Hamiltonian perspective on parquet theory

Our many-body system has the second-quantized Hamiltonian

	$\displaystyle{\cal H}$	$\displaystyle=$	$\displaystyle\sum_{k}\varepsilon_{k}a^{*}_{k}a_{k}$		(4)
			$\displaystyle+{1\over 2}{\sum_{k_{1}k_{2}k_{3}k_{4}}}\!\!\!\!\!^{\prime}{\langle k_{1}k_{2}|V|k_{3}k_{4}\rangle}a^{*}_{k_{1}}a^{*}_{k_{2}}a_{k_{3}}a_{k_{4}};$
			$\displaystyle{\langle k_{1}k_{2}|V|k_{3}k_{4}\rangle}\equiv\delta_{s_{1}s_{4}}\delta_{s_{2}s_{3}}V({\bf k}_{1}-{\bf k}_{4})$

in terms of one-particle creation operators $a^{*}$ and annihilation operators $a$. The first right-hand term is the usual total kinetic energy, the second term is the pairwise interaction. For simplicity we discuss a spin- (or isospin-) independent scalar $V$ but this can be relaxed without invalidating the argument for pair interactions. Here, again for simplicity, we address a spatially uniform system for which momentum is a good quantum number; index $k$ stands for the wave vector and spin pair $({\bf k},s)$, writing $a^{*}_{k}$ as the creation operator with $a_{k}$ the annihilation operator, both satisfying fermion anticommutation. The summation ${\sum}^{\prime}_{k_{1}k_{2}k_{3}k_{4}}$ comes with the restriction $k_{1}+k_{2}=k_{3}+k_{4}$. In a neutral uniform Coulomb system, potential terms with $k_{2}-k_{3}=0=k_{4}-k_{1}$ are canceled by the background and are excluded.

The ground-state energy resulting from the full Hamiltonian includes a correlation component $\Phi[V]$, the essential generator for the diagrammatic expansions that act as vocabulary to the grammar of the analysis. Here we go directly to $\Phi[V]$ and for full discussion of the interacting ground-state structure we refer to the classic literature. [2, 17] The correlation energy can always be written as a coupling-constant integral

$\displaystyle\Phi[V]\equiv{1\over 2}\int^{1}_{0}{dZ\over Z}G[ZV]\!:\!\Lambda[ZV;G]\!:\!G[ZV]$

(5)

in which $G[V]$ is the complete renormalized two-point Green function of the system, describing propagation of a single particle in the presence of all the rest, and $\Lambda[V;G]$ is the fully renormalized four-point scattering amplitude whose internal structure manifests all the possible modes by which the propagating particles (via $G$) interact via $V$. Single dots “$\cdot$” and double dots “:” denote single and double internal integrations respectively, over frequency, spin and wave vector, rendering $G\!:\!\Lambda\!:\!G$ an energy expectation value.

The renormalized Green function satisfies Dyson’s equation

$\displaystyle G[V]$

$\displaystyle=$

$\displaystyle G^{(0)}+G^{(0)}\!\cdot\!\Sigma[V;G]\!\cdot\!G[V]$

(6)

with $G^{(0)}$ as the noninteracting Green function $G^{(0)}_{k}(\omega)\equiv(\omega-\varepsilon_{k})^{-1}$ with $\Sigma[V;G]$ as the self-energy. Equation (6) links back to the correlation-energy functional self-consistently through the variation that defines the self-energy

$\displaystyle\Sigma[V;G]$

$\displaystyle\equiv$

$\displaystyle\frac{\delta\Phi[V]}{\delta G[V]}=\Lambda[V;G]\!:\!G[V].$

(7)

We recall the basic requirements on $\Lambda$. In the expansion to order $n$ in $V$ within any particular linked structure of $G\!:\!\!\Lambda\!\!:\!G$ reduced to its bare elements, there will be $2n$ bare propagators. The integral effectively treats each $G^{(0)}$ as distinguishable, and there is a $2n$-fold ambiguity as to which bare propagator should be the seed on which the given contribution is built up. That is, integration replicates the same graph $2n$ times from any particular $G^{(0)}$ in the integral; but the structure contributes once only in $\Phi$. The coupling-constant formula removes the multiplicity to all orders.

The essential feature of $\Lambda$ in the exact correlation energy functional is the following symmetry: consider the skeleton $G^{(0)}\!:\!\!\Lambda[V;G^{(0)}]\!\!:\!G^{(0)}$. Removal of any $G^{(0)}$ from the skeleton, at any order in $V$, must result in the same unique variational structure; all lines are equivalent, The same applies when all bare lines are replaced with dressed ones. [1] This is due to unitarity and ultimately to the Hermitian character of the Hamiltonian. It also follows that $\Lambda$ must be pairwise irreducible: removing any two propagators $G$ from $G\!:\!\Lambda\!:\!G$ cannot produce two unlinked self-energy insertions, or else there would be inequivalent $G$s in a contribution of form $G\!:\!\!\Lambda_{1}\!\!:\!GG\!:\!\!\Lambda_{2}\!\!:\!G$. These conditions impose a strongly restrictive graphical structure upon the four-point scattering kernel entering into the self-energy.

Other than the generic symmetry of $G$ in $\Phi$, the variational relationships among $\Lambda$, $\Sigma$ and $G$ do not depend on topological specifics. Those relationships were thus adopted as defining criteria by Baym and Kadanoff [1, 2] for constructing conserving approximations: the $\Phi$-derivable models. Choosing a subset of skeleton diagrams from the full $\Phi[V]$ with every $G^{(0)}$ topologically equivalent and replacing these with dressed lines guarantees unitarity of the effective model $\Lambda$ and secures microscopic conservation not only at the one-body level but also for the pairwise dynamic particle-hole response under an external perturbation.

$\Phi$ derivability necessarily entails an infinite-order approximation to the correlation structure in terms of the bare potential. While a finite choice of skeleton diagrams of $\Phi$ fulfills formal conservation, it must still lead to an infinite nesting of bare interactions linked by pairs of renormalized $G$s. Self-consistency in Eqs. (6) and (7) is a fundamental feature of all $\Phi$-derivable models.

