Completing the force law on a quantized vortex in a superfluid: the Iordanskii-force coefficient and dissipative forces beyond strong pinning, homogeneity, and scalar order parameters

Thouless, Ao and Niu (Phys. Rev. Lett. 76, 3758 (1996)) derived an exact expression for the nondissipative transverse (Magnus) force on a quantized vortex moving through a uniform background, valid for neutral superfluids with a scalar complex order parameter and a pinning potential strong enough to prevent vortex detachment. They explicitly left open the coefficient of the transverse term proportional to the relative velocity $\mathbf{v}_n-\mathbf{v}_s$ (the Iordanskii-type contribution), the structure of the dissipative forces, and extensions to disordered backgrounds, non-scalar order parameters, and weak pinning. The 1996-2013 literature settled the central coefficient question for vortices pinned to a strong pinning center in the hydrodynamic and low-temperature quantum regimes: a consensus emerged that the Iordanskii force $f_I=-\rho_n\boldsymbol\kappa\times(\mathbf{v}_R-\mathbf{v}_L)$ exists, with the original discrepancy traced to the implicit assumption of zero normal-fluid circulation. What remains open is the completion of the force law in the regimes and settings excluded by the 1996 derivation and the 2000-2001 consensus analysis: the transverse and dissipative forces on a vortex with a free (unpinned) core, including the collisionless-to-hydrodynamic transition region for which no theory exists; the regime dependence and non-universality of the Iordanskii-force coefficient in dirty, near-critical, lattice, and relativistic multi-component systems; the exact treatment of disordered backgrounds; the weak-pinning and depinning dynamics; and the still-missing direct experimental isolation of the Iordanskii force.

Background

A quantized vortex is a line defect of a superfluid or type-II superconductor around which the superfluid velocity circulates by a fixed quantum $\kappa=h/m$ (with $m$ the mass of the condensed boson or of the fermion pair), a consequence of the single-valuedness of the macroscopic order parameter. In the two-fluid description of a finite-temperature superfluid, an inviscid superfluid component of mass density $\rho_s$ and velocity $\mathbf{v}_s$ coexists with a viscous normal component of mass density $\rho_n$ and velocity $\mathbf{v}_n$ that carries the thermal excitations (in helium-4: phonons, the sound-wave quanta, and rotons, the short-wavelength quasiparticles associated with the minimum of the excitation spectrum); $\rho=\rho_s+\rho_n$ is the total mass density.

A vortex moving at velocity $\mathbf{v}_V$ through the fluid experiences a force per unit length transverse to its motion, the analogue of the lift (Magnus) force of classical aerodynamics, which the textbook Kutta-Joukowski theorem of fluid mechanics gives as density times circulation times relative velocity. Vinen used this force in the experiment that first detected single circulation quanta in liquid helium-4, and Hall and Vinen built it into the theory of mutual friction in rotating superfluids: the force balance on a vortex is closed by a mutual-friction force parameterized by a longitudinal (dissipative) coefficient $D$ and a transverse coefficient $D'$. The transverse quasiparticle contribution itself, produced by the asymmetric scattering of thermal excitations off the vortex velocity field in close analogy with the Aharonov-Bohm effect (the quantum phase shift acquired by a wave encircling a confined flux), was found for phonons by Iordanskii and, for rotons, by Lifshitz and Pitaevskii; for noninteracting phonons and rotons the scattering theory yields $D'=-\rho_n\kappa$. In Fermi superfluids there is a third transverse contribution: quasiparticle states bound in the vortex core can undergo spectral flow, the gradual transfer of bound core states across the energy gap as the vortex moves, which transfers momentum to the vortex (Volovik; Kopnin). Barenghi, Donnelly and Vinen argued that the total transverse force on a vortex in helium-4 is

$$\mathbf{F}=\rho_s\boldsymbol\kappa\times(\mathbf{v}_V-\mathbf{v}_s)+\rho_n\boldsymbol\kappa\times(\mathbf{v}_n-\mathbf{v}_V),$$

i.e. the superfluid Magnus force plus a transverse term proportional to the velocity $\mathbf{v}_n-\mathbf{v}_V$ of the normal component relative to the vortex.

Ao and Thouless connected the Magnus force with a Berry phase, the geometric phase accumulated by the many-body wave function when the vortex is adiabatically transported around a closed loop, and Thouless, Ao and Niu (1996) turned this into an exact result: for a single vortex forced through an infinite, homogeneous, neutral superfluid with a scalar complex order parameter, pinned by a potential strong enough that the vortex neither tunnels off the pinning site nor is detached by absorbing phonons, the nondissipative transverse force proportional to the vortex velocity is exactly the Magnus force $\rho_s\boldsymbol\kappa\times\mathbf{v}_V$ (with $\mathbf{v}_V\to\mathbf{v}_V-\mathbf{v}_s$ following from Galilean invariance, the invariance of the dynamics under a common boost of all components), and localized core states do not change it. The authors were explicit about what the argument could not deliver: the coefficient of the term proportional to $\mathbf{v}_n-\mathbf{v}_s$ could not be obtained without imposing separate background velocities on the two components; the energy-conserving (dissipative) terms in their perturbative expression were left unevaluated, their extraction requiring a Kubo-type limiting procedure, the careful order-of-limits familiar from linear-response theory; and no generalization was offered to disordered backgrounds (important for real superconductors), to non-scalar (multi-component or spin-triplet) order parameters, or to the weak-pinning regime.

