The equation of motion of a vortex in a type-II superconductor contains a nondissipative transverse force per unit length, writable as an effective density times K×V\mathbf{K}\times\mathbf{V} (quantized circulation vector K\mathbf{K}, vortex velocity V\mathbf{V}), which fixes the Hall response of the mixed state. Geller, Wexler, and Thouless [Phys. Rev. B 57, R8119 (1998)] derived an essentially exact value for this force, f=ρ K×Vf=\rho\,\mathbf{K}\times\mathbf{V} with ρ\rho the total mass density, for a translationally invariant, disorder-free superconductor in the superclean limit, and themselves flagged the central open issue: their model yields an infinite normal conductivity, and the neglect of disorder may be responsible for the appearance of ρ\rho rather than the superfluid density ρs\rho_s. How impurity scattering and finite normal-component conductivity modify this force — whether the effective coefficient interpolates between the total-density and superfluid-density limits, and what the quantitative consequences are for the measurable Hall response — remains unsettled: the spectral-flow theory predicts a crossover controlled by the product of the core-state level spacing and the quasiparticle relaxation time; the microscopic influence-functional treatment by Ao and Zhu finds the force to be impurity-insensitive and proportional to ρs\rho_s; and Sonin's scattering-based analyses dispute the very existence of steady spectral flow. The short-coherence-length regime, where the vortex core size is comparable to the crystal lattice constant and the coarse-graining argument of the original derivation fails, is essentially untreated. Reviews through 2024 still list these transverse-force terms as debatable.

Background

A type-II superconductor admits magnetic flux in the form of quantized vortex lines — tube-like regions of suppressed superconducting order parameter, each carrying one flux quantum Φ0=hc/2e\Phi_0=hc/2e — which fill the mixed state between the lower and upper critical fields. The macroscopic electrodynamics of the mixed state (flux flow, i.e. the steady drift of vortices under a driving current, which generates a resistive voltage; dissipation; and the Hall effect) is governed by the equation of motion of a single vortex line. For a straight vortex at position R(t)\mathbf{R}(t) moving with velocity V=dR/dt\mathbf{V}=d\mathbf{R}/dt, this equation is conventionally written as M d2R/dt2=−η V−γ V×z^+fL+fpM\,d^2\mathbf{R}/dt^2=-\eta\,\mathbf{V}-\gamma\,\mathbf{V}\times\hat{\mathbf{z}}+\mathbf{f}_L+\mathbf{f}_p, where MM is the vortex mass per unit length, η\eta the viscous damping coefficient, γ\gamma the coefficient of the nondissipative transverse force per unit length, fL\mathbf{f}_L the Lorentz force exerted by the transport current, fp\mathbf{f}_p the pinning force from disorder, and z^\hat{\mathbf{z}} the direction of the circulation vector K\mathbf{K} (a vector along the vortex whose magnitude equals the quantized circulation, h/2mh/2m for Cooper pairs of mass 2m2m). The coefficient γ\gamma is what fixes the Hall response: in flux flow, the component of vortex motion transverse to the driving current, controlled by γ\gamma, generates the transverse (Hall) electric field.

In the two-fluid description, the liquid mass density far from the vortex core splits into a superfluid part ρs\rho_s and a normal (quasiparticle) part ρn\rho_n, with total ρ=ρs+ρn\rho=\rho_s+\rho_n. For a neutral superfluid, Thouless, Ao, and Niu (1996) used a Berry-phase argument (the geometric phase accumulated by the many-body wavefunction as the vortex is transported) to derive the transverse force per unit length f=ρs K×Vf=\rho_s\,\mathbf{K}\times\mathbf{V} exactly, independent of the microscopic structure of the vortex core. Microscopic work on superconductors, in contrast, had long found additional transverse forces tied to the vortex core: the quasiparticle states bound in the core (Caroli–de Gennes–Matricon states, with level spacing ω0∼Δ2/EF\omega_0\sim\Delta^2/E_F, where Δ\Delta is the superconducting gap and EFE_F the Fermi energy) contain an anomalous spectral branch that crosses zero energy, and as the vortex moves, quasiparticles can be pumped along this branch between the sides of the Fermi surface — a process called spectral flow — transferring momentum to the vortex. Whether spectral flow produces an extra nondissipative transverse force, and how large it is, was and remains the heart of a decades-long controversy (reviewed by Sonin, 2023).