The authoritative references for Kraichnan Hamiltonians are the original papers of Kraichnan. [14, 15] Here we follow the more recent paper by one of us, hereafter called KI. [16] As per the Introduction, Kraichnan’s construction proceeds by two ensemble-building steps. First, one generates an assembly of $N$ functionally identical distinguishable copies of the exact Hamiltonian, Eq. (4). The total Hamiltonian is

	$\displaystyle{\cal H}_{N}$	$\displaystyle=$	$\displaystyle\sum^{N}_{n=1}\sum_{k}\varepsilon_{k}a^{*(n)}_{k}a^{(n)}_{k}$		(9)
			$\displaystyle+{1\over 2}\sum^{N}_{n=1}{\sum_{k_{1}k_{2}k_{3}k_{4}}}\!\!\!\!\!^{\prime}{\langle k_{1}k_{2}|V|k_{3}k_{4}\rangle}a^{*(n)}_{k_{1}}a^{*(n)}_{k_{2}}a^{(n)}_{k_{3}}a^{(n)}_{k_{4}};~{}~{}~{}~{}~{}$

the creation and annihilation operators with equal index $n$ anticommute as normal; for values of $n$ that differ, they commute. At this point one goes over to a collective description of the $N$-fold ensemble by Fourier transforming over index $n$. For integer $\nu$ define the collective operators

$\displaystyle a^{*[\nu]}_{k}$

$\displaystyle\equiv$

$\displaystyle N^{-1/2}\sum^{N}_{n=1}e^{2\pi i\nu n/N}a^{*(n)}_{k}~{}~{}{\rm and}~{}~{}a^{[\nu]}_{k}\equiv N^{-1/2}\sum^{N}_{n=1}e^{-2\pi i\nu n/N}a^{(n)}_{k}.$

(10)

These preserve anticommutation up to a term strongly suppressed by mutual interference among unequal phase factors and at most of vanishing order $1/N$. The argument is a random-phase one: [18]

	$\displaystyle[a^{*[\nu]}_{k},a^{[\nu^{\prime}]}_{k^{\prime}}]_{+}$	$\displaystyle=$	$\displaystyle\frac{1}{N}\sum_{n,n^{\prime}}e^{2\pi i(\nu n-\nu^{\prime}n^{\prime})/N}[a^{*(n)}_{k},a^{(n^{\prime})}_{k^{\prime}}]_{+}$		(11)
		$\displaystyle=$	$\displaystyle\frac{1}{N}\sum_{n}e^{2\pi i(\nu-\nu^{\prime})n/N}[a^{*(n)}_{k},a^{(n)}_{k^{\prime}}]_{+}+\frac{2}{N}\sum_{n\neq n^{\prime}}e^{2\pi i(\nu n-\nu^{\prime}n^{\prime})/N}a^{*(n)}_{k}a^{(n^{\prime})}_{k^{\prime}}$		(12)
		$\displaystyle=$	$\displaystyle\delta_{kk^{\prime}}\delta_{\nu\nu^{\prime}}+{\cal O}(N^{-1}).$		(13)

Similarly for $[a^{[\nu]}_{k},a^{[\nu^{\prime}]}_{k^{\prime}}]_{+}$. In practice, any term in the Wick expansion of physical expectations that links dissimilar elements $n\neq n^{\prime}$ will not contribute in any case. The transformation yields the new representation

	$\displaystyle{\cal H}_{N}$	$\displaystyle=$	$\displaystyle\sum^{N}_{\nu=1}\sum_{k}\varepsilon_{k}a^{*[\nu]}_{k}a^{[\nu]}_{k}+{1\over 2N}{\sum_{k_{1}k_{2}k_{3}k_{4}}}\!\!\!\!\!^{\prime}~{}~{}\sum^{N}_{\nu_{1}\nu_{2}\nu_{3}\nu_{4}}\delta_{\nu_{1}+\nu_{2},\nu_{3}+\nu_{4}}~{}{\langle k_{1}k_{2}|V|k_{3}k_{4}\rangle}~{}a^{*[\nu_{1}]}_{k_{1}}a^{*[\nu_{2}]}_{k_{2}}a^{[\nu_{3}]}_{k_{3}}a^{[\nu_{4}]}_{k_{4}}$		(14)
		$\displaystyle\equiv$	$\displaystyle\sum_{\ell}\varepsilon_{k}a^{*}_{\ell}a_{\ell}+{1\over 2N}{\sum_{\ell_{1}\ell_{2}\ell_{3}\ell_{4}}}\!\!\!\!^{\prime}~{}{\langle k_{1}k_{2}|V|k_{3}k_{4}\rangle}~{}a^{*}_{\ell_{1}}a^{*}_{\ell_{2}}a_{\ell_{3}}a_{\ell_{4}}$		(15)

where in the last right-hand expression we condense the notation so $\ell\equiv(k,\nu)$ and the restriction on the sum now comprises $\nu_{1}+\nu_{2}=\nu_{3}+\nu_{4}$ (modulo $N$) as well as the constraint on the momenta; equivalently, $\ell_{1}+\ell_{2}=\ell_{3}+\ell_{4}$.

FIG. 1. Construction of the Kraichnan Hamiltonian. (a) The exact many-body Hamiltonian is embedded in a large sum of $N$ identical but distinguishable duplicates. A Fourier transform over the identifying index $n=1,2,...N$ is performed. To each physical interaction potential ${\langle k_{1}k_{2}|V|k_{3}k_{4}\rangle}$ a new parameter $\varphi_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}$ is attached, labeled by the new Fourier indices and transforming in them as does $V$ in its physical indices. (b) The modified Hamiltonian is again embedded in a large sum of $M$ replicas, but each replica is now assigned a unique set of factors $\varphi$. The resulting Kraichnan Hamiltonian remains Hermitian. Setting every instance of $\varphi$ to unity recovers the exact expectations resulting from the original, physical Hamiltonian. If values are specifically structured but otherwise randomly assigned to the $M$-fold ensemble $\{\varphi\}$, only a selected subset of the physical correlations survives while the rest are suppressed by random phasing.

