# Variational structure of Luttinger–Ward formalism and bold diagrammatic expansion for Euclidean lattice field theory

## Abstract

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.

- Authors:

- Univ. of California, Berkeley, CA (United States). Dept. of Mathematics; Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division
- Univ. of California, Berkeley, CA (United States). Dept. of Mathematics

- Publication Date:

- Research Org.:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1421895

- Alternate Identifier(s):
- OSTI ID: 1485086

- Grant/Contract Number:
- AC02-05CH11231; SC0017867

- Resource Type:
- Published Article

- Journal Name:
- Proceedings of the National Academy of Sciences of the United States of America

- Additional Journal Information:
- Journal Volume: 115; Journal Issue: 10; Journal ID: ISSN 0027-8424

- Publisher:
- National Academy of Sciences, Washington, DC (United States)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICS AND COMPUTING

### Citation Formats

```
Lin, Lin, and Lindsey, Michael. Variational structure of Luttinger–Ward formalism and bold diagrammatic expansion for Euclidean lattice field theory. United States: N. p., 2018.
Web. doi:10.1073/pnas.1720782115.
```

```
Lin, Lin, & Lindsey, Michael. Variational structure of Luttinger–Ward formalism and bold diagrammatic expansion for Euclidean lattice field theory. United States. doi:10.1073/pnas.1720782115.
```

```
Lin, Lin, and Lindsey, Michael. Tue .
"Variational structure of Luttinger–Ward formalism and bold diagrammatic expansion for Euclidean lattice field theory". United States. doi:10.1073/pnas.1720782115.
```

```
@article{osti_1421895,
```

title = {Variational structure of Luttinger–Ward formalism and bold diagrammatic expansion for Euclidean lattice field theory},

author = {Lin, Lin and Lindsey, Michael},

abstractNote = {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.},

doi = {10.1073/pnas.1720782115},

journal = {Proceedings of the National Academy of Sciences of the United States of America},

number = 10,

volume = 115,

place = {United States},

year = {2018},

month = {2}

}

DOI: 10.1073/pnas.1720782115

#### Figures / Tables:

*A*and

*B*) and second-order (

*C*and

*D*) contribution to the self-energy for interaction of the form Eq. 10.

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*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*