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

Edited by George Papanicolaou, Stanford University, Stanford, CA, and approved January 24, 2018 (received for review November 29, 2017)

## Significance

Many-body perturbation theory is one of the pillars of quantum many-body physics and has been used extensively to predict ground-state and excited-state electronic properties of real materials in the past few decades. Nonetheless, few practically used methods in many-body perturbation theory have been justified on a rigorous basis. We present a variational formalism that can be used for the rigorous study of a number of many-body perturbation methods in Euclidean lattice field theory. In particular, this perspective allows us to justify the widely used bold Feynman diagrammatic expansion, without relying on formal arguments such as partial resummation of Feynman diagrams to infinite order.

## 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.

- many-body perturbation theory
- Feynman diagrams
- lattice field theory
- Luttinger–Ward formalism
- Green’s function

The Luttinger–Ward (LW) formalism (1) is an important component of Green’s function theories in quantum many-body physics. The LW functional

In this work, we provide a rigorously justified LW formalism, in the context of the Euclidean lattice field theory [such as the

For a general interaction form (not necessarily the quartic interaction), we prove using Legendre duality that there exists a universal functional of the Green’s function, denoted

After proper discretization, a Euclidean lattice field theory can be described by the partition function**2** by a quadratic form given by a symmetric matrix **1**. Here **1** is very general and can represent interaction terms that are quartic, beyond quartic, or even nonpolynomial in classical and quantum statistical mechanics (e.g., refs. 11, 20, and 21). For the integral to be well defined, we assume that

Let **1** can be viewed as a functional of

The derivative of

Let *Theorems 1* and *2*.

### Theorem 1 (Variational Structure).

*The Gibbs free energy* *can be expressed variationally via**where* *defined as**is the Legendre dual of* *with respect to the inner product* *. Here* *is the entropy and is defined as* *. The mapping* *is a bijection* *with the inverse given by*

Note that

The last statement of *Theorem 1* suggests that

### Theorem 2 (LW Functional).

*The LW functional in* *Eq.***7** *is universal*, *satisfies* *for noninteracting systems*, *and extends continuously up to the boundary of* *The self-energy functional is defined as *.

*The solution of the Dyson equation*

*in*

*is the unique minimizer of the free energy in*

*Eq*.

**4**.

According to the preceding discussion, for **8** can be written equivalently as**1** diverges in noninteracting limit. On the other hand, the Dyson equation in the form of Eq. **8** is more general and is valid for any

When the self-energy functional **8** can be solved to obtain **8** can also be used in the reverse direction to compute *Theorem 1* resolves this paradox, and

Although the dependence of the LW functional on the interaction

So far we have considered the LW formalism for any interaction that satisfies the strong growth condition. To draw a closer connection with the diagrammatic expansion used in quantum many-body physics, we now restrict our attention to the quartic interaction**10** can mimic a short-range interaction as well as a long-range (such as Coulomb) interaction in its second quantized form (13). One can derive an exact correspondence between the Feynman diagrammatic expansions in this lattice field theory and those in condensed matter physics (11), neglecting the particle-hole distinction.

For fixed interaction *Theorem 3* shows that the bold diagrammatic expansion of the LW functional at **11** to be asymptotic means that the error of the *Theorem 3* is to show that the bold diagrammatic expansion is well defined without any reference to the noninteracting Green’s function

### Theorem 3 (Bold Diagrammatic Expansion).

*For any interaction* *satisfying the strong growth condition*, *the LW functional and the self-energy have the following asymptotic series expansions*:*Moreover*, *for* *of the form* *Eq*. **10**, *the coefficients of the asymptotic series satisfy**and* *consists of all one-particle irreducible skeleton diagrams of order *.

For example, when the **10**, one can show that the self-energy obtained from bold diagrammatic expansion up to second order is

Compared with the Feynman diagrams for condensed matter systems, we find that not coincidentally, Fig. 1 *A* and *B* corresponds to the Hartree and Fock exchange diagrams, respectively, and Fig. 1 *C* and *D* corresponds to the ring and second-order exchange diagrams, respectively. The only difference is that the lines in the diagrams in Fig. 1 do not possess directions, due to the absence of any distinction between creation and annihilation operators.