Performing averages given the extended Kraichnan Hamiltonian of Eq. (15), as it stands, simply recovers the exact expectations for the originating one; any cross-correlations between distinguishable members are identically zero. However, embedding the physical Hamiltonian within a collective description opens a novel degree of freedom for treating interactions. Figure 1 summarizes the whole process. From now on we concentrate on the interaction part of Eq. (15), denoted by ${\cal H}_{i;N}$, since the one-body part conveys no new information. We will not consider issues of convergence here; for particular implementations they are carefully discussed in the original papers. [14, 15]

We can modify the behavior of the interaction part, keeping it Hermitian, by adjoining a factor $\varphi_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}$ such that

	$\displaystyle{\cal H}_{i;N}[\varphi]$	$\displaystyle\equiv$	$\displaystyle{1\over 2N}{\sum_{\ell_{1}\ell_{2}\ell_{3}\ell_{4}}}\!\!\!\!^{\prime}~{}{\langle k_{1}k_{2}|V|k_{3}k_{4}\rangle}~{}\varphi_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}$		(17)
			$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times a^{*}_{\ell_{1}}a^{*}_{\ell_{2}}a_{\ell_{3}}a_{\ell_{4}}.$

The expression remains Hermitian if and only if the factor has the same symmetry properties as the potential under exchange of its indices. Thus

$\displaystyle\varphi_{\nu_{4}\nu_{3}|\nu_{2}\nu_{1}}=\varphi^{*}_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}};~{}~{}~{}\varphi_{\nu_{2}\nu_{1}|\nu_{4}\nu_{3}}=\varphi_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}.$

(18)

The additional procedure of taking expectations will no longer match those for the exact physical Hamiltonian unless, evidently, $\varphi$ is unity. The crux, however, is that all identities among expectations, dependent on causal analyticity, will still be strictly respected. The Kraichnan Hamiltonian remains well formed in its own right (its Fock space is complete) except that now it describes an abstract system necessarily different from the physical one that motivated it. The task is to tailor it to recover the most relevant aspects of the real physics in reduced but tractable form.

The last step in the logic considers the much larger sum $\mathbb{H}$ of collective Hamiltonians all of the form of Eq. (17), with an interaction part $\mathbb{H}_{i}$ encompassing a distribution $\{\varphi\}$ of coupling factors prescribed by a common rule:

$\displaystyle\mathbb{H}_{i}\equiv\sum_{\{\varphi\}}{\cal H}_{i;N}[\varphi].$

(19)

In Eq. (19) the sum ranges over the prescribed couplings. Each Hamiltonian in the family is Hermitian, so $\mathbb{H}$ must be also. As stated, all physical quantities – with one exception – preserve their canonical interrelationships as their expectations run through $\varphi$.

The exception is for identities relying explicitly on the completeness of Fock space associated with $\mathbb{H}$; for, Kraichnan’s ensemble averaging destroys completeness owing to its decohering action. Consider, symbolically, the ensemble projection operator

$\displaystyle\mathbb{P}\equiv\prod_{\{\varphi\}}\sum_{\Psi[\varphi]}\Psi[\varphi]\Psi^{*}[\varphi].$

Overwhelmingly the orthonormal eigenstates of $\mathbb{H}$ will be products of correlated, highly entangled, Kraichnan-coupled superpositions of states in the Fock space of each collective member over the distribution $\{\varphi\}$. Any expectation over the K-couplings, directly for $\mathbb{P}$, will cause only those terms to survive whose components in every factor $\varphi_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}$ within $\Psi[\varphi]$ find a counterpart in $\Psi^{*}[\varphi]$; see Eq. (27) below for the structure of $\varphi$. Any other legitimate but off-diagonal cross-correlations interfere mutually and are suppressed. Numerically, the integrity of the Kraichnan projection operator $\mathbb{P}$ is not preserved. [20]

Among other things, this loss of coherence leads to a clear computational distinction, within the same $\Phi$-derivable approximation, of static (instantaneous) correlation functions over against dynamic ones. Given that distinction, the dynamic and static response functions will still keep their canonical definitions and the sum-rule relations among them are still preserved. [16, 21]

To align the forthcoming presentation to the notion of crossing symmetry [22] for fermion interactions, we take the further step of antisymmetrizing the potential $V$. This is readily done in the interaction Hamiltonian, which now reads

$\displaystyle{\cal H}_{i;N}[\varphi]$

$\displaystyle\equiv$

$\displaystyle\!\!{1\over 2N}\!{\sum_{\ell_{1}\ell_{2}\ell_{3}\ell_{4}}}\!\!\!\!^{\prime}\!\varphi_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}{\langle k_{1}k_{2}|\overline{V}|k_{3}k_{4}\rangle}a^{*}_{\ell_{1}}a^{*}_{\ell_{2}}a_{\ell_{3}}a_{\ell_{4}}$

(20)

where

$${\langle k_{1}k_{2}|\overline{V}|k_{3}k_{4}\rangle}\equiv\frac{1}{2}({\langle k_{1}k_{2}|V|k_{3}k_{4}\rangle}-{\langle k_{2}k_{1}|V|k_{3}k_{4}\rangle}).$$

Invocations of the pair potential will now refer to Eq. (LABEL:kII11). Care has to be taken with signs for composite “direct” and “exchange” objects that turn out actually to be mixtures of both (yet still needing to be topologically distinguished), to make sure the accounting for $V$ itself stays consistent.