Geller, Wexler, and Thouless (1998) extended the exact Berry-phase-type derivation to the charged case. They modeled the superconductor microscopically with a current–current (Darwin) interaction that reproduces ordinary electrodynamics to order v2/c2v^2/c^2 and proved in a random-phase approximation (a standard scheme that resums the collective screening response of the particles) that their Hamiltonian exhibits a Meissner effect (expulsion of magnetic field) with the conventional London penetration depth λ\lambda, the scale over which magnetic fields are screened in a superconductor. Because the gauge-invariant canonical momentum density vanishes far from the vortex in the charged case, the same line-integral construction that yields ρs\rho_s for the neutral superfluid yields the total density for the superconductor: the transverse force per unit length is f=ρ K×Vf=\rho\,\mathbf{K}\times\mathbf{V}, valid in the superclean limit — the regime in which the core-level spacing ω0\omega_0 exceeds the impurity-limited quasiparticle relaxation rate 1/τ1/\tau (with τ\tau the relaxation time), so that the core spectrum is discrete and spectral flow is blocked. They furthermore showed that the result survives a periodic crystal potential, but only under the coarse-graining condition ξ≫b\xi\gg b, where ξ\xi is the superconducting coherence length (which sets the vortex core size) and bb is the lattice constant; the demonstration expands the many-body states in Wannier functions (localized orbitals of the band) and uses effective-mass envelope functions, which describe only coarse-grained quantities.

The authors themselves flagged what is missing: the model contains no disorder, so its normal component has infinite conductivity, and, quoting the paper, 'this neglect of disorder may be responsible for the ρ dependence of the transverse force instead of the usual ρs'. The unresolved target is the transverse-force law for realistic samples — finite impurity scattering and finite normal conductivity, from the dirty to the superclean regime — together with its observable Hall consequences and the regime of coherence length comparable to the lattice constant, where no coarse-grained derivation exists.

Problem Statement

The problem, flagged in the concluding discussion of Geller, Wexler, and Thouless (Phys. Rev. B 57, R8119 (1998)) and taken up by the subsequent literature, is to determine the nondissipative transverse force per unit length on a quantized vortex in a type-II superconductor outside the superclean, disorder-free, translationally invariant limit in which they derived f=ρ K×Vf=\rho\,\mathbf{K}\times\mathbf{V} with ρ\rho the total mass density. Its three components jointly serve this single objective — the transverse-force law of realistic samples. They are: (1) Derive the effective transverse-force coefficient, in two-fluid language an effective density generalizing the coefficient of K×V\mathbf{K}\times\mathbf{V}, for a superconductor with finite impurity scattering and finite normal-component conductivity, as a function of the quasiparticle relaxation time τ\tau, temperature, and magnetic field, covering the dirty (ω0τ≪1\omega_0\tau\ll 1), crossover (ω0τ∼1\omega_0\tau\sim 1), and superclean (ω0τ≫1\omega_0\tau\gg 1) regimes; in particular, establish whether and how it interpolates between the total density ρ\rho of the superclean limit and the superfluid density ρs\rho_s, and determine which of the three mechanisms (spectral flow through the anomalous core branch, quasiparticle scattering, and the topological/Berry-phase contribution) operate in each regime. (2) Work out the quantitative consequences for the measurable flux-flow Hall response of the mixed state — explicitly the flux-flow Hall angle or, equivalently, the Hall conductivity as a function of τ\tau, temperature, and magnetic field — quantitatively enough to discriminate between the competing existing predictions, separating the total-density (ρ\rho) and superfluid-density (ρs\rho_s) predictions. (3) Determine the transverse force in the regime where the coherence length (vortex core size) ξ\xi is comparable to the lattice constant bb, where the envelope-function coarse-graining of the original derivation fails. Component (1) additionally requires reproducing the established limiting results — the superclean f=ρ K×Vf=\rho\,\mathbf{K}\times\mathbf{V} of the source paper and the neutral-superfluid f=ρs K×Vf=\rho_s\,\mathbf{K}\times\mathbf{V} of Thouless, Ao, and Niu — and either reconciling or refuting each of the three mechanisms named in it (spectral flow, quasiparticle scattering, and the topological/Berry-phase contribution); refuting one mechanism settles only that sub-claim and leaves the interpolation between ρ\rho and ρs\rho_s required by component (1) to be established.

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