Interestingly, the relation Eq. **12** was originally assumed to be true to obtain a formal derivation of the LW functional (1, 22). Our proof here does not rely on such formal manipulation, but instead only on the transformation rule (*Proposition 4*) below and the quartic nature of the interaction

Finally, we remark that certain properties in the Euclidean setting, such as the concavity of the free energy functional, can noticeably fail in the non-Euclidean setting. Indeed, the original setting for the LW formalism is a field theory described by the fermionic coherent state path integral represented by Grassmann variables. The free energy functional is nonconcave and the induced Legendre correspondence may not be one to one. This leads to the failure of the LW formalism observed in refs. 4 and7⇓⇓–10, and the full picture of the LW functional remains to be revealed. Intriguingly, the LW formalism may also be seen as an expansion of a static density matrix formalism (16, 19), which itself does enjoy convexity properties and hence well-defined Legendre duality. However, the density matrix formalism is not induced in the same way by a field theory and does not enjoy even formally properties such as the diagrammatic expansion. The Euclidean field theory setting can then be viewed as combining the best of both worlds, in that it enjoys the convexity properties needed for the nonperturbative definition of the Legendre dual functionals, as well as the formal properties convenient for systematic approximation as in diagrammatic expansions and DMFT.

This work also opens up several immediate research directions. By making a connection between quantum many-body physics and Euclidean lattice field theory, it lowers the barrier for quantitatively assessing the effectiveness of bold diagrammatic schemes and other numerical schemes based on many-body perturbation theory such as the GW theory (2). Possible topics to be developed include the effectiveness of self-consistent many-body perturbation theories and the effectiveness of the vertex correction methods in the GW

### Outline of the proof of Theorem 1:

First, we reformulate the computation of the Gibbs free energy *Proof of the Classical Gibbs Variational Principle in Theorem 1*.

Next, we split the infimum in Eq. **13** as**5**, we obtain the variational formulation in Eq. **4**. Now Eq. **4** means precisely that

One can further prove that *Proof That* F *Is Concave in Theorem 1* and *Proof That* F *Diverges to* −∞ *at the Boundary of* S++n *in Theorem 1*). Based on these facts, we have that

The Legendre duality suggests that **5** of

### Outline of the proof of *Theorem 2*:

The differentiability of the LW functional on **4** can be written as**8**, and the uniqueness of the solution follows from that of the minimizer in *Theorem 1*.

We now establish that unlike *Proof of the Transformation Rule (Proposition 4)*).

### Proposition 4 (Transformation Rule).

*Let* *and let* *be the interaction term. Let* *denote an invertible matrix in* *as well as the corresponding linear transformation* *. Then*

Define a *Sketch of the Proof of the Continuous Extension of the LW Functional in Theorem 2*.Q.E.D.

### Outline of the proof of *Theorem 3*:

For a given Green’s function **11**. This existence proof is nonconstructive, and hence the series coefficients still need to be determined. We abbreviate the notation for the series coefficients via

*Theorem 3* then consists of identifying that these coefficients are given by the bold diagrammatic expansion using **12**. Since the series expansion is valid only in the asymptotic sense, for any finite

A difficulty in proving *Theorem 3* is that, although we wish to use the technique that resums bare self-energy diagrams to bold self-energy diagrams, *Lemma 5* below identifies a bare propagator *Proof of the Resummation Step in Theorem 3*). After this, the relation Eq. **12** is a consequence of the transformation rule (*Proof of the Expansion Coefficients of the LW Functional in Theorem 3*).

It remains to introduce the aforementioned *Lemma 5* and explain how it allows us to identify a bare propagator (dependent on *Lemma 5* says that *Proof of Lemma 5*):

### Lemma 5.

*is the self-energy at* *induced by the modified interaction* *. In other words*, *is the exact self-energy corresponding to a noninteracting Green’s function**and the interaction*

Note carefully that *Lemma 5* is a nonperturbative fact and is valid for all

Then one finds, by swapping **17**, that*Proof of the Resummation Step in Theorem 3*), this establishes the expansion for the self-energy.Q.E.D.

## Acknowledgments

We thank Fabien Bruneval, Roberto Car, Garnet Chan, Lek-Heng Lim, Sohrab Ismail-Beigi, Nicolai Reshetikhin, and Chao Yang for helpful discussions. This work was partially supported by the National Science Foundation (NSF) under Grant DMS-1652330, by the Department of Energy under Grants DE-SC0017867 and DE-AC02-05CH11231 (to L.L.), and by the NSF Graduate Research Fellowship Program under Grant DGE-1106400 (to M.L.).

## Footnotes

↵

^{1}L.L. and M.L. contributed equally to this work.- ↵
^{2}To whom correspondence should be addressed. Email: linlin{at}math.berkeley.edu.

Author contributions: L.L. and M.L. designed research, performed research, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1720782115/-/DCSupplemental.

Published under the PNAS license.

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