Explicit construction of the two-photon vertex in Nambu-Tuan's gauge-invariant theory of the magnetic-field dependence of the superconducting energy gap

In 1962 Nambu and Tuan derived gauge-invariant formulas for the reduction of the BCS energy gap of a bulk superconductor by a static magnetic field, to second order in the field strength, proving gauge invariance by Ward identities in Nambu's two-component Green's-function formalism. They acknowledged three unresolved issues: extension of the bulk formulas to finite geometries, explicit verification of the inferred structure of the two-photon vertex $\Gamma_{\mu\nu}$ , and reconciliation with the earlier Gupta-Mathur calculation. The present problem takes up the second item: whether the two-photon vertex $\Gamma_{\mu\nu}$ at second order in the vector potential has the inferred structure and satisfies the second-order Ward identity.

Background

In the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity, electrons near the Fermi surface form Cooper pairs and the electronic excitation spectrum acquires an energy gap $2\Omega$ ( $\Omega$ is the gap parameter), which is reduced by an applied magnetic field before superconductivity is destroyed at the critical field. A long-standing difficulty of the microscopic theory was gauge invariance: the naive quasiparticle response to the electromagnetic vector potential $A$ is not invariant under the gauge transformation $A\to A+\partial\chi$ , and the physical transverse response (the Meissner effect, the expulsion of magnetic fields from the superconductor) is recovered only when the collective excitations of the condensate are retained, chiefly the Anderson-Bogoliubov phase mode, a sound-wave-like oscillation of the pair phase with dispersion $q_0^2=v^2q^2$ ( $q_0$ the frequency, $q$ the wave vector, $v$ the sound velocity). Nambu (Phys. Rev. 117, 648 (1960)) reformulated BCS theory with a two-component (Nambu-Gorkov) spinor combining the electron field with its time-reversed partner, a self-energy $\Omega\tau_1+p\tau_3$ built from Pauli matrices $\tau_i$ , and showed that the electromagnetic vertex functions obey Ward identities, integral relations between Green's functions and vertices that encode current conservation, with the collective modes coupling to the longitudinal part of the vertex. Gupta and Mathur (Phys. Rev. 121, 107 (1961)), working from G. Wentzel's manifestly gauge-invariant reformulation of BCS theory, computed the magnetic-field dependence of the gap and obtained a gap shift depending directly on the vector potential $A$ . Nambu and Tuan (Phys. Rev. 128, 2622 (1962)) recalculated the gap reduction for a bulk superconductor at zero temperature to second order in the field strength, obtaining simple formulas in the long-wavelength limit $q\xi_0\ll 1$ (the local or London limit) and the short-wavelength limit $q\xi_0\gg 1$ (the nonlocal or Pippard limit), where $\xi_0=v_F/\pi\Omega_0$ is the coherence length, $\Omega_0$ the zero-field gap parameter, and $v_F$ the Fermi velocity; a characteristic feature of their small- $q$ result is that the gap shift depends on the field strength $H$ rather than on $A$ itself. Their proof of gauge invariance proceeds by showing that the one- and two-photon vertex functions, $\Gamma_\mu$ and $\Gamma_{\mu\nu}$ (the quasiparticle electromagnetic vertices carrying one and two photon insertions), satisfy the Ward identities as solutions of the underlying integral equations. For $\Gamma_\mu$ the requisite collective-mode structure had been established in Nambu's earlier work; for $\Gamma_{\mu\nu}$ , which enters the gap's field dependence at second order in $A$ , they could only infer the necessary structure: terms of the form $\tau_1 h_\mu h_\nu/(q_0^2-v^2q^2)$ , with $h_\mu=(v^2\mathbf{q},q_4)$ the photon four-vector of the source's convention (spatial part $v^2\mathbf{q}$ , fourth component $q_4$ ), coupled to the collective states, together with a $\tau_3\delta_{\mu\nu}q^2$ -type term that produces the field dependence of the gap, which must combine with the longitudinal contributions into the transverse tensor $\tau_3(\delta_{\mu\nu}q^2-q_\mu q_\nu)$ . They stated that they had not verified this by explicit calculation. The paper's concluding remarks further acknowledged that the bulk-medium formulas do not directly apply to the thin films and colloidal particles used in D. H. Douglass's electron-tunneling experiments (Phys. Rev. Lett. 6, 346 (1961)), and that the disagreement with Gupta and Mathur was unresolved.

Problem Statement

The problem, stated in Section 4 of Nambu and Tuan, Phys. Rev. 128, 2622 (1962), is to verify the inferred structure of the two-photon vertex. Within Nambu's gauge-invariant two-component (Nambu-Gorkov) formulation of BCS superconductivity at zero temperature, determine whether (i) the two-photon vertex $\Gamma_{\mu\nu}(p,p;q,-q)$ , the quasiparticle electromagnetic vertex at second order in the vector potential, contains terms of the inferred form $\tau_1 h_\mu h_\nu/(q_0^2-v^2q^2)$ , where $h_\mu=(v^2\mathbf{q},q_4)$ is the photon four-vector of the source's convention (spatial part $v^2\mathbf{q}$ , fourth component $q_4$ ), coupled to the collective (Anderson-Bogoliubov) states, together with the $\tau_3\delta_{\mu\nu}q^2$ -type term that produces the field dependence of the energy gap obtained in Section 3 of the source; and (ii) these terms combine with the longitudinal contributions to satisfy the second-order Ward identity, yielding the transverse combination $\tau_3(\delta_{\mu\nu}q^2-q_\mu q_\nu)$ that underlies the gauge invariance of the second-order gap-reduction formulas (Eqs. (3.12) and (3.25) of the source) for a bulk superconductor. A result that contradicts the inferred structure is an admissible resolution of the question. The two companion issues raised in the same concluding remarks, the application of the bulk formulas to finite geometries and the reconciliation with the Gupta-Mathur result, are out of scope here; the objective is solely the completion of the Ward-identity proof that the source's authors explicitly left unverified.

References

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