CPHT, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, Route de Saclay, F-91128 Palaiseau, France.
Phys Rev Lett. 2019 Aug 16;123(7):070405. doi: 10.1103/PhysRevLett.123.070405.
Quasiperiodic systems offer an appealing intermediate between long-range ordered and genuine disordered systems, with unusual critical properties. One-dimensional models that break the so-called self-dual symmetry usually display a mobility edge, similarly as truly disordered systems in a dimension strictly higher than two. Here, we determine the critical localization properties of single particles in shallow, one-dimensional, quasiperiodic models and relate them to the fractal character of the energy spectrum. On the one hand, we determine the mobility edge and show that it separates the localized and extended phases, with no intermediate phase. On the other hand, we determine the critical potential amplitude and find the universal critical exponent ν≃1/3. We also study the spectral Hausdorff dimension and show that it is nonuniversal but always smaller than unity, hence showing that the spectrum is nowhere dense. Finally, applications to ongoing studies of Anderson localization, Bose-glass physics, and many-body localization in ultracold atoms are discussed.
准周期系统在长程有序和真正无序系统之间提供了一个吸引人的中间状态，具有不同寻常的临界性质。打破所谓自对偶对称性的一维模型通常会显示出迁移率边缘，类似于在严格高于二维的维度上的真正无序系统。在这里，我们确定了浅一维准周期模型中单粒子的临界局域化性质，并将其与能谱的分形特征联系起来。一方面，我们确定了迁移率边缘，并表明它将局域化相和扩展相分开，没有中间相。另一方面，我们确定了临界势幅度，并发现了普遍的临界指数 ν≃1/3。我们还研究了谱豪斯多夫维数，并表明它不是普遍的，但总是小于 1，因此表明谱在任何地方都不是密集的。最后，讨论了对正在进行的安德森局域化、玻色玻璃物理和超冷原子中多体局域化的应用。
