Green functions in the renormalized many-body perturbation theory
Václav Janiš
备注：
Lecture notes to Online Autumn School on Correlated Electrons: Simulating Correlations with Computers, Jülich, 20-24 September 2021, (https://www.cond-mat.de/events/correl21/)
I review the way the many-body Green functions are used to renormalize the perturbation theory of correlated fermions. The Green functions are introduced to implement systematically dynamical corrections to the static mean-field theory. The renormalizations enter the perturbation theory via self-consistent evaluations of one-particle and two-particle Green functions. The one-particle self-consistency is discussed within the Baym-Kadanoff construction. We point to an inherent dichotomy of the Baym-Kadanoff approach in the relation between one and two-particle Green functions. They are the Schwinger-Dyson equation, directly derived from the generating Luttinger-Ward functional, and the Ward identity. The latter imposes a severe restriction on the two-particle irreducible vertex and is not guaranteed in the Baym-Kadanoff approach. No approximate theory is capable to obey both relations. I further discuss the two-particle approach with two-particle renormalizations that offers a way to reconcile the Schwinger-Dyson equation and the Ward identity. The two-particle irreducible vertex from the critical Bethe-Salpeter equation becomes a generating functional in the two-particle renormalization of the perturbation theory. The irreducible vertex is used in the Ward identity to determine the order parameter beyond the critical point and in the Bethe-Salpeter equation representing the vertex of the Schwinger-Dyson equation to determine the spectral self-energy in such a way that the thermodynamic critical behavior from both equations is qualitatively the same. Finally, I discuss a static mean-field-like approximation with a renormalized effective interaction determined self-consistently that leads to a qualitatively correct description of strong electron correlations in impurity and lattice models.
Total energies from variational functionals of the Green function and the renormalized four-point vertex
We derive variational expressions for the grand potential or action in terms of the many-body Green function $G$ which describes the propagation of particles and the renormalized four-point vertex $Γ$ which describes the scattering of two particles in many-body systems. The main ingredient of the variational functionals is a term we denote as the $Ξ$ -functional which plays a role analogously to the usual $Φ$ -functional studied by Baym (G.Baym, Phys.Rev. 127, 1391 (1962)) in connection with the conservation laws in many-body systems. We show that any $Ξ$ -derivable theory is also $Φ$ -derivab…
Fundamental limitations of single-particle Green's-function zeroes as probes of many-body topology
Institut de Physique Théorique, Université Paris Saclay, CEA CNRS（巴黎萨克雷大学理论物理研究所）；Center for Computational Quantum Physics, Flatiron Institute（平顿研究所计算量子物理中心）
The exact reduced density-matrix functional is derived from the Luttinger-Ward functional of the single-particle Green's function. Thereby, a formal link is provided between diagrammatic many-body approaches using Green's functions on the one hand and theories based on many-body wave functions on the other. This link can be used to explicitly construct approximations for the density-matrix functional that are equivalent to standard diagrammatic re-summation techniques and to non-perturbative dynamical mean- field theory in particular. Contrary to functionals of the Green's-function, the exact …
Theory of Strongly Correlated Electron Systems. II.Intersite Coulomb Interaction and the Approximation of Renormalized Fermions in Total Energy Calculations
The many-body Green function of degenerate systems
本文导出了简并系统多体物理中演化算符的严格非微扰绝热近似。这种近似是用来解决长期存在的问题的初始状态的选择H0导致本征态的H0+V的退化系统。这些初始状态是P0 V P0的本征态，其中P0是H0的简并本征空间上的投影。利用这个结果给出了简并系统的格林函数、统计格林函数和非平衡格林函数的正确定义。这些格林函数的收敛性。
Training biases in machine learning for the analytic continuation of quantum many-body Green's functions
我们解决的问题解析连续的频率格林函数，这是至关重要的多体物理，使用机器学习的基础上的多级残差神经网络。我们专门解决潜在的偏见，可以引入由于使用人工创建的频谱函数，用于训练神经网络。我们还基于Monte Carlo dropout实现了预测谱函数的不确定性估计，这允许识别预测可能不准确的频率区域，并且我们研究了噪声的影响，特别是对于训练期间的噪声水平与实际数据不同的情况。我们的分析表明，这种方法确实可以实现高质量的预测，与广泛使用的最大熵方法相当或更好，但目前由于缺乏可用于训练的真实数据，进一步的改进受到限制。我们还基准我们的方法，通过将其应用到SrVO $_3$的情况下，其中一个准确的谱函数已获得从动态平均场论使用求解器，直接在实际频率轴上工作。
Efficient ab initio many-body calculations based on sparse modeling of Matsubara Green's function
本课堂讲稿回顾了最近提出的基于格林函数数据压缩的有效从头计算多体计算的稀疏建模方法。稀疏建模技术是基于一个紧凑的正交基，一个中间表示（IR）的基础上，时间和松原格林函数。基于IR基的稀疏采样方法可以有效地求解图解方程。我们描述了IR基的基本性质，稀疏采样方法及其在基于GW近似和Migdal-Eliashberg理论的从头计算中的应用。我们还描述了一个数值库的IR基础和稀疏采样方法，稀疏IR，并提供其示例代码。这份课堂讲稿是在日文评论文章的基础上做了重大修改[H. Shinaoka等人，Solid State Physics 56（6），301（2021）]。
Protocols for a many-body phase microscope: From coherences and d-wave superconductivity to Green's functions
TU Dortmund University（多特蒙德工业大学）；Kyoto University（京都大学）；Ludwig-Maximilians-Universität München（慕尼黑大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）
