Extending the Ao–Thouless–Zhu nonlinear Schrödinger equation for superconductors beyond zero temperature, clean s-wave pairing, and slow variation

In 1995 Ao, Thouless, and Zhu derived a Galilean-invariant nonlinear Schrödinger equation (NLSE) for the superconducting condensate — the Cooper-pair center-of-mass wavefunction whose squared modulus is the superfluid density — at zero temperature for a clean s-wave superconductor, using a Hartree–Fock–Bogoliubov factorization of the density matrix and a Born–Oppenheimer separation of fast pair formation from slow condensate motion. The equation contains pseudo-potentials $a_0$ and $\mathbf{a}$, the response of the pair's internal degrees of freedom to the condensate motion, which were left as formal expressions; its derivation rests on explicit restrictions: quasiparticles exponentially rare (near zero temperature), clean s-wave singlet pairing, order-parameter and gauge-field variations slow on the coherence-length and pair scales (effectively restricting the charged equation to type-II superconductors), and neglect of corrections beyond the factorization. The open problem is to extend this effective equation beyond those restrictions — to finite temperature with quasiparticle back-action, to quantitative determination of $a_0$ and $\mathbf{a}$ including non-singlet pairing and dirty or strongly anisotropic materials, to regimes of rapid spatial or temporal variation (sharp interfaces, strong disorder, cores of vortices with very short coherence length), and to type-I regimes where gauge fields vary on the Cooper-pair scale — and to connect the extended equation to low-temperature vortex dynamics (vortex mass, dissipation) in high-$T_c$ oxide superconductors.

Background

A superconductor below its critical temperature possesses off-diagonal long-range order (ODLRO), quantified by a macroscopic Cooper-pair wavefunction, which underlies zero resistance, flux quantization, and the Josephson effects. The static Ginzburg–Landau (GL) equation, valid near the transition temperature $T_c$, describes the spatial structure of this order parameter and was later given a microscopic basis in the Bardeen–Cooper–Schrieffer (BCS) theory of paired electrons. For time-dependent phenomena a time-dependent Ginzburg–Landau (TDGL) equation is available near $T_c$, but its form at low temperature was long unsettled: naive TDGL equations are not Galilean invariant (they do not keep their form when the whole system is viewed from a uniformly moving frame), and their relaxational structure conflicts with the nondissipative dynamics expected near zero temperature.

Ao, Thouless, and Zhu (1995) addressed this at $T\approx 0$ for a clean s-wave superconductor in two steps. First, the $N$-electron density matrix is factorized in the manner of Hartree–Fock–Bogoliubov (HFB): two-body reduced density matrices are written in terms of the one-body density matrix plus a pair wavefunction $\Psi$ (the ODLRO object), an approximation of the Dirac type familiar from time-dependent Hartree–Fock theory; this yields a closed self-consistent equation set, valid at any temperature. Second, because the BCS energy gap makes quasiparticle excitations exponentially rare at low temperature, the pair wavefunction is split into a slow center-of-mass part $\psi(\mathbf{R},t)$ and a fast internal part $\phi(\mathbf{r},\ldots)$ describing pair formation — a Born–Oppenheimer separation of the kind used for electrons and nuclei in molecules. The result is a nonlinear Schrödinger equation for the condensate,

$$i\hbar\,\frac{\partial\psi}{\partial t} = \frac{1}{4m}\left[-i\hbar\nabla_{\mathbf{R}} + \mathbf{a}(\mathbf{R},\psi,\psi^*)\right]^2\psi + \left[a_0(\mathbf{R},\psi,\psi^*) + V(\mathbf{R},\psi,\psi^*)\right]\psi,$$

with minimal electromagnetic coupling $-i\hbar\nabla_{\mathbf{R}} \to -i\hbar\nabla_{\mathbf{R}} - 2e\mathbf{A}$ and $a_0 \to a_0 + 2eA_0$. Here $\psi$ is normalized to the Cooper-pair density (proportional to the superfluid density, not to the gap amplitude or the condensate fraction), $2m$ and $2e$ are the pair mass and charge, $V$ is the Born–Oppenheimer potential (twice the chemical potential plus small corrections), and $a_0$ and $\mathbf{a}$ are pseudo-potentials — Berry-connection-like responses of the fast internal pair dynamics to the slow condensate motion, given only as formal integral expressions involving $\phi$. For a uniform s-wave state both vanish; for a deformed (nonuniform) pair $a_0 \neq 0$ while $\mathbf{a} = 0$; for non-singlet pairing the authors pointed out that the pseudo-vector $\mathbf{a}$ may be nonzero. The equation is Galilean invariant and unifies Josephson relations, the Magnus (transverse) force on a vortex — proportional to the superfluid density — and the Bogoliubov–Anderson collective phase mode (velocity $v_F/\sqrt{3}$ in the neutral case, becoming the plasma mode when charged), where $v_F$ is the Fermi velocity.

