The clean ss-wave BCS superconductor at T=0T=0 admits a remarkably simple low-energy description: Aitchison, Ao, Thouless, and Zhu showed that the low-frequency, long-wavelength phase dynamics of the pair field is equivalent, once the terms required by Galilean invariance are included, to a time-dependent nonlinear Schrödinger Lagrangian (TDNLSL), and to two coupled TDNLSL's when the pair-field modulus is allowed to vary. That derivation was truncated at leading nontrivial order in the derivative-expansion parameters ∂t/∣Δ0∣\partial_t/|\Delta_0| and vF∇/∣Δ0∣∼ξ∇v_F\nabla/|\Delta_0|\sim\xi\nabla, and at quadratic order in the relative amplitude fluctuation, and it left the Schrödinger-field amplitude ρ\sqrt{\rho} without any microscopic relation to the pair-field modulus ∣Δ∣|\Delta|. This task asks for the controlled quantitative completion of that program: compute the explicit higher-order time- and space-derivative terms and the nonlinear amplitude interactions of the amplitude-inclusive effective Lagrangian, and apply the extended Lagrangian to the detailed vortex-core structure and the near-core amplitude and phase profiles beyond the exponential far-core approximation available at leading order. Later literature has resolved several neighboring sectors (next-to-leading-order Goldstone-field Lagrangian across the BCS–BEC crossover, fourth-order phase-only actions, clean and dirty finite-temperature effective actions), but the amplitude-inclusive coupled-TDNLSL extension and its vortex-core application remain open.

Background

Bardeen–Cooper–Schrieffer (BCS) theory describes conventional superconductivity as a condensate of Cooper pairs, encoded near the Fermi surface in a complex pair field Δ(x,t)=∣Δ∣eiθ\Delta({\bf x},t)=|\Delta|e^{i\theta} carrying charge −2e-2e. Integrating the microscopic fermions out of the BCS Lagrangian gives an effective action Seff=−i Trln⁡G−1−1g∫d4x ∣Δ∣2S_{\rm eff}=-i\,{\rm Tr}\ln G^{-1}-\frac{1}{g}\int d^4x\,|\Delta|^2, where GG is the Nambu Green function (the 2×22\times2 matrix propagator mixing electrons and holes) of the paired fermions. In a clean (impurity-free) superconductor at zero temperature the low-energy dynamics is carried by slow variations of the phase θ\theta — the Goldstone mode of the spontaneously broken global U(1)U(1) symmetry, known here as the Bogoliubov–Anderson sound mode — while the amplitude fluctuation of ∣Δ∣|\Delta| is a gapped collective mode (in modern terminology the Higgs mode, with weak-coupling mean-field gap 2∣Δ0∣2|\Delta_0|, where Δ0\Delta_0 is the equilibrium gap; its dynamics was first obtained by Abrahams and Tsuneto). The relevant microscopic scales are N(0)N(0), the single-spin density of states at the Fermi surface; the Fermi velocity vFv_F; and the coherence length ξ∼vF/∣Δ0∣\xi\sim v_F/|\Delta_0| (up to numerical factors), the spatial scale over which the pair amplitude can vary.

The static Ginzburg–Landau theory is reliable only near the critical temperature TcT_c. For time-dependent phenomena at low temperatures, many proposed time-dependent Ginzburg–Landau (TDGL) equations and field-theoretic effective actions coexisted without consensus. A key constraint is Galilean invariance: the microscopic electron Lagrangian is invariant under Galilean boosts, so any effective theory must be too, and this fixes which combinations of θ˙\dot\theta and ∇θ\nabla\theta may appear (for example the Galilean scalar θ˙+(∇θ)2/4m\dot\theta+(\nabla\theta)^2/4m); naive expansions in the real and imaginary parts of Δ\Delta violate this. Aitchison, Ao, Thouless, and Zhu settled the T=0T=0 question: the low-frequency, long-wavelength phase dynamics of a clean ss-wave (isotropic-gap) BCS superconductor is equivalent to a time-dependent nonlinear Schrödinger Lagrangian (TDNLSL) — the Lagrangian of an interacting Schrödinger field, Lψ=iψ∗ψ˙−14m∣∇ψ∣2−V(∣ψ∣2)L_\psi=i\psi^*\dot\psi-\frac{1}{4m}|\nabla\psi|^2-V(|\psi|^2) in their normalization, the same structure as the Gross–Pitaevskii theory of a Bose superfluid such as liquid helium — once the Galilean-invariance-required terms are included. The hydrodynamic phenomenology of superfluids at T=0T=0 (continuity equation, Bernoulli and Euler relations, momentum density proportional to number current) and the Magnus force (the transverse lift force on a vortex moving in a superflow) then follow directly. A density-matrix derivation of the same nonlinear Schrödinger equation by three of the same authors, a concurrent derivation by Stone from Fermi-surface bosonization (a field-theoretic method that re-expresses the low-energy fermionic degrees of freedom near the Fermi surface in terms of bosonic fields), and an earlier symmetry-based proposal of these hydrodynamic relations by Greiter, Wilczek, and Witten all agree. When the modulus ∣Δ∣|\Delta| is allowed to vary, the effective Lagrangian becomes that of two coupled TDNLSL's, for fields ψ1,2\psi_{1,2} of charge −2e-2e joined by a mass term; the gapped amplitude mode resides in the relative field.

