Numerical investigations of ensembles of Boguliubov-deGennes Hamiltonians

无序金属和半导体的物理学不断带来惊喜，因此仍然是当代凝聚态物理学中重要的活跃领域之一。近年来，无序和相互作用之间的相互作用再次成为关注的焦点，例如，无序超导体与超导体-绝缘体结的联系。尽管付出了巨大的努力，但对这些系统的理论理解仍然有限，其中一个原因是，在最有趣的参数状态下的分析计算很难以系统控制的方式进行，即无序、相互作用和动能强度相似的状态，这就是这里提出的项目发挥作用的地方。在第一个资助期，可以开发一个代码，允许在有吸引力的Hubbard模型和Boguliubov-deGennes理论的框架内对无序和相互作用进行极其有效的自一致的数值处理。可以生成自洽随机矩阵的大集合，并根据其统计性质对其进行系统评估。因此，第一次有可能用数值方法接近重要的中间参数区，并证实那里长期以来怀疑的超导岛的形成。在第二个资助期，我们建议利用这一成功，研究将在当前实验中确定的物理观测值，其中一些仍处于规划阶段。具体来说，一个强烈的兴趣是在局域谱函数的间隙形式和理解明显的“无间隙超导性”，这是最近观察到的实验。我们将测试的解释性假设涉及自由磁矩，它不可避免地由于无序而产生，我们的计算能够证明它的存在。另一个研究主题将是超导刚度，以及它对磁场和温度的依赖，最近也可以进行高精度实验。最后，作为项目的最后一个阶段，应该用磁场、温度和无序强度的轴来创建相图。一方面，该图是关于确定假定的“失效超导体”可能存在的状态。另一方面，一个美国印第安人小组最近提出了一个相图，然而，它不能充分准确地解释局部矩的形成。具体来说，排斥性相互作用的重要性尚未得到认识。在后续项目中，这些研究将进一步深化和扩大，以包括一个具有吸引和排斥相互作用的扩展模型。

The physics of disordered metals and semiconductors continues to hold surprises and therefore is still one of the important active fields in contemporary condensed matter physics. In recent years, the interplay between disorder and interaction has come into focus again, e.g., in disordered superconductors in connection with the superconductor-insulator junction. Despite intensive efforts, the theoretical understanding of such systems is still limited, among other reasons because analytical calculations in the most interesting parameter regime can hardly be carried out in a systematically controlled manner, i.e. the regime in which disorder, interaction and kinetic energy of similar strength, This is where the project proposed here comes into play.In the first funding period, a code could be developed that allows for an extremely efficient self-consistent numerical treatment of disorder and interaction within the framework of the attractive Hubbard model and Boguliubov-deGennes theory. Large ensembles of self-consistent random matrices can be generated and systematically evaluated with regard to their statistical properties. It was thus possible for the first time to numerically access the important intermediate parameter regime and to confirm the long suspected formation of superconducting islands there. For the second funding period, we propose to exploit this success and study physical observables that are to be determined in current experiments, some of which are still in planning stage. Specifically, a strong interest is in the form of the gap in the local spectral function and in understanding the apparent “gapless superconductivity”, which was recently observed experimentally. The explanatory hypothesis we will test involves free magnetic moments, which inevitably arise due to the disorder and whose existence our calculations are able to demonstrate. Another subject of investigation will be the superconducting stiffness, as well as its dependence on magnetic field and temperature, which recently became accessible also to high precision experiments.Finally, as last stage of the project a phase diagram should be created with axes of magnetic field, temperature and disorder strength. On the one hand, the diagram is about identifying a regime in which the putative "failed superconductors" could reside. On the other hand, an American-Indian team recently proposed a phase diagram, which, however, could did not account sufficiently accurately for the formation of local moments. Specifically, the importance of repulsive interactions has not been recognized. In the follow-up project, these investigations are to be deepened and expanded to include an extended model with attractive and repulsive interaction.