The derivation carries explicit restrictions that delimit the present problem: (i) near zero temperature, with quasiparticle excitations exponentially rare — the finite-temperature regime, where a normal component exists and quasiparticles can act back on the condensate, is not addressed; (ii) the Born–Oppenheimer/gradient expansion requires order-parameter and electromagnetic variations slow on the scale of the coherence length $\xi$ (the pair size) and frequencies small compared with the energy gap — rapid variations, such as near sharp interfaces, strong disorder, or the cores of vortices with very short coherence length, lie outside its validity; (iii) corrections beyond the HFB factorization (exchange and band-structure effects) are neglected, and $a_0$ and $\mathbf{a}$ are never evaluated quantitatively; (iv) clean s-wave singlet pairing is assumed "for definiteness" — non-singlet pairing, dirt, and strong anisotropy change the pair's internal structure and hence the pseudo-potentials in undetermined ways; (v) the electromagnetic coupling assumes gauge potentials smooth over the Cooper-pair size, which the authors note restricts the charged equation to type-II behavior (a type-II superconductor has GL parameter $\kappa = \lambda/\xi > 1/\sqrt{2}$, with $\lambda$ the London penetration depth; type-I behavior, very short coherence lengths, or strong local fields violate the assumption). The companion derivation by Aitchison, Ao, Thouless, and Zhu (1995) confirmed the same equation from an effective-Lagrangian route, made the expansion basis explicit ($\omega \ll |\Delta_0|$, $k\xi \ll 1$, with $\Delta_0$ the gap), and noted that including appropriate higher-derivative terms should make predictions about vortex-core structure possible — a program that was not carried out.

The application motivating much of this is low-temperature vortex dynamics, including in the high-temperature (cuprate) oxide superconductors: the NLSE implies a transverse (Magnus) force on a moving vortex proportional to the superfluid density, whose consequences for the Hall effect in the mixed state would allow a direct measurement of the superfluid density against the London penetration depth.

Problem Statement

The problem is to turn the 1995 nonlinear Schrödinger equation for the superconducting condensate from a formal zero-temperature, clean-limit result into a quantitatively determined and validated effective equation of motion for realistic superconductors. All components below serve this single objective — a microscopic, Galilean-invariant condensate equation with quantitatively known coefficients — and share one derivation structure (the Hartree–Fock–Bogoliubov factorization, the Born–Oppenheimer separation, and the pseudo-potentials $a_0$ and $\mathbf{a}$); they are jointly motivated by that objective, but each is independently answerable and none is mandatory.

1. Finite temperature: derive the equation of motion for the condensate when quasiparticle excitations are populated, including their back-action on the condensate, recovering the 1995 equation in the $T \to 0$ limit and connecting to two-fluid (normal-plus-superfluid) physics.

2. Pseudo-potentials: evaluate $a_0$ and $\mathbf{a}$ quantitatively — including corrections beyond the HFB factorization (exchange, band-structure) — at least in representative regimes, and determine their structure and magnitude for non-singlet (non-s-wave) pairing and for dirty or strongly anisotropic materials, where the pseudo-vector $\mathbf{a}$ is expected to be nonzero and to reflect the pair's deformed internal structure.

3. Validity boundary: determine how the equation fails, and what replaces it, when the slow-variation hypothesis breaks down — sharp interfaces, strong disorder, vortex cores with very short coherence length, and temporal variations comparable to the gap.

4. Charged regimes beyond type II: derive the condensate dynamics when electromagnetic gauge fields vary on the Cooper-pair scale (type-I behavior, very short coherence lengths, strong local fields).

5. Vortex application: connect the extended equation to low-temperature vortex dynamics — vortex mass and dissipation mechanisms — and to measurable transport (for instance the Hall effect) in high-$T_c$ oxide superconductors, recovering as limiting cases the existing microscopic results for homogeneous s-wave type-II superconductors (vortex friction and transverse force at finite temperature with impurities; the microscopic vortex mass).

The extended equation must reduce to the 1995 equation — Galilean-invariant, with the identification of $|\psi|^2$ with the superfluid (not condensate-fraction) density, and with minimal electromagnetic coupling — in the $T \to 0$, clean, slow-variation limit.

References

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