This description is a derivative expansion in ∂t/∣Δ0∣\partial_t/|\Delta_0| and vF∇/∣Δ0∣∼ξ∇v_F\nabla/|\Delta_0|\sim\xi\nabla, valid for ω≪∣Δ0∣\omega\ll|\Delta_0| and k≪ξ−1k\ll\xi^{-1}, truncated at leading nontrivial order; the amplitude sector is truncated at quadratic order in the relative amplitude ϵ=δ∣Δ∣/(3∣Δ0∣)\epsilon=\delta|\Delta|/(\sqrt{3}|\Delta_0|), which already yields the gapped amplitude mode and a static amplitude solution decaying exponentially over ξ/6\xi/6 — the behavior expected far from a vortex core (in a type-II superconductor, the vortex core is the region of suppressed order parameter of size ∼ξ\sim\xi; in the microscopic description it hosts bound quasiparticle states, the Caroli–de Gennes–Matricon states). The source states that including the appropriate higher-derivative terms 'should make possible some predictions about the vortex core structure', and stresses that the Schrödinger wavefunction ψ\psi is distinct from the pair field Δ\Delta: they share only the phase θ\theta, and 'there is no simple relation between ∣Δ∣|\Delta| and ρ\sqrt{\rho}', where ρ=∣ψ∣2\rho=|\psi|^2 is the electron-number density conjugate to θ\theta.

Later literature resolved several neighboring sectors. For the phase-only, neutral Goldstone field, Mañes and Valle computed the next-to-leading-order effective Lagrangian at T=0T=0 across the entire BCS–BEC crossover (the continuous evolution, with increasing attraction, from weakly bound, strongly overlapping Cooper pairs to a Bose–Einstein condensate of tightly bound dimers governed by a Gross–Pitaevskii equation), with all one-loop coefficient functions in closed form; Benfatto, Toschi, and Caprara had earlier derived the phase-only effective action to fourth order, including the anharmonic vertices required by Galilean invariance. Van Otterlo, Golubev, Zaikin, and Blatter derived the effective dynamical theory of BCS superconductors at both T∼TcT\sim T_c and T≪TcT\ll T_c, in both clean and impurity-dominated (dirty) limits, in manifestly gauge-invariant form. De Palo, Castellani, Di Castro, and Chakraverty derived the T=0T=0 effective action with the density conjugate to the phase across the crossover. The amplitude-mode (Higgs) dynamics and its damping are extensively developed by microscopic approaches (gauge-invariant kinetic theory, the pseudospin representation of the BCS degree of freedom, pump–probe spectroscopy, and Gaussian fluctuations) and reviewed by Shimano and Tsuji; vortex-core structure is computed by self-consistent Bogoliubov–de Gennes methods (the mean-field treatment of the microscopic quasiparticles in a spatially nonuniform paired state; Gygi and Schlüter). None of these extends the coupled TDNLSL of the source beyond its stated truncations or supplies the ∣Δ∣|\Delta|–ρ\sqrt{\rho} relation.

Problem Statement

For the clean ss-wave BCS superconductor at T=0T=0 (the neutral theory; when electromagnetic coupling is needed it is restored, as in the source, by the minimal substitutions θ˙→θ˙−2eA0\dot\theta\to\dot\theta-2eA_0 and ∇θ→∇θ+2eA\nabla\theta\to\nabla\theta+2e{\bf A} with e>0e>0), extend the effective Lagrangian of Aitchison, Ao, Thouless, and Zhu beyond its stated truncations by one further order in the joint derivative-and-amplitude expansion fixed in (1) below, so as to extend its quantitative validity beyond the leading-order regime ω≪∣Δ0∣\omega\ll|\Delta_0|, k≪ξ−1k\ll\xi^{-1}. The requested results, which jointly serve this single objective of a controlled quantitative effective theory, are:

(1) The explicit form and magnitude of the higher-order time- and space-derivative terms of the amplitude-inclusive effective Lagrangian — extending the coupled TDNLSL from its leading nontrivial order in ∂t/∣Δ0∣\partial_t/|\Delta_0| and vF∇/∣Δ0∣∼ξ∇v_F\nabla/|\Delta_0|\sim\xi\nabla to next-to-leading order in this derivative expansion, whose order is fixed by the Galilean/Schrödinger counting ∂t∼∇2\partial_t\sim\nabla^2 (the two small parameters ∂t/∣Δ0∣\partial_t/|\Delta_0| and (ξ∇)2(\xi\nabla)^2 are the same order, because under the source's Galilean transformation (16) θ˙∼mv2\dot\theta\sim mv^2 while ∇θ∼mv\nabla\theta\sim mv), so that the leading order comprises the two-derivative kinetic terms together with the non-derivative potential and next-to-leading order means all terms of one further order, the four-derivative structures ∇4\nabla^4, ∂t∇2\partial_t\nabla^2 and ∂t2\partial_t^2; and including the nonlinear amplitude interactions (powers of the relative amplitude ϵ=δ∣Δ∣/(3∣Δ0∣)\epsilon=\delta|\Delta|/(\sqrt{3}|\Delta_0|) through quartic order O(ϵ4)O(\epsilon^4), required at leading order in derivatives), with Galilean (and gauge) invariance maintained at each order.

(2) Application of the extended effective Lagrangian to the vortex-core problem: the detailed vortex-core structure and the spatial profiles of the amplitude and phase fields near the core, beyond the exponential far-field solution (decay length ξ/6\xi/6) available at leading order, computed at the accuracy of the extended next-to-leading-order Lagrangian and hence controlled at core distances rr with ξ/r≪1\xi/r\ll1; the innermost region r≲ξr\lesssim\xi, where the gradient expansion breaks down, is not required.

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