Zero radius of convergence of the many-fermion linked-cluster expansion: extending Cooper's strongly coupled result to the full many-body problem and the fate of normal solutions at repulsive coupling

Leon N. Cooper (Phys. Rev. 122, 1021 (1961)) analyzed a 'strongly coupled' many-fermion system — the limit in which the variation of the single-particle kinetic energy is neglected and the interaction is restricted to a thin shell around the Fermi surface — and proved that the linked-cluster (connected-diagram) expansion of the vacuum expectation value of the resolvent operator has zero radius of convergence as a power series in the coupling constant, although the full unlinked resolvent series and the linked expansion of the evolution operator have finite radii. He left open whether this divergence extends to the full many-body problem with kinetic energy, what it implies physically — in particular whether repulsive interactions still admit a physically interesting class of 'normal' (Fermi-liquid-like) solutions (e.g. through a phenomenon such as zero sound), or whether the divergence signals qualitative changes such as a pairing instability — and how ground-state quantities, obtainable only as limits and not as convergent coupling-constant series, should be computed. Six decades of subsequent work has clarified fragments: Kohn and Luttinger argued, within perturbative pairing theory, that the normal state is unstable even for purely repulsive interactions; the Gell-Mann–Brueckner logarithm (recently proven rigorously in the mean-field regime) establishes non-analyticity for Coulomb systems; renormalized perturbation expansions have been proven convergent for two-dimensional models with counterterms; and modern diagrammatic-series studies find bare series that can converge to a normal branch while skeleton series can converge misleadingly. No general theorem settles the convergence of the linked-cluster expansion, or the existence of normal ground states, for many-fermion systems with short-range interactions in the thermodynamic limit.

Background

In a degenerate many-fermion system — a gas of spin-1/2 particles filling all single-particle momentum states up to the Fermi surface at zero temperature — the central quantitative problem is the shift of the ground-state energy and excitation spectrum produced by two-body interactions. A convenient exact object is the resolvent operator $R(s)=(H-s)^{-1}$ for the full Hamiltonian $H=H_0+gV$ (kinetic energy plus interaction, with the coupling constant $g$ written explicitly): its vacuum expectation value $(R(s))_0$, averaged over the unperturbed ground state, is analytic in the complex $s$ plane except for simple poles at the exact energy levels, so its analytic structure encodes the spectrum. Expanding $(R(s))_0$ in powers of $g$ and evaluating matrix elements of $V^n$ by Wick's theorem (the standard combinatorial rule for expectations of products of free-field operators) yields a representation as a sum of Feynman-like diagrams. Connected ('linked') diagrams — those that cannot be cut into two without severing an internal line — contribute terms proportional to the volume $N$ of the system, while disconnected diagrams carry higher powers of $N$. The linked-cluster theorem, established for the many-fermion ground state by Goldstone and by Hugenholtz, states that the ground-state energy shift per particle is given exactly by the sum of linked diagrams; this Goldstone series is the foundation of many-body perturbation theory (MBPT) in nuclear, condensed-matter, and quantum-chemical applications. Its formal derivation, however, assumes convergence, and Kohn and Luttinger had shown in 1960 that the Brueckner–Goldstone formalism (the linked-cluster expansion in its original nuclear-matter formulation) can break down when the unperturbed spectrum or the interaction lacks spherical symmetry.

Cooper (1961) therefore analyzed a solvable limit that keeps the many-body character but removes all convergence assumptions: the strongly coupled system, in which the variation of the kinetic energy is neglected (the kinetic-energy operator is replaced by its constant expectation value, so every unperturbed level is exactly degenerate and the interaction produces the entire level structure) and the interaction is restricted to a thin shell of relative thickness $\delta/k_F\ll 1$ around the Fermi surface, which supplies the cutoff otherwise provided by energy denominators. Diagonalizing the resulting finite-dimensional interaction matrix, Cooper expressed $(R(s))_0$ as the Laplace transform of the exponential of a function linear in the volume, and proved two things: the linked-cluster series of the resolvent expectation value, $B_L(s)=1+\sum_n (g/s)^n S_n$ (where $S_n$ sums connected vacuum diagrams of order $n$), has zero radius of convergence as a power series in $g$ whenever the spectrum contains more than one level, whereas the linked-cluster expansion of the evolution operator $U(-it)=e^{-Ht}$ — the object underlying the grand partition function — has a finite radius of convergence. Consequently the ground-state energy can be obtained as a limit but not as a convergent power series in the coupling constant, and, in the paper's words, a nontrivial interaction never results in a 'normal' system.

Here 'normal' refers to a system qualitatively similar to the ideal Fermi gas — the situation described by Landau's Fermi-liquid theory, in which low-energy excitations remain well-defined quasiparticles. The archetype of a non-normal state is the superconducting state of Bardeen–Cooper–Schrieffer (BCS) theory, in which an arbitrarily weak attraction between fermions of opposite spin and momentum reorganizes the ground state into a pair condensate; Cooper was one of the authors of that theory. Zero sound is a collective density oscillation, predicted by Landau, that a repulsive Fermi liquid sustains through repeated particle-hole scattering; in perturbation theory it appears as the collective pole obtained by resumming the ring (bubble) diagram subseries. The physical reading of the divergence result is thus tied to the classic instabilities of the Fermi surface: pairing for attraction, and — the case the paper explicitly left open — possible qualitative changes for repulsion.

