Self-consistent treatment of disorder-induced interacting criticality

该项目的目标是实现对无序诱导相互作用临界的自洽场(SCF)理论的更好理解。根据Altland-Zirnbauer对称性分类，相应的SCF-随机哈密顿量系综可以被分类，值得注意的是，SCF-条件可以在SCF-哈密顿量的单粒子态之间诱导额外的关联，从而产生新的相图。物理上，超临界流体系综是重要的，因为它们为无序费米子系统中相互作用效应的微扰分析提供了参考点。它们也很重要，因为理解平均场行为是揭示相互作用介导的量子涨落效应的先决条件。最后，SCF-随机哈密顿量为数学物理的基础研究提供了一个丰富的和大部分未被探索的领域。本项目的重点是在排斥或配对相互作用存在的情况下无序电子的Hartree-Fock理论和Boguluibov-deGennes理论。我们将通过数值和理论相结合的方法来解决相应的问题。一方面，我们将利用核多项式方法来实现和改进这些问题的数值编码。因此，可以为非常大的系统大小的全自旋和无自旋模型生成大量的数值数据。另一方面，我们将发展弱无序区SCF-哈密顿量的分析理论(非线性Sigma模型类型)。通过协调一致的工作，将模拟数据与理论结果对比分析。理想情况下，数值和场论这两种方法是相辅相成的。数值计算甚至可以精确地处理强无序效应，但本身只能产生有限的理解，因为对于这个问题，需要解析公式。例如，可以用场论方法来生成这样的公式；然而，在良好的控制下，这只能在相对较小的扇区参数空间中实现。通过将这两种方法结合起来，并且只有通过结合，才能在全参数空间中实现对物理学的更深层次的理解。两个项目团队以前已经通过这种方式成功地进行了合作，这种方式取得的经验非常令人鼓舞。

The project goal is to achieve an improved understanding of self-consistent field (scf) theories of disorder-induced interacting criticality. The corresponding ensembles of scf-random Hamiltonians can be categorised acording to the Altland-Zirnbauer symmetry classification.Remarkably, the scf-condition can induce extra correlations among single-particle states of the scf-Hamiltonian that can result in new phase diagram. Physically, scf-ensembles are important because they provide the reference point for a perturbative analysis of interaction effects in disordered fermion systems. Also they are important because understanding the mean-field behavior is a prerequisite for revealing the effect of interaction-mediated quantum fluctuations. Finally, scf-random Hamiltonians provide a rich and largely unexplored field for fundamental research in mathematical physics.The focus of this project is on the Hartree-Fock theory and the Boguluibov-deGennes theory of disordered electrons in the presence of repulsive or pairing interactions. We will solve the corresponding problems by combined numerical and theoretical efforts. On the one hand, we will implement and improve numerical codes for these problems employing the kernel-polynomial-method. Thus, extensive numerical data can be generated for spin-full and spin-less models of very large system sizes. On the other hand, we will develop the analytical theory (of the nonlinear sigma model type) for scf-Hamiltonians in the weak disorder regime. In a concerted effort the simulation data will be contrasted against and analysed with the help of the theoretical results. Both approaches, numerics and field theory, complement each other ideally. Numerics allows to treat even strong-disorder effects exactly, but can only generate a limited understanding per se, because for this analytical formulae are required. Such formulae can be generated, e.g., with field-theoretic approaches; hower, with good control this can be achieved only in a relatively small sector parameter space. By combining both approaches, and only by combining, a deeper understanding of the physics in the full parameter space can be achieved. Both project teams have already successfully collaborated in this way before and the experiences made in this way are very encouraging.