Validity and well-posedness of Nambu's generalized Hartree-Fock theory of superconductivity: omitted diagrams beyond weak coupling and existence of solutions of the nonlocal Eliashberg-Nambu equations

Nambu (Phys. Rev. 117, 648 (1960)) reformulated strong-coupling superconductivity as a self-consistent generalized Hartree-Fock scheme for nonlocal (energy-dependent) normal and anomalous self-energies, and proved gauge invariance of the electromagnetic response to first order in the external field via vertex corrections tied to the self-energy by Ward identities. He explicitly acknowledged that the scheme is uncontrolled: the importance of the diagram classes neglected beyond it, of order $g^{2}N$ and higher ($g$ the electron-phonon coupling, $N$ the density of states), is unknown and rests on the weak-coupling assumption $g^{2}N\ll1$; and the existence of solutions to the nonlocal self-consistency equations 'may be difficult to establish'. The spin-paramagnetism and nonlinear-gauge-invariance strands of Nambu's acknowledged limitations are not part of this question; the open problem is the controlled status of the approximation itself: (a) quantify the omitted vertex corrections and their effect on the self-energy, gap, critical temperature, and collective modes, in particular beyond the weak-coupling regime $g^{2}N\ll1$ on which the scheme rests; and (b) establish existence of solutions to the coupled nonlocal integral equations, which the paper concedes 'may be difficult to establish'.

Background

Superconductivity in ordinary metals arises from an effective attraction between electrons mediated by lattice vibrations (phonons). In BCS theory, electrons near the Fermi surface form bound Cooper pairs, and the pairing opens an energy gap in the quasiparticle excitation spectrum (a quasiparticle being the low-energy excitation of the interacting electron system, behaving approximately like a free particle with a renormalized energy). BCS treats the pairing interaction in a Hartree-Fock-like way: each electron moves in the self-consistent average field of the others. Because the phonon-mediated attraction is retarded (phonons respond much more slowly than electrons), a more accurate treatment keeps the electron self-energy, the energy shift of an electron due to the surrounding phonons and electrons, as a function of both momentum and frequency rather than a constant; such an operator is called nonlocal. The pair potential, the anomalous self-energy that plays the role of the energy gap, is likewise frequency dependent.

Nambu's 1960 paper reformulated this strong-coupling problem using techniques from quantum electrodynamics. In the Nambu formalism, electron and hole operators are combined into a two-component spinor and the normal and anomalous propagators are packed into a single $2\times2$ matrix Green's function; the self-energy is a matrix whose scalar part renormalizes the dispersion and whose off-diagonal part is the gap. Nambu wrote down the self-consistent integral equations for these nonlocal self-energies (the integral forms of his Eqs. (3.7)-(3.8); the same scheme, developed in parallel, is now called the Eliashberg equations), recovered the gap equation and the quasiparticle picture, and showed that gauge invariance survives the quasiparticle approximation. Gauge invariance here means that physical predictions do not depend on the arbitrary split of the electromagnetic field into scalar and vector potentials; it underlies the Meissner effect, the expulsion of static magnetic fields from a superconductor. It is maintained because the vertex part, the function describing how a quasiparticle couples to an external field, receives correction diagrams from the phonon and Coulomb interactions, and these vertex corrections are locked to the self-energy by Ward identities, relations expressing charge conservation between the derivative of the self-energy and the vertex at zero momentum transfer. The vertex integral equation also admits homogeneous solutions describing collective excitations (bound particle-hole pair modes, which in the charged system become the plasma oscillation).

