Aubry-Andre Anderson模型：磁性杂质与分形谱的耦合
The Aubry-Andre Anderson model: Magnetic impurities coupled to a fractal spectrum
摘要：
利用数值重整化群（NRG）研究了一维准晶中磁性杂质的Anderson模型。重点是阐明Aubry-Andre（AA）哈密顿量在临界点的物理性质，该哈密顿量具有多重分形波函数的分形谱，从而得到AA Anderson（AAA）杂质模型，该模型具有通过杂质位置的多重分形局域态密度定义的与能量相关的杂化函数.我们首先研究了一类具有均匀分形杂化函数的Anderson杂质模型，NRG可以在任意低温下求解。在近藤尺度$T_K$以下，这些模型接近分形强耦合固定点，其中杂质热力学性质与$\log_b T$一起振荡，大约由谱的分形维数确定的负平均值。分维也进入$T_K$对近藤交换耦合$J_K$的幂律依赖关系。为了治疗AAA模型，我们结合了核多项式方法（KPM）的NRG，形成一个有效的方法，可以治疗主机没有平移对称性的KPM展开顺序设置的温度范围。上述分形强耦合不动点是通过临界AAA模型在忽略波函数对杂交贡献的简化处理中达到的。温度平均的属性是那些预期的数值确定的分形维数的$0.5$。在AA临界点，杂质热力学性质变为负的和振荡的。在样本平均的情况下，近藤温度的平均值和中值对具有不同分维特征的指数的$J_K$表现出幂律依赖性。我们将这些签名归因于杂质探测分形强耦合不动点随温度降低的分布。
英文摘要：
The Anderson model for a magnetic impurity in a one-dimensional quasicrystal is studied using the numerical renormalization group (NRG). The main focus is elucidating the physics at the critical point of the Aubry-Andre (AA) Hamiltonian, which exhibits a fractal spectrum with multifractal wave functions, leading to an AA Anderson (AAA) impurity model with an energy-dependent hybridization function defined through the multifractal local density of states at the impurity site. We first study a class of Anderson impurity models with uniform fractal hybridization functions that the NRG can solve to arbitrarily low temperatures. Below a Kondo scale $T_K$, these models approach a fractal strong-coupling fixed point where impurity thermodynamic properties oscillate with $\log_b T$ about negative average values determined by the fractal dimension of the spectrum. The fractal dimension also enters into a power-law dependence of $T_K$ on the Kondo exchange coupling $J_K$. To treat the AAA model, we combine the NRG with the kernel polynomial method (KPM) to form an efficient approach that can treat hosts without translational symmetry down to a temperature scale set by the KPM expansion order. The aforementioned fractal strong-coupling fixed point is reached by the critical AAA model in a simplified treatment that neglects the wave-function contribution to the hybridization. The temperature-averaged properties are those expected for the numerically determined fractal dimension of $0.5$. At the AA critical point, impurity thermodynamic properties become negative and oscillatory. Under sample-averaging, the mean and median Kondo temperatures exhibit power-law dependences on $J_K$ with exponents characteristic of different fractal dimensions. We attribute these signatures to the impurity probing a distribution of fractal strong-coupling fixed points with decreasing temperature.
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