FIG. 2. Scheme for the self-consistent Hartree-Fock interaction energy, derived by insertion into Eq. (LABEL:kII11) of the Kraichnan coupling $\varphi^{\rm HF}_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}\equiv\delta_{\nu_{1}\nu_{4}}\delta_{\nu_{2}\nu_{3}}$. Dots: the antisymmetrized pair interaction. Broken lines: the originating potential. Full lines: one-body propagators. (a) Contributions to the interaction energy. (b) Self-consistency is made evident in the Dyson equation for the single-particle propagator $G$, where $G^{(0)}$ is the noninteracting counterpart. Nesting of $G\!:\!\overline{V}$ in the self-energy contribution means that the bare potential is present to all orders, albeit as a highly reduced subset of the physically exact self-energy.

We end the review of the Kraichnan formalism by recalling the simplest example for $\varphi$ generating the exchange-corrected random-phase, or Hartree-Fock, approximation. This choice is $\varphi^{\rm HF}_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}\equiv\delta_{\nu_{1}\nu_{4}}\delta_{\nu_{2}\nu_{3}}$. [16] The diagrammatic outcome of this non-stochastic Ansatz is illustrated in Figure 2. The expectation ${\langle{\cal H}_{i;N}[\varphi^{\rm HF}V]\rangle}$ of the interaction energy over $\varphi^{\rm HF}$ consists, almost trivially, of a pair of one-body Hartree-Fock Green functions attached to a single node representing $\overline{V}$.

The physically richer stochastic definitions of $\varphi$, due originally to Kraichnan [14, 15] were generalized and adapted in KI. [16] They will again be used in the next Section to build up a Kraichnan Hamiltonian for the parquet-generating correlation energy functional and all objects derived from it variationally.

In simplest form, the parquet equations take the bare interaction $\overline{V}$ and from it build up all possible iterations that require propagation of pairs of particles from one interaction to the next. This excludes any contributions to the interaction energy functional in which no interaction nodes are directly linked by such a pair; they cannot be broken down into simpler particle-pair processes. Two examples are shown in Fig. 3. We comment later on how these can always be added legitimately but ad hoc to the minimal $\Phi$ functional of immediate interest.

FIG. 3. Two skeleton diagrams for the exact correlation energy not reducible to pairwise-only propagation. (a) Next-order component, beyond first and second in $\overline{V}$, fulfilling the symmetry for $\Phi$ derivability but with no two nodes directly linked by a pair of single-particle propagators. (b) Next-higher-order term. Such terms are not generated by any Kraichnan formulation of the pairwise-only parquet Hamiltonian but can be added freely, albeit only ad hoc, to its $\Phi$-derivable functional.

There are three possible choices of randomized K-couplings for $\varphi$, each corresponding to the three channels included in parquet: the $s$ channel explicitly selects propagation of pairs of particles, the $t$ channel covers particle-hole pair propagation associated with long-range screening in the random-phase approximation, and its complement the $u$ channel describes the Hartree-Fock-like exchange counterpart to $t$. Recall that while antisymmetrization of the bare potential from $V$ to $\overline{V}$ superposes the actual $t$ and $u$ contributions, one has to continue distinguishing their diagrammatic representations topologically (and their relative sign) to preserve the quantitative outcomes of the Hamiltonian, Eq. (LABEL:kII11).

For each possible channel we define a stochastic coupling:

	$\displaystyle s{\rm~{}channel:~{}~{}}\sigma_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}$	$\displaystyle\equiv$	$\displaystyle\exp[\pi i(\xi_{\nu_{1}\nu_{2}}-\xi_{\nu_{3}\nu_{4}})];$		(22)
	$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}\xi_{\nu\nu^{\prime}}\in[-1,1]~{}~{}{\rm and}$		$\displaystyle\!\!\!\!\xi_{\nu^{\prime}\nu}=\xi_{\nu\nu^{\prime}},$		(23)
	$\displaystyle t{\rm~{}channel:~{}~{}}~{}\tau_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}$	$\displaystyle\equiv$	$\displaystyle\exp[\pi i(\zeta_{\nu_{1}\nu_{4}}+\zeta_{\nu_{2}\nu_{3}})];$		(24)
	$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}\zeta_{\nu\nu^{\prime}}\in[-1,1]~{}~{}{\rm and}$		$\displaystyle\!\!\!~{}\zeta_{\nu^{\prime}\nu}=-\zeta_{\nu\nu^{\prime}},$		(25)
	$\displaystyle u{\rm~{}channel:~{}~{}}\upsilon_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}$	$\displaystyle\equiv$	$\displaystyle\exp[\pi i(\vartheta_{\nu_{1}\nu_{3}}+\vartheta_{\nu_{2}\nu_{4}})];~{}~{}~{}~{}~{}~{}$		(26)
	$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}\vartheta_{\nu\nu^{\prime}}\in[-1,1]~{}~{}{\rm and}$		$\displaystyle\!\!\!\!\vartheta_{\nu^{\prime}\nu}=-\vartheta_{\nu\nu^{\prime}}.$		(27)

Their full outworkings are detailed in KI. The uniformly random numbers $\xi,\zeta,\vartheta$ are independently distributed; expectations over them mutually decouple and all factors conform to Eq. (18). Each is designed so that, in the stochastic average of the diagrammatic expansion of $\Phi$, product chains whose phases cancel identically from start to finish are immune to the averaging. All other product chains fail to cancel. Being stochastic they interfere destructively, vanishing in the limit of an arbitrarily large ensemble.

With respect to $t$ and $u$ channels, note that an exchange of labels $1\leftrightarrow 2$ or $3\leftrightarrow 4$ effectively swaps the definitions and thus the actions of their K-couplings. This is consistent with the physics of these channels as mutual exchange counterparts.

In the modality of Eq. (27), $\sigma$ generates the particle-particle Brueckner-ladder functional while the ring approximation is generated by $\tau$. Last, $\upsilon$ also results in a Brueckner-like functional where particle-hole ladders replace particle-particle ones. [16] There are no other options for pairwise propagation, just as with parquet.

None of the K-couplings of Eq. (27), alone or in twos, can cover all conceivable scattering arrangements strictly between particle and/or hole propagator pairs. All three must combine sequentially in all possible ways, while preventing any potential replication of terms if two or more K-couplings led to the survival of identical $\Phi$ terms. To first and second order in $\overline{V}$ one can show that all three elementary couplings generate identical contributions, inducing overcounting which would propagate throughout the nesting of self-energy insertions.

