The open question is whether the adiabatic-phase Magnus force on a slowly moving vortex in a homogeneous two-dimensional superconductor film at finite temperature, Fm=ρs2ℏqv(vs−r˙0)×z^\mathbf{F}_m=\frac{\rho_s}{2}\hbar q_v(\mathbf{v}_s-\dot{\mathbf{r}}_0)\times\hat{z}, is exact, and what its complete composition is: whether vortex-core bound states (including any spectral flow), normal-fluid scattering (the Iordanskii force), or the conservative force proposed by Ferrell add to, remove, or modify terms at finite temperature. The adiabatic-phase derivation of Ao, Niu and Thouless, 'Implications of adiabatic phases for a vortex in a superconductor film' (Physica B 194-196, 1453 (1994)), obtains this force by extending the Berry phase of a pure many-body state to a mixed-state density matrix with the two-fluid model, neglecting vortex-core contributions to the geometric phase and taking the two-particle reduced density matrix to equal its vortex-free form away from the core, and finds that neither the Iordanskii nor the Ferrell force seems to appear in this approach. The question became a three-way controversy (Ao-Thouless-Niu Berry-phase; Kopnin-Salomaa spectral flow; Sonin Iordanskii-force) that remains without a cross-school accepted resolution through at least 2023; a two-fluid reanalysis of the transverse force (Thouless, Geller, Vinen, Fortin and Rhee, Physical Review B 63, 224504 (2001)) is among the prior analyses. The complete force composition for a finite-temperature superconducting film is neither resolved nor refuted, and the exactness of the adiabatic-phase result for nonzero normal-fluid velocity remains contested.

Background

A vortex in a type-II superconductor is a line, or in a thin film a point, around which the phase of the superconducting order parameter winds by 2π2\pi times an integer winding number qvq_v; each vortex carries one quantum of magnetic flux, and the motion of vortices controls the dissipation and Hall response of the mixed state (the vortex-carrying phase of a type-II superconductor, stable between the lower and upper critical magnetic fields). In the late 1980s and early 1990s, measurements of the flux-flow Hall effect (the transverse voltage generated when a transport current drives vortices across the film) in conventional and high-temperature-superconductor films showed sign reversals and anomalously small Hall angles, motivating competing microscopic theories of the nondissipative transverse force on a moving vortex. The Magnus force is the hydrodynamic transverse force on a circulation-carrying object moving relative to a background flow: a vortex at r0\mathbf{r}_0 moving with velocity r˙0\dot{\mathbf{r}}_0 in a background superfluid velocity vs\mathbf{v}_s feels a force perpendicular to vs−r˙0\mathbf{v}_s-\dot{\mathbf{r}}_0. What fixes the proportionality coefficient in a superconductor, the full electron density, the superfluid density, or terms involving vortex-core physics, was disputed. Related contested objects are the Iordanskii force, a transverse force on a vortex proposed by Iordanskii (1964) for rotating superfluids, arising from scattering of normal (thermal) quasiparticles, and a conservative transverse force proposed by Ferrell (1992) to explain Hall-voltage sign reversal in thin films. In Fermi superfluids and superconductors, quasiparticle bound states localized in the vortex core (Caroli-de Gennes-Matricon states) can undergo spectral flow, a shift of bound-state occupation as the vortex moves; Kopnin and Salomaa (1991) argued that this transfers momentum and produces an additional transverse force. Ao and Thouless (1993) introduced a Berry-phase derivation of the force: a Berry phase is the phase a quantum state accumulates when the parameters of its Hamiltonian are changed slowly around a closed cycle, depending only on the path in parameter space; for a vortex transported adiabatically around a closed loop, the many-body wavefunction acquires a phase proportional to the loop area, which is equivalent to a transverse force. They concluded that the force is the Magnus force fixed by the density far from the vortex, independent of core structure. Thouless, Ao and Niu (1996) extended the claim to neutral superfluids, stating that the total nondissipative transverse force is the Magnus force proportional to the superfluid density and that contributions of the localized core states do not change this force. The 1994 paper is the finite-temperature member of this series. It replaces the zero-temperature wavefunction by the finite-temperature density matrix, decomposes the one-particle reduced density matrix into normal and superfluid parts using the two-fluid model (the Gorter-Casimir phenomenological division of the fluid into coexisting normal and superfluid components, the superfluid part carrying off-diagonal long-range order, meaning non-decaying far-off-diagonal elements of the one-particle density matrix, the property Yang identified as defining a superfluid), and evaluates the Berry phase for an adiabatically transported vortex. Neglecting the vortex-core region (within roughly the coherence length ξ\xi, the scale of the vortex core) and taking the two-particle reduced density matrix to equal its vortex-free form outside the core, it obtains the Berry phase Θ=−2πqvρs2S(Γ)\Theta=-2\pi q_v\frac{\rho_s}{2}S(\Gamma) with S(Γ)S(\Gamma) the signed loop area, the potential energy U(r0)=−ρs2ℏqvr0⋅(vs×z^)U(\mathbf{r}_0)=-\frac{\rho_s}{2}\hbar q_v\mathbf{r}_0\cdot(\mathbf{v}_s\times\hat{z}), and hence the Magnus force Fm=ρs2ℏqv(vs−r˙0)×z^\mathbf{F}_m=\frac{\rho_s}{2}\hbar q_v(\mathbf{v}_s-\dot{\mathbf{r}}_0)\times\hat{z}, the zero-temperature result with the temperature-dependent superfluid electron number density ρs\rho_s. It concludes that neither the Iordanskii nor the Ferrell force seems to appear in this approach, that disorder does not affect the existence of the result except for a possible reduction of the superfluid density due to the localization of some Cooper pairs and a pinning force due to the introduced inhomogeneity, and that the electromagnetic field leaves the result unchanged in the regime where the magnetic-field effect is negligible. The derivation rests on two approximations made without error estimates — the core contributes negligibly to the adiabatic phase, and the two-particle reduced density matrix is close to its vortex-free value away from the core — together with the two-fluid ansatz itself, the assumption that the normal-fluid wavevectors kn\mathbf{k}_n and the phase gradient ∇ϕ\nabla\phi do not alter appreciably over a distance of order ξ\xi, and the direct substitution of the density matrix into the pure-state Berry-phase formula; the paper also notes that the exact Hamiltonian for vortex dynamics is unknown.

