Proper completions of truncated many-body perturbation expansions: existence, uniqueness, construction, and EPV-free computation

Stölan and Cooper (1966) showed that when a truncated subset of connected vacuum diagrams is exponentiated, the discarded exclusion-principle-violating (EPV) processes no longer cancel and the resulting many-body amplitudes become meaningless: for the strong-coupling BCS reduced (pairing) Hamiltonian, the ladder-only sum of Fukushima and Fukuda grows factorially, violates rigorous bounds, and corresponds to no Hamiltonian. They proposed to repair such approximations by adjoining a 'proper completion' — a set of graphs representing only EPV processes that restores exact EPV cancellation — and showed, for that one model, that the BCS graph set is the proper completion of the ladder graphs and that in the strong-coupling limit the proper completion, if it exists, is unique. The open problem is the general program: for an arbitrary truncated subset of connected diagrams of a general many-fermion Hamiltonian, settle existence and uniqueness of the proper completion, give a general construction algorithm, provide a tractable and convergent scheme for the EPV-free vacuum amplitude $\langle 0|U(-it)|0\rangle_{\mathrm{EPV}^{(0)}}$, and reconcile the completion with ordering in the small parameter that defined the truncation. Only the authors' own 1971–1977 sequels engage the construction, while modern coupled-cluster, CEPA, and RPA-exchange lines solve neighboring instances within ansatz-specific frameworks; the general questions remain unanswered.

Background

Many-body perturbation theory (MBPT) computes ground states and spectra of interacting fermion systems by expanding in the interaction. Splitting the Hamiltonian as $H=T+H_1$ (kinetic plus interaction parts) and taking the unperturbed ground state $|0\rangle$ to be the filled Fermi sea, one studies the vacuum amplitude of the evolution operator $U(-it)=\exp(-H_1 t)$ with $t$ real,

$$\langle 0|U(-it)|0\rangle=\sum_m |\langle 0|m\rangle|^2 e^{-E_m t}=\exp[B_0(-Vt)],$$

where the sum runs over exact eigenstates $|m\rangle$ and $B_0(-Vt)=\sum_n \frac{S_n}{n!}(-Vt)^n$ is a series in the coupling $V$ built only from the connected (linked) vacuum-to-vacuum diagrams $S_n$ of order $n$. The moments $\langle 0|H_1^n|0\rangle$ decompose into products of connected parts — the linked-cluster structure rooted in Wick's theorem (the standard combinatorial rule for evaluating expectation values of products of fermion creation and annihilation operators) and in Goldstone's proof of the linked-cluster (Brueckner many-body) theory. The ground-state energy is read off from the large-$t$ asymptotics of $B_0(-Vt)$, which must be dominated by a term linear in $t$.

Wick's-theorem sums over the labels of intermediate particle and hole lines disregard the restriction that intermediate states be allowed by the Pauli exclusion principle. Individual diagrams therefore contain exclusion-principle-violating (EPV) processes — contributions in which an intermediate state would hold two fermions in the same single-particle level; H. P. Kelly introduced the abbreviation in atomic MBPT. In the complete expansion such contributions cancel exactly among all graphs, a consequence of fermion anticommutation. But when one truncates — keeping only a selected subset $\{S_n^{(0)}\}$ of the connected diagrams (chosen, for example, because they dominate in a small parameter such as the inverse system volume $\Omega$) and exponentiating that subset — the canceling partners are discarded and uncancelled EPV contributions proliferate at high order.

Stölan and Cooper analyzed this failure for the BCS reduced Hamiltonian (the pairing model that keeps only the interaction of time-reversed pairs $({\bf k}\uparrow,-{\bf k}\downarrow)$) in the strong-coupling limit, in which kinetic-energy variation across a thin shell around the Fermi surface is neglected and the exact answer is known. Re-examining a calculation of Fukushima and Fukuda, who kept only ladder graphs (chains of successive particle–particle scattering vertices) on the grounds that ladders scale as $\Omega^0$ while omitted graphs are $O(\Omega^{-1})$, they showed that the ladder sum $S_n^{\mathrm{FF}}=(n-1)!N^n$ ($N$ the number of pair levels) implies $B_0(-Vt)=-\ln(1+NVt)$: wrong asymptotics, a wrong ground state for attractive potentials, and a violation of a rigorous upper bound on $|\langle 0|H_1^n|0\rangle|$ that grows only combinatorially in $n$ (a bound of the type introduced in Cooper's linked-cluster convergence analysis). Hence the selected graph set 'does not correspond to an actual Hamiltonian'. Other natural subclasses (ascending ladders, etc.) fail similarly.