Nambu was explicit that this construction rests on uncontrolled ground. In his concluding remarks he asks what will happen if one seeks corrections to the Hartree-Fock approximation by including processes or diagrams not considered there, noting that the importance of the neglected terms, of order $g^{2}N$ and higher, is not known, $g$ being the electron-phonon coupling and $N$ the density of states at the Fermi surface, so that $g^{2}N$ is the dimensionless coupling strength; the whole scheme additionally assumes weak coupling $g^{2}N\ll1$. In deriving the nonlocal equations he concedes that the existence of a solution to the self-consistency problem 'may be difficult to establish'. Two further acknowledged limitations concerned spin paramagnetism (the spin susceptibility of a superconductor, probed by the Knight shift, the shift of the nuclear magnetic resonance frequency caused by electron spin polarization): his vertex solution did not give a finite spin paramagnetism at $0$ K, in apparent tension with contemporary experiments, and he asked for theory and experiment to be developed further as a crucial test of BCS; and the gauge-invariance proof was only to first order in the external field, higher orders being explicitly 'beyond the scope of this paper'.

The modern descendants of these questions are standard objects. Migdal's theorem states that vertex corrections to the electron-phonon interaction are suppressed by the adiabatic ratio of phonon to electron energy scales, $\omega_D/E_F$ (Debye frequency to Fermi energy), which justifies neglecting them in conventional superconductors; when that protection fails (strong coupling, low density, low dimension, molecular or non-phonon bosons), one speaks of the breakdown of Migdal-Eliashberg theory, and this is actively studied both analytically and with determinant quantum Monte Carlo, a numerical method that treats interacting lattice models essentially exactly by sampling auxiliary fields (typically on the Holstein model, electrons coupled to dispersionless local phonons). A stylized kernel in which the effective interaction is proportional to $|\omega_n-\omega_m|^{-\gamma}$, where $\omega_n$ are Matsubara frequencies (the discrete imaginary frequencies at which finite-temperature perturbation theory is formulated), defines the gamma model, a standard testbed for rigorous work. The mathematical-physics program on BCS theory proves theorems about the instantaneous (non-retarded) BCS gap equation; the retarded, nonlocal Nambu-Eliashberg system has only very recently begun to be treated rigorously.

Problem Statement

Let $g$ be the electron-phonon coupling constant and $N(0)$ the electronic density of states at the Fermi surface, so that $\lambda\sim g^{2}N(0)$ is the dimensionless coupling. Nambu's scheme determines the renormalization function $Z(p,p_{0})$ (which fixes the normal self-energy) and the pair potential $\phi(p,p_{0})$, both nonlocal, that is, dependent on both momentum $p$ and Matsubara frequency $p_{0}$ (the discrete imaginary frequency of finite-temperature perturbation theory), as solutions of the coupled nonlinear integral equations (3.7)-(3.8) of Nambu (1960), equivalently the Eliashberg equations, with a phonon kernel and a Coulomb term; the electromagnetic vertex parts satisfy Bethe-Salpeter-type integral equations (the standard ladder-summed integral equations for the two-particle vertex) whose solutions obey Ward identities and include homogeneous collective-mode solutions. The problem concerns two facets of this one approximation scheme, acknowledged as separate open limitations in different parts of Nambu's paper:

(a) Validity. Determine what happens when corrections to the generalized Hartree-Fock approximation are included: quantify the importance and effects of the diagram classes not considered in Nambu's scheme, of order $g^{2}N$ and higher, on its physical predictions—the self-energy, the gap, the critical temperature $T_c$, and the collective modes—in particular outside the weak-coupling regime $g^{2}N\ll1$ that the scheme assumes, within the paper's own phonon-mediated formulation.

(b) Well-posedness. Establish the existence of a solution with nonvanishing pair potential $\phi$ (a superconducting, broken-symmetry branch) to the coupled nonlocal self-consistency equations of Nambu's Section 3, equivalently the Eliashberg equations, for the phonon plus screened-Coulomb kernel of the paper's own formulation—the question the paper itself concedes 'may be difficult to establish'. Uniqueness of solutions, the classification of normal versus superconducting branches, and the convergence of the iterative procedures used in numerical practice are closely related questions that lie outside the scope of this problem.

The two facets are distinct limitations acknowledged separately in the paper (the concluding remarks and Section 3, respectively) and are independently addressable.

References

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