The solution to overcounting is to combine the couplings of Eq. (27) to inhibit any concurrency. We propose the candidate parquet K-coupling to be

	$\displaystyle\varphi$	$\displaystyle\equiv$	$\displaystyle 1-(1-\sigma)(1-\tau)(1-\upsilon)~{}~{}{\rm so}$		(28)
	$\displaystyle\varphi_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}$	$\displaystyle=$	$\displaystyle\sigma_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}+\tau_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}+\upsilon_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}$		(33)
			$\displaystyle-~{}\sigma_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}\tau_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}$
			$\displaystyle-~{}\tau_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}\upsilon_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}$
			$\displaystyle-~{}\upsilon_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}\sigma_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}$
			$\displaystyle+~{}\sigma_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}\tau_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}}\upsilon_{\nu_{1}\nu_{2}|\nu_{3}\nu_{4}},$

preserving overall the Hermitian property specified by Eq. (18).

In any diagram expanded to a given order in $\overline{V}$, the products of K-couplings in Eq. (33) may or may not resolve into a set of elementary closed cycles whose multiplicative chain is identically unity when $\varphi$-averaged (this means, by way of definition, that any sub-chain hived off within an elementary cycle would necessarily vanish through phase interference). Chains not resolving into a set of independent closed cycles over the contribution will be quenched to vanish in the Kraichnan expectation.

In Fig. 4(a) we show schematically the structures of the three possible pairwise multiple-scattering combinations contributing explicitly to the correlation energy functional $\Phi$, Fig. 4(b). The K-coupling $\sigma$ leads to the particle-particle Brueckner t-matrix $\Lambda_{s}$, while $\tau$ leads to the screened interaction $\Lambda_{t}$ and lastly $\upsilon$ is the $t$-exchange complement leading to the particle-hole Brueckner-type ladder $\Lambda_{u}$; this carries an implicit sign change relative to $\Lambda_{t}$ owing to the difference of one fermion loop count. From Eq. (33) all processes combine so the renormalized one-body propagators $G$ carry self-energy insertions to all orders in which $s$-, $t$- and $u-$ processes act synergetically, not competing in parallel but entering sequentially.

FIG. 4. (a) Definition of the fundamental all-order $s,t$ and $u$ interactions. Dots: antisymmetrized pair potential. (b) Symbolic definition of $\Phi$, the correlation energy functional (weightings induced by Eq. (5) are understood), following Kraichnan averaging over all K-couplings $\sigma,\tau,\upsilon$ as in Eq. (33) to remove overcounting when different K-couplings lead to identical diagrams. Although the skeleton graphs for $\Phi$ appear simple, their complexity is hidden within the self-consistent nesting of self-energy insertions in the propagators (solid lines) according to Eqs. (5)–(7). Since the $stu$ correlation energy is identical to that of the fluctuation-exchange model [6, 23], the Kraichnan construction already subsumes the essence of parquet. The combinatorial $stu$ structure is fully revealed only when the response to an external perturbation is extracted (see following).

Should two or even three channels have coincident closed cycles, the structure of $\varphi$ makes certain that the net contribution from this coincidence is always precisely unity; Eq. (33) ensures that there is no overcounting if $GG$ pairings from different channels gave rise to the same diagrammatic structure. Such terms can turn up only once in their locations within the expansion for $\Phi$, including iteratively in the self-energy parts.

All allowed pairwise-only combinations of scatterings, and only those, survive the expectation over $\{\varphi\}$ to lead to a legitimate $\Phi$-derivable correlation term with all its symmetries and conserving properties. [14, 16, 2] The individual energy functional of each component Hamiltonian in Eq. (19), being exact in its particular configuration prior to averaging, automatically has these symmetries in the renormalized expansion of Eq. (5). These are inherited by the diagrammatic structure of every term that survives the taking of expectations and ultimately by the complete averaged $\Phi$.

Having arrived at the maximally paired structure of $\Phi$ in Fig. 4(b) given the K-couplings of Eq. (33), the work of obtaining the parquet equations from it has been done, in one sense, in the analysis detailed by Bickers [6] for the equivalent heuristic FLEX model. However, the Hamiltonian prescription’s ramifications lead beyond the derivation of classic parquet.

FIG. 5. Systematic removal of a propagator $G$ internal to the self-energy $\Sigma[\varphi\overline{V};G]=\Lambda\!:\!G$, after Baym and Kadanoff, [1, 2] generates the primitive scattering kernel $\Lambda^{\prime}$. Removal of $G(32)$, solid line, simply regenerates $\Lambda$. Removing any internal $G$ other than $G(32)$ yields additional terms required for $\Phi$ derivability (microscopic conservation). Top line: beyond the $s$-channel ladder $\Lambda_{s}$ the non-crossing symmetric $t$-like term $\Lambda_{s;t}$ and $u$-term $\Lambda_{s;u}$ are generated. Middle line: generation of $\Lambda_{t}$ and the non-symmetric $\Lambda_{t;s}$ and $\Lambda_{t;u}$. Bottom line: generation of $\Lambda_{u}$ with $\Lambda_{u;t}$ and $\Lambda_{u;s}$.

The goal, then, is to reconstruct a parquet-like scattering amplitude $\Gamma$ using the ingredients provided by the Kraichnan machinery. We are still left to show its relation to the scattering function $\Lambda$, generator of the $\Phi$-derivable diagrammatic $stu$ expansion. While the renormalized structure of $\Lambda$ seems sparse compared with $\Gamma$ for parquet, [6] the structure for actual comparison is not $\Lambda$ but begins with the variation

$\displaystyle\Lambda^{\prime}$

$\displaystyle\equiv$

$\displaystyle\frac{\delta\Sigma}{\delta G}=\frac{\delta^{2}\Phi}{\delta G\delta G},$

(34)

which in fact is the source of the Ward-Pitaevsky identities. [19] One goes from there to set up the complete scattering interaction $\Gamma^{\prime}$ for the total system response to a perturbation.