The conclusion of the 1961 paper leaves three questions open: First, whether the arguments generalize: comparison of the strongly coupled and full linked-cluster expansions 'seems to make it unlikely that one should converge if the other does not', but 'this has not yet been done'; the many-body resolvent is in any case not expected to admit an expansion in powers of $1/s$, which decouples the two limits. Second, for repulsive potentials it is 'not known' whether the actual solution differs qualitatively from the normal one: the divergence may coexist with a physically interesting class of normal solutions (e.g. through a phenomenon such as zero sound) or may indicate qualitative changes not yet understood. Third, since the ground-state energy shift cannot be calculated as a convergent power series in the coupling constant, alternative non-perturbative approximations must be developed, ideally exploiting the analytic structure revealed by the analysis (the paper suggests a connection with the theory of entire functions of finite order (an entire function is one analytic everywhere in the complex plane; finite order measures its growth at infinity, and Hadamard's factorization theorem reconstructs such a function from its zeros) and Dirichlet series, since the zeros and growth of the evolution-operator expectation value encode the degeneracy and spacing of the energy spectrum).

Several later developments frame the modern form of these questions. The Kohn–Luttinger mechanism shows that the sharp Fermi surface itself generates an oscillatory long-range effective interaction, so that even purely repulsive forces destabilize the normal state toward Cooper pairing at high angular momentum and exponentially small temperature scales. The Gell-Mann–Brueckner high-density electron-gas result, $\epsilon_c = A\ln r_s + C + O(r_s^3)$ (with $r_s$ the density parameter, the mean inter-electron spacing in units of the Bohr radius, so small $r_s$ means high density), obtained by summing the most divergent ring diagrams, shows that for the Coulomb gas the coupling-constant series does not converge to an analytic function — the logarithm is the non-analytic remnant of a divergent diagram subseries; this formula has recently been proven rigorously in the mean-field regime by Christiansen, Hainzl, and Nam. In the opposite direction, rigorous renormalized perturbation theory — in which counterterms (subtractions added to the Hamiltonian to cancel divergent channels) remove the Cooper logarithm — has been proven by Feldman, Knörrer, and Trubowitz to converge with nonzero radius for two-dimensional models with short-range interactions. Finally, the modern diagrammatic-Monte-Carlo program, which samples Feynman series to all orders, has found that bare series can converge to a physical normal branch in strongly correlated regimes while partially resummed (skeleton) series can converge misleadingly to unphysical branches — so the convergence status of fermionic diagrammatic series remains an actively investigated, unsettled question.

Problem Statement

Determine whether the zero-radius-of-convergence result that Cooper proved for the strongly coupled many-fermion system extends to the full many-body problem, and what it means physically. Concretely, for a many-fermion system with the kinetic-energy variation restored (a non-degenerate unperturbed spectrum) and a nonzero short-range, translation-invariant, spin-independent two-body interaction of finite range and finite, nonzero scattering length in the thermodynamic limit, the problem asks:

1. (Analytic core) Prove or refute: the linked-cluster expansion of the vacuum expectation value of the resolvent operator — as a power series $B_L(s)=1+\sum_n g^n b_n(s)$ in the coupling constant $g$ at fixed $s$, where $b_n(s)$ sums the connected vacuum diagrams of order $n$ carrying their full $(H_0-s)^{-1}$ energy denominators (in the strongly coupled limit this reduces to the source's $B_L(s)=1+\sum_n (g/s)^n S_n$, with $S_n$ the connected vacuum diagrams of order $n$) — has zero radius of convergence, as it does in the strongly coupled limit. As the source notes, the many-body resolvent operator is not likely in any case to have an expansion in powers of $1/s$, and the expansion in $g$ is not related so easily to it as in the strongly coupled case; the ground-state energy shift is connected to $B_L(s)$ through Hugenholtz's limit expression, but it cannot itself be calculated as a power series in $g$. A refutation of the claim — a model of the stated class whose linked-cluster expansion has nonzero radius of convergence — resolves this component and removes the premise of components 2 and 3, which are stated for the divergent case.

2. (Physical content) For repulsive interactions in particular, decide whether the actual solution differs qualitatively from the 'normal' one: whether the divergence may coexist with a physically interesting class of normal solutions (e.g. through a phenomenon such as zero sound), or whether it indicates qualitative changes in the system that are not yet understood (e.g. a pairing instability of the Kohn–Luttinger type). Concretely: is there a concrete repulsive-interaction model of the stated class whose linked-cluster expansion has zero radius of convergence while its ground state is normal, or does no such model exist?

3. (Constructive corollary) Since the ground-state energy shift is obtainable only as a limit and not as a convergent coupling-constant series, develop approximations to the many-fermion system quite different from those usually made, namely a concrete non-perturbative scheme together with a demonstration of its convergence or analytic structure on at least one model of the stated class.

These three components form one scientific objective, exactly as posed in the source: establish the analytic structure in the coupling constant of the many-fermion linked-cluster expansion, read off its physical consequences for the existence of normal states, and develop non-perturbative approximations.

References

1. Leon N. Cooper, 'Theory of Strongly Coupled Many-Fermion Systems. I. Convergence of Linked-Cluster Expansions', Physical Review 122, 1021-1028 (1961). DOI: 10.1103/PhysRev.122.1021, https://doi.org/10.1103/PhysRev.122.1021
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