Problem Statement

For a vortex moving slowly in a homogeneous two-dimensional superconductor film at finite temperature, determine the complete nondissipative transverse force on the vortex — the total transverse force entering the vortex equation of motion — in the regime where the magnetic-field effect is negligible and the normal-fluid wavevector kn\mathbf{k}_n and the superfluid phase gradient ∇ϕ\nabla\phi do not alter appreciably over a distance of order the coherence length ξ\xi. Throughout this task the name 'the adiabatic-phase Magnus force' denotes the full relative-velocity term Fm=ρs2ℏqv(vs−r˙0)×z^\mathbf{F}_m=\frac{\rho_s}{2}\hbar q_v(\mathbf{v}_s-\dot{\mathbf{r}}_0)\times\hat{z} of the source derivation, covering both its r˙0\dot{\mathbf{r}}_0-part and its vs\mathbf{v}_s-part; in a narrower convention used in part of the superconductivity literature only the r˙0\dot{\mathbf{r}}_0-part is called the Magnus force and the vs\mathbf{v}_s-part the Lorentz force. Determine whether the adiabatic-phase result Fm=ρs2ℏqv(vs−r˙0)×z^\mathbf{F}_m=\frac{\rho_s}{2}\hbar q_v(\mathbf{v}_s-\dot{\mathbf{r}}_0)\times\hat{z} (with z^\hat{z} the unit vector normal to the film) receives further contributions from vortex-core bound states (including any spectral flow), from normal-fluid scattering (the Iordanskii force), or from the conservative force proposed by Ferrell, and how such contributions would modify its superfluid-density coefficient and its temperature dependence. The force composition may also depend on the normal-fluid flow state (vanishing, uniform, or logarithmically increasing with a nonlinear cutoff at large distances). The exactness of the adiabatic-phase result for nonzero normal-fluid velocity is contested (see, e.g., Thouless et al. 2001).

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