The full outcome of the derivation of $\Gamma^{\prime}$ is the dynamical theory of Baym and Kadanoff; [2] in it, conservation entails the additional family of non-parquet diagrams $\Lambda^{\prime\prime}=\Lambda^{\prime}-\Lambda$ shown in Fig. 5. These contribute to every order of iteration. The topologies contained in $\Lambda^{\prime\prime}$ are not explicit in the renormalized $\Lambda$ embedded within $\Phi[\varphi\overline{V};G]$. Not being crossing symmetric they are not permitted, much less generated, within parquet. As with the normal parquet structures that we aim to exhibit from the stochastic Hamiltonian construction, the apparently extra correlation effects, actually mandated by conservation, remain virtual in the renormalized summation for $\Phi$ until elicited by an external probe.

Figure 5 details how functional differentiation gives rise to the non-parquet terms, typical of all $\Phi$-derivable descriptions. We want to trace how the purely parquet crossing symmetric $\Gamma$ diagrams make up a nontrivial component of the complete set for $\Gamma^{\prime}$, the total Baym-Kadanoff response kernel. $\Gamma$ is not equivalent to $\Gamma^{\prime}$; it is a proper subset. [6]

We emphasize the necessary presence, for $\Phi$ derivability, of the non-symmetric components $\Lambda^{\prime\prime}$. These are the approximate system’s attempt to match its $u$ terms, for example, with partner terms topologically like the two complementary channels $t$ and $s$; the same applies correspondingly to the primary $s$ and $t$ terms. However, the question is less why they break antisymmetry but how their presence fits into the cancellation of terms for conservation to govern the model’s response.

To unpack the nested correlations hidden in the renormalized form of $\Phi$ we turn to the full Kraichnan Hamiltonian prior to averaging and derive the response to a one-body nonlocal perturbation ${\langle k^{\prime}|U|k\rangle}$, which generally will have a time dependence also. [1, 2] External perturbations do not couple to the collective index $\nu$ but physically only to labels $k$. The interaction Hamiltonian in Eq. (LABEL:kII11) is augmented:

$\displaystyle{\cal H}_{i;N}[\varphi;U]$

$\displaystyle\equiv$

$\displaystyle\sum_{ll^{\prime}}{\langle k^{\prime}|U|k\rangle}a^{*}_{l^{\prime}}a_{l}+{\cal H}_{i;N}[\varphi;U=0].~{}~{}~{}$

(35)

Response to a local field is generated by setting ${\langle k^{\prime}|U|k\rangle}\to U(q)\delta_{k^{\prime},k+q}$, dynamically linking (contracting) the propagators that terminate and start at $U$.

Physical expectations are taken next, while retaining the individual K-couplings $\varphi$ to keep track of all pair processes. We use matrix notation with repeated indices to expand the intermediate sums.

The two-body Green function is $\delta G/\delta U$. [1] Working from Eq. (6), vary $G^{-1}$ for

	$\displaystyle\delta G^{-1}(12)$	$\displaystyle=$	$\displaystyle-\delta U(12)-\delta\Sigma[\varphi,G](12)~{}~{}~{}{\rm or}$
	$\displaystyle G^{-1}(12^{\prime})\delta G(2^{\prime}1^{\prime})G^{-1}(1^{\prime}2)$	$\displaystyle=$	$\displaystyle\delta U(12)+\frac{\delta\Sigma(12)}{\delta G(43)}\frac{\delta G(43)}{\delta U(56)}~{}~{}{\rm so}$
	$\displaystyle\frac{\delta G(21)}{\delta U(56)}\equiv G(25)G(61)$		$\displaystyle\!\!\!\!\!\!+~{}G(21^{\prime})G(2^{\prime}1)\Lambda^{\prime}(1^{\prime}3|2^{\prime}4)\varphi_{1^{\prime}3|2^{\prime}4}\frac{\delta G(43)}{\delta U(56)}$

where $\varphi$ explicitly partners the effective interaction $\Lambda^{\prime}$. There is no overcounting of the primary $s,t$ and $u$ contributions of $\Lambda$ since, once a line in any self-energy insertion is opened, it will not reconnect to its originating structure, joining instead a new and different (ultimately closed) loop. Symbolically, with $I$ the two-point identity,

	$\displaystyle[II-GG\!:\!\Lambda^{\prime}\varphi]\!:\!\frac{\delta G}{\delta U}$	$\displaystyle=$	$\displaystyle GG~{}~{}{\rm so}$		(36)
	$\displaystyle\frac{\delta G}{\delta U}$	$\displaystyle=$	$\displaystyle[II-GG\!:\!\Lambda^{\prime}\varphi]^{-1}\!:\!GG$		(38)
		$\displaystyle=$	$\displaystyle GG+GG\!:\![II-GG\!:\!\Lambda^{\prime}\varphi]^{-1}\Lambda^{\prime}\varphi\!:\!GG.$		(40)

Recalling Eq. (5), the form of the generating kernel $\Lambda$ (without the non-crossing-symmetric components from Fig. 5) can be read off from the structure of $\Phi$ as in Fig. 4, with the subsidiary kernels $\Lambda_{s},\Lambda_{t}$ and $\Lambda_{u}$:

	$\displaystyle\Lambda$	$\displaystyle=$	$\displaystyle\Lambda_{s}+\Lambda_{t}-\Lambda_{u}~{}~{}{\rm where}$		(41)
	$\displaystyle\Lambda_{s}$	$\displaystyle=$	$\displaystyle\overline{V}+{\phi}^{-1}\overline{V}\sigma\!:\!GG\!:\!\Lambda_{s}\varphi;$		(43)
	$\displaystyle\Lambda_{t}$	$\displaystyle=$	$\displaystyle\overline{V}+{\phi}^{-1}\overline{V}\tau\!:\!GG\!:\!\Lambda_{t}\varphi;$		(44)
	$\displaystyle\Lambda_{u}$	$\displaystyle=$	$\displaystyle\overline{V}+{\phi}^{-1}\overline{V}\upsilon\!:\!GG\!:\!\Lambda_{u}\varphi.$		(45)

To put the interactions on the same representational footing as $\overline{V}$, we factor out the outermost K-coupling, ${\phi}$. Intermediate chains that cancel right across will finally cancel with ${\phi}^{-1}$ appropriate to each channel. In Eq, (45) the $u$-channel term of $\Lambda$, being the exchange of the $t$-channel, carries the sign tracking the structural antisymmetry of $\Lambda_{t}$ on swapping particle (or hole) end points and restoring to $V$ its proper weight of unity in the intermediate summations.