They proposed, first, a validity criterion — an approximation should not lead to the inclusion of large numbers of uncancelled EPV processes — and, second, a repair construction. Define the EPV-free function $\langle 0|U(-it)|0\rangle_{\mathrm{EPV}^{(0)}}$ by exponentiating the truncated set and subtracting all EPV contributions. A proper completion of $\{S_n^{(0)}\}$ is a set $\{S_n^{(1)}\}$ such that (a) every process represented by the graphs of $\{S_n^{(1)}\}$ violates the exclusion principle, and (b) the completed set $\{S_n^{(0)}+S_n^{(1)}\}$, inserted into the standard exponentiation, reproduces exactly this EPV-free function. Since any set of connected graphs generated by a Hamiltonian — however truncated the Hamiltonian — yields a $U$-matrix amplitude free of EPV processes, proper completion converts an arbitrary graph set into one that might come from a Hamiltonian. For the strong-coupling BCS reduced Hamiltonian they showed that the full graph set splits into ladder graphs plus nonladder graphs whose characteristic parts all violate statistics (including 'bubbles', i.e. passive unexcited particle loops), so the BCS graph set is the proper completion of the ladders; and in this strong-coupling limit the proper completion, if it exists, is unique.

What the paper leaves open, in its own words, is everything beyond this one worked example: no guarantee that the construction 'can always be carried out'; uniqueness only in the strong-coupling limit; the EPV-free function 'is of course harder to calculate than the original', with no computational scheme or convergence control; and the completion 'is not characterized by the small parameter', which leaves the parameter dependence of the completed set 'obscured'.

The underlying concern is alive in modern electronic-structure theory, though addressed with different machinery. Coupled-cluster methods build the wavefunction as an exponential $e^{T}|\Phi_0\rangle$ of connected excitation operators, which enforces size extensivity (correct linear scaling of the energy with particle number) and a consistent order-by-order EPV accounting; coupled-electron-pair approximations (CEPA) add explicit EPV correction terms; and in the random phase approximation (RPA) — the resummation of direct ring diagrams without exchange — uncanceled EPV contributions bias correlation energies and pair densities, motivating screened-exchange corrections — second-order screened exchange (SOSEX) and adjacent-pairs exchange (APX), which exchange particle/hole pairs inside the direct-ring diagrams — designed to cancel the largest EPV sets. None of these lines uses or develops the proper-completion formalism, and none answers its general questions.

Problem Statement

The problem, stated in the source's framework and notation, is to establish the proper-completion program of Stölan and Cooper (1966) in general, beyond the single worked example of the strong-coupling BCS reduced Hamiltonian. Let $H_1$ be the interaction Hamiltonian of a many-fermion system; let $\{S_n\}$ be the full set of connected vacuum-to-vacuum graphs, $S_n$ collecting those of order $n$; and let $\{S_n^{(0)}\}$ be an arbitrary truncated subset selected by some approximation scheme (typically by a small parameter). Let $\langle 0|U(-it)|0\rangle_{\mathrm{EPV}^{(0)}}$ be the EPV-free function obtained, as defined above, by exponentiating the truncated set and subtracting all EPV contributions. The joint objective is to convert the EPV-cancellation validity criterion into a general, usable method for constructing physically consistent diagrammatic approximations, which comprises the following five sub-questions:

(1) Existence. Does a proper completion $\{S_n^{(1)}\}$ — a set whose graphs represent only exclusion-principle-violating processes and whose adjunction makes the recombined exponential exactly EPV-free — exist for general Hamiltonians and general truncated subsets? If not universally, under what precise conditions on the Hamiltonian and on $\{S_n^{(0)}\}$ does it exist or fail?

(2) Uniqueness. Uniqueness is established, as a statement conditional on existence, only in the strong-coupling limit of the BCS reduced Hamiltonian; is the proper completion unique at finite coupling, for other Hamiltonians, and for general truncations?

(3) Construction. Is there a general algorithm that, given the Hamiltonian and $\{S_n^{(0)}\}$, constructs $\{S_n^{(1)}\}$?

(4) Calculability. Is there a tractable computational scheme, with convergence control, for the EPV-free amplitude $\langle 0|U(-it)|0\rangle_{\mathrm{EPV}^{(0)}}$ and for the energies extracted from its long-time asymptotics — given that this function is harder to calculate than the naive exponentiation of the truncated set and that no scheme or error control exists in the source?

(5) Parameter consistency. Since the completion is not characterized by the small parameter that selected $\{S_n^{(0)}\}$, how can the completed expansion be organized so that ordering in that parameter remains controlled?

These five questions state the open program beyond the single worked example, and each is independently reportable: a definite answer to any one — positive or negative — is a self-contained partial result. A general construction for (3) subsumes the affirmative branch of (1), while (1) also asks for the precise conditions under which no completion exists; and (2) is posed conditionally on existence. Any answer must reproduce the strong-coupling BCS case, in which the proper completion of the ladder graphs is the set of nonladder BCS graphs and the completion is unique. The problem concerns arbitrary Hamiltonians, truncations, and all orders.

References

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