具有一般关联无序的长程模型中的遍历性破缺相图与分形维数
Ergodicity-breaking phase diagram and fractal dimensions in long-range models with generically correlated disorder
AI总结：
本文提出一种计算长程关联无序模型中局域化转变、遍历性破缺转变及分形维数下界的方法，并揭示关联无序与幂律跃迁的相互作用可在全参数范围内产生两类分形相。
摘要：
具有关联无序的模型在物理学中相当常见。在其中一些模型中，例如Aubry-André（AA）模型，局域化相图可以通过关于傅里叶变换的（自）对偶性得到。在另一些模型中，例如全连接平移不变的Rosenzweig-Porter（TI RP）系综或多体局域化的希尔伯特空间结构，则需要发展更复杂且通常是唯象的方法来寻找局域化转变。此外，这类模型不仅包含局域化，还包含遍历性破缺转变，从而产生具有非平凡分形维数$D_q$的非遍历扩展相态。在这项工作中，我们提出了一种方法，用于计算上述两种转变以及与物理观测量相关的分形维数$D_2$和$D_\infty$的下界。为了验证该方法，我们将其应用于一类长程（自）对偶模型，这些模型通过在位无序关联和跃迁项的幂律依赖关系在AA模型和TI RP模型之间插值，因此超出了先前发展方法的适用范围。我们表明，关联无序与幂律衰减跃迁项之间的相互作用导致在整个参数范围内出现两种类型的分形相，即使没有任何AA势的准周期性。上述方法的解析结果与广泛的数值计算完全一致。
英文摘要：
Models with correlated disorders are rather common in physics. In some of them, like the Aubry-André (AA) model, the localization phase diagram can be found from the (self)duality with respect to the Fourier transform. In the others, like the all-to-all translation-invariant Rosenzweig-Porter (TI RP) ensemble or the Hilbert-space structure of the many-body localization, one needs to develop more sophisticated and usually phenomenological methods to find the localization transition. In addition, such models contain not only localization but also the ergodicity-breaking transition, giving way to the non-ergodic extended phase of states with non-trivial fractal dimensions $D_q$. In this work, we suggest a method to calculate both the above transitions and a lower bound to the fractal dimensions $D_2$ and $D_\infty$, relevant for physical observables. In order to verify this method, we apply it to the class of long-range (self-)dual models, interpolating between AA and TI RP ones via both power-law dependences of on-site disorder correlations and hopping terms, and, thus, being out of the validity range of the previously developed methods. We show that the interplay of the correlated disorder and the power-law decaying hopping terms leads to the emergence of the two types of fractal phases in an entire range of parameters, even without having any quasiperiodicity of the AA potential. The analytical results of the above method are in full agreement with the extensive numerical calculations.
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