From Eq. (40) the complete four-point amplitude is defined:

	$\displaystyle\Gamma^{\prime}$	$\displaystyle\equiv$	$\displaystyle{\phi}^{-1}\Lambda^{\prime}\varphi\!:\![II-GG\!:\!\Lambda^{\prime}\varphi]^{-1}$		(46)
		$\displaystyle=$	$\displaystyle{\phi}^{-1}[II-\Lambda^{\prime}\varphi\!:\!GG]^{-1}\!:\!\Lambda^{\prime}\varphi\!,$		(47)

In terms of $\Gamma^{\prime}$ the conserving two-body Green function becomes

$\displaystyle\frac{\delta G}{\delta U}$

$\displaystyle=$

$\displaystyle GG\!:\![II+\Gamma^{\prime}\varphi\!:\!GG].$

(48)

At this stage we specialize to the crossing symmetric sub-class of the expansion dictated by Eq. (47). After dropping $\Lambda^{\prime\prime}$ all the crossing-symmetric terms are gathered. The Equation is truncated and defines the now crossing symmetric kernel

	$\displaystyle\Gamma$	$\displaystyle\equiv$	$\displaystyle{\phi}^{-1}\Lambda\varphi\!:\![II-GG\!:\!\Lambda\varphi]^{-1}$		(49)
		$\displaystyle=$	$\displaystyle{\phi}^{-1}[II-\Lambda\varphi\!:\!GG]^{-1}\!:\!\Lambda\varphi\!,$		(50)

keeping in mind that the crossing symmetric $\Lambda$ consists only of the primary structures embedded in $\Phi$. While $\Gamma$ inherits antisymmetry, it forfeits conservation at the two-body level that is guaranteed for $\Gamma^{\prime}$. [24]

As with Eq. (47) above, Eq. (50) sums $\Gamma$ differently from parquet; but the underlying architecture of $\Gamma$ is the same. In the Equation, $s$-, $t$- and $u$-processes combine in all possible ways while inhibited from acting concurrently. The resolution of $\Gamma$ becomes a bookkeeping exercise: to make a systematic species by species inventory of all its permissible pair-only scattering sequences, irreducible in the parquet sense, within each channel, finally to weave these into all possible reducible contributions.

In Kraichnan’s description one resums $\Gamma$ by tracking how selective filtering works through the three possible K-couplings, while in the parquet approach one enforces, on the intermediate $GG$ pairs, the three distinct modes of momentum, energy and spin transfer characterizing the $s,t$ and $u$ channels. Our operation is the same as the pairwise topological argument for $\Gamma$ in FLEX, detailed in Ref. [6]. To achieve it, the first set of equations isolates the components that are not further reducible within each particular channel:

	$\displaystyle\Gamma_{s}$	$\displaystyle\equiv$	$\displaystyle\overline{V}+{\phi}^{-1}\Gamma\tau\!:\!GG\!:\!\Gamma_{t}\varphi-{\phi}^{-1}\Gamma\upsilon\!:\!GG\!:\!\Gamma_{u}\varphi;$		(51)
	$\displaystyle\Gamma_{t}$	$\displaystyle\equiv$	$\displaystyle\overline{V}-{\phi}^{-1}\Gamma\upsilon\!:\!GG\!:\!\Gamma_{u}\varphi+{\phi}^{-1}\Gamma\sigma\!:\!GG\!:\!\Gamma_{s}\varphi;$		(52)
	$\displaystyle\Gamma_{u}$	$\displaystyle\equiv$	$\displaystyle\overline{V}+{\phi}^{-1}\Gamma\sigma\!:\!GG\!:\!\Gamma_{s}\varphi+{\phi}^{-1}\Gamma\tau\!:\!GG\!:\!\Gamma_{t}\varphi.~{}~{}~{}~{}~{}~{}$		(53)

Manifestly, the components of $\Gamma_{s}$ couple only via $t$ or $u$, excluding any $s$-channel processes where cutting a pair sequence $\sigma GG$ yields two detached diagrams. Thus, taking the Kraichnan expectation of $(\Gamma_{s}-\overline{V})\sigma$, nothing survives; and so on for the other channels. The arrangement generates every legitimate convolution involving internally closed cycles of propagation through every channel within $\Gamma$ while ensuring irreducibility of the three component kernels.

Finally the complete $\Gamma$ is assembled:

	$\displaystyle\Gamma$	$\displaystyle=$	$\displaystyle\overline{V}+{\phi}^{-1}\Gamma\sigma\!:\!GG\!:\!\Gamma_{s}\varphi+{\phi}^{-1}\Gamma\tau\!:\!GG\!:\!\Gamma_{t}\varphi$		(55)
			$\displaystyle~{}~{}~{}~{}~{}~{}-{\phi}^{-1}\Gamma\upsilon\!:\!GG\!:\!\Gamma_{u}\varphi.$

In the Kraichnan average only the pairwise terms we have highlighted will survive. Then Eqs. (53) and (55) become identical to the FLEX parquet equations. [6] For each channel the total amplitude can also be recast to show its reducibility explicitly:

