Department of Mathematics, University of California, Berkeley, CA 94720;
Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720.
Proc Natl Acad Sci U S A. 2018 Mar 6;115(10):2282-2286. doi: 10.1073/pnas.1720782115. Epub 2018 Feb 20.
The Luttinger-Ward functional was proposed more than five decades ago and has been used to formally justify most practically used Green's function methods for quantum many-body systems. Nonetheless, the very existence of the Luttinger-Ward functional has been challenged by recent theoretical and numerical evidence. We provide a rigorously justified Luttinger-Ward formalism, in the context of Euclidean lattice field theory. Using the Luttinger-Ward functional, the free energy can be variationally minimized with respect to Green's functions in its domain. We then derive the widely used bold diagrammatic expansion rigorously, without relying on formal arguments such as partial resummation of bare diagrams to infinite order.
Luttinger-Ward 泛函早在五十年前就被提出，并被广泛应用于量子多体系统的格林函数方法中。然而，最近的理论和数值证据对 Luttinger-Ward 泛函的存在性提出了挑战。我们在欧几里得格点场论的框架下，提供了一个严格证明的 Luttinger-Ward 形式理论。通过 Luttinger-Ward 泛函，我们可以在其定义域内对格林函数进行自由能变分最小化。然后，我们推导出了广泛使用的粗粒图式展开式，而无需依赖于裸图式的部分重求和到无穷阶等形式论证。
