state conduction properties of noninteracting electrons in aperiodic but non-random one-dimensional models with chiral symmetry, and make comparisons against Anderson models with non-deterministic disorder. The first model we consider is the Fibonacci lattice, which is a paradigmatic model of quasicrystals; the second is the Riemann lattice, which we define inspired by Dyson's proposal on the possible connection between the Riemann hypothesis and a suitably defined quasicrystal. Our analysis is based on Kohn's many-particle localization tensor defined within the modern theory of the insulating state. In the Fibonacci quasicrystal, where all single-particle eigenstates are critical (i.e., intermediate between ergodic and localized), the noninteracting electron gas is found to be a conductor at most electron densities, including the half-filled case; however, at various specific fillings $ρ$, including the values $ρ= 1/g^n$, where $g$ is the golden ratio and $n$ is any integer, the gas turns into an insulator due to spectral gaps. Metallic behaviour is found at half-filling in the Riemann lattice as well; however, in contrast to the Fibonacci quasicrystal, the Riemann lattice is generically an insulator due to single-particle eigenstate localization, likely at all other fillings. Its behaviour turns out to be alike that of the off-diagonal Anderson model, albeit with different system-size scaling of the band-centre anomalies. The advantages of analysing the Kohn's localization tensor instead of other measures of localization familiar from the theory of Anderson insulators (such as the participation ratio or the Lyapunov exponent) are highlighted.
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