	$\displaystyle\Gamma$	$\displaystyle=$	$\displaystyle\Gamma_{s}+{\phi}^{-1}\Gamma\sigma\!:\!GG\!:\!\Gamma_{s}\varphi$
		$\displaystyle=$	$\displaystyle\Gamma_{t}+{\phi}^{-1}\Gamma\tau\!:\!GG\!:\!\Gamma_{t}\varphi$
		$\displaystyle=$	$\displaystyle\Gamma_{u}-{\phi}^{-1}\Gamma\upsilon\!:\!GG\!:\!\Gamma_{u}\varphi.$

In establishing full parquet the $\Phi$-derivable FLEX approximation has been taken as a suitable entry point for successive iterations aimed at approaching the full structure; but the initial self-energy $\Lambda\!:\!G$ is considered to fall short of a maximally correlated parquet. It is deemed necessary to feed the FLEX-derived crossing symmetric $\Gamma$ in Eq. (55) back into $\Sigma$ in Eq. (7) via the replacements [6] (ensuring that $G\!\!:\!\Gamma\!\!:\!G$ does not double up on terms previously included)

	$\displaystyle\Sigma(13)$	$\displaystyle\leftarrow$	$\displaystyle{\widehat{\Gamma}}(12|34)G(42)~{}~{}\text{in which}~{}~{}$		(61)
	$\displaystyle{\widehat{\Gamma}}(12|34)$	$\displaystyle\leftarrow$	$\displaystyle\overline{V}(12|34)$		(64)
			$\displaystyle+~{}\Gamma(12|3^{\prime}4^{\prime})G(4^{\prime}2^{\prime})G(3^{\prime}1^{\prime})\overline{V}(1^{\prime}2^{\prime}|34),~{}~{}~{}~{}~{}$

with the nonconforming piece, $\Gamma^{\prime\prime}=\Gamma^{\prime}-\Gamma$, naturally absent. Substitution of ${\widehat{\Gamma}}$ for $\Lambda$ in the self-energy assumes that no distinction should be made between the approximate self-energy kernel and the approximate two-body response kernel: that, as in the exact theory, they are one and the same. [5, 6] As a generator of new primitively irreducible structures Eq. (64) can be iterated at will.

In view of how the generic parquet Equations (58) and (60) always build up from at least the leading primitive irreducible, namely $\overline{V}$, any resulting $\Gamma$ must always incorporate $\Lambda$ from Eq. (45). Reopening lines in the self-energy ${\widehat{\Gamma}}[V,G]\!:\!G$ is always going to regenerate pieces including the unwanted non-parquet term $\Lambda^{\prime\prime}$.

To compare the behaviors of the different self-energy kernels for $stu$ and standard parquet, we use a result of Luttinger and Ward. [17] Equation (47) of that Reference provides an alternative formulation of the correlation energy when $\Lambda$ in Eq. (5) is the exact $\Gamma$ interaction:

	$\displaystyle\Phi[V;G]$	$\displaystyle=$	$\displaystyle-{\langle\ln(I-G^{(0)}\!\cdot\!\Sigma~{}\cdot)\rangle}-G[V]\!:\!\Sigma$		(66)
			$\displaystyle+~{}\int^{1}_{0}\frac{dz}{2z}G[V]\!:\!\Gamma[zV;G[V]]\!:\!G[V].$

The difference between the coupling-constant integral on the right-hand side of this identity and its counterpart in Eq. (5) is that the former keeps track only of the combinatorial factors for the $V$s in the original linked skeleton diagrams $\Gamma[V;G^{(0)}]$ but now with $G[V]$, containing $V$ at full strength, in place of each bare line $G^{(0)}$. By contrast, in the integral on the right-hand side of Eq. (5) the coupling factor attaches to all occurrences of $V$; that is, including those within $G[V]$ itself.

The correlation energy as given in Eq. (66) leads to two identities. Exploiting the equivalence of all propagators in the closed structure $G\!\!:\!\Gamma\!\!:\!G$ within the integral, varying on both sides with respect to the self-energy gives

	$\displaystyle\frac{\delta\Phi}{\delta\Sigma}$	$\displaystyle=$	$\displaystyle(I-G^{(0)}\!\cdot\!\Sigma)^{-1}\!\cdot\!G^{(0)}-G-\frac{\delta G}{\delta\Sigma}\!:\!\Sigma$		(68)
			$\displaystyle~{}~{}~{}~{}~{}~{}+\frac{\delta G}{\delta\Sigma}\!:\!\Gamma[V;G]\!:\!G$
		$\displaystyle=$	$\displaystyle-\frac{\delta G}{\delta\Sigma}\!:\!(\Sigma-\Gamma[V;G]\!:\!G)$		(69)
		$\displaystyle=$	$\displaystyle 0$		(70)

on using Eq. (7). The vanishing of this derivative establishes the correlation energy as an extremum with respect to perturbations, as these add linearly to $\Sigma$.

Next,

	$\displaystyle\frac{\delta\Phi}{\delta G}$	$\displaystyle=$	$\displaystyle{\Bigl{(}(I-G^{(0)}\!\cdot\!\Sigma)^{-1}\!\cdot\!G^{(0)}-G\Bigr{)}}\!\cdot\!\frac{\delta\Sigma}{\delta G}$		(72)
			$\displaystyle~{}~{}~{}~{}~{}~{}+~{}\Gamma[V;G]\!:\!G$
		$\displaystyle=$	$\displaystyle\Sigma.$		(73)

Consistency with Eq. (7) is confirmed.

We look at how Eq. (66) works in the $\Phi$-derivable case. Since it applies canonically in the case of the full Kraichnan Hamiltonian, the form survives the expectation over the $stu$ couplings, as will the form of the variational derivatives; the skeletal topology of the integrals on the right sides of Eqs. (5) and (66) is the same. In the stochastic expectations on the right-hand side of Eq. (66), $\Gamma$ goes over to the reduced $stu$ structure $\Lambda$ depicted in Fig. 4. This is because, in the coupling-constant integral, the pattern of surviving and suppressed products of factors $\varphi$ is identical with that leading to Eq. (5).