The following decade turned the first gap into the Iordanskii force controversy: Wexler argued from a Laughlin-style thermodynamic argument plus Galilean invariance that no transverse force proportional to the normal velocity exists; Sonin, Stone and Volovik derived the Iordanskii force from scattering theory and its gravitational Aharonov-Bohm analogue; comment-and-reply exchanges appeared in Physical Review Letters in 1998. The consensus that emerged around 2000-2001 (Geller, Thouless, Rhee and Vinen; Thouless, Geller, Vinen, Fortin and Rhee; and Sonin) is that the exact 1996 derivation had implicitly assumed zero circulation of the normal component around the vortex; once that assumption is dropped, the transverse quasiparticle force is rehabilitated. For a vortex pinned to a wire or comparably strong pinning center, with noninteracting phonon or roton scattering, the accepted coefficient is the Iordanskii value $f_I=-\rho_n\boldsymbol\kappa\times(\mathbf{v}_R-\mathbf{v}_L)$, where $\mathbf{v}_R$ is the excitation drift velocity near the core and $\mathbf{v}_L$ the vortex velocity; quantum-mechanical calculations by Thompson and Stamp at low temperature recover the same form in both the classical and the quantum regime. Two-fluid hydrodynamics around a pinned core adds logarithmic structure: a longitudinal viscous force scaling as $u/\ln u$ and a transverse contribution scaling as $u/(\ln u)^2$ for normal-flow speed $u$, with the logarithms cut off at the Oseen length, the scale beyond which nonlinear advection dominates viscous diffusion in the flow past the vortex. In the superclean limit of a type-II superconductor (quasiparticle mean free path longer than the vortex-core size, so that impurity scattering does not suppress spectral flow) the transverse force is $\rho\boldsymbol\kappa\times\mathbf{V}$ with $\rho=\rho_n+\rho_s$.

What is not settled is the force law outside these regimes: for a vortex whose core is free rather than wire-pinned (the collisionless-to-hydrodynamic transition region around a free core has no theory at all, in the assessment of Sergeev's 2023 review), in dirty, disordered, lattice, near-critical, or non-scalar systems where the accepted coefficient is explicitly not universal (Sonin), in the weak-pinning regime, and in the absence of any direct experimental isolation of the Iordanskii contribution. Gavassino, Antonelli and Haskell (2021) propose that the mutually incompatible values reported for the transverse coefficient - the full value $-\rho_n\kappa$, the logarithmically suppressed two-fluid values, and zero - each hold in a different dynamical regime of the fluid system.

Problem Statement

Determine the transverse and longitudinal forces on a quantized vortex in a finite-temperature superfluid in the regimes and settings excluded by the 1996 derivation of Thouless, Ao and Niu and by the 2000-2001 consensus analysis; that is, complete the microscopic vortex force law. Throughout, $\rho_s$ and $\rho_n$ are the superfluid and normal mass densities, $\kappa=h/m$ the circulation quantum, $\mathbf{v}_s$ and $\mathbf{v}_n$ the superfluid and normal velocities, $\mathbf{v}_V$ (also written $\mathbf{v}_L$) the vortex velocity, $D$ and $D'$ the longitudinal (dissipative) and transverse mutual-friction coefficients, $\mathbf{v}_R$ the quasiparticle drift velocity near the vortex core, and $u$ the normal-flow speed.

(1) Free-core force law. Give a controlled theory of the transverse (Iordanskii-type) and dissipative forces on a vortex whose core is not pinned to a wire or strong external potential, including the transition region between the collisionless inner region (of order the quasiparticle mean free path) and the outer hydrodynamic region, for which no theory currently exists. Determine the Iordanskii-force coefficient as a function of the dynamical regime.

(2) Beyond homogeneous, clean, scalar systems. Extend the exact determination of the transverse force to disordered backgrounds (relevant to real superconductors) and to non-scalar (multi-component or spin-triplet) order parameters beyond the clean-limit cases already treated.

(3) Weak pinning. Determine the vortex force law and dynamics when the pinning potential is too weak for the perturbative treatment of the vortex displacement, that is, in the regimes of tunneling-induced depinning and time-dependent core detachment.

(4) Experimental isolation. A direct measurement separating the Iordanskii force from the Magnus force and the scattering drag, for example in superfluid helium-4 or in a Bose-Einstein condensate, remains unavailable.

The common scientific objective is a single regime-resolved microscopic force law for a quantized vortex, valid across pinning strength, disorder, order-parameter structure, and the collisionless-to-hydrodynamic crossover. Any answer must be consistent with the already-established results in overlapping regimes: the exact Magnus force $\rho_s\boldsymbol\kappa\times\mathbf{v}_V$ for a strongly pinned vortex, the Iordanskii value $D'=-\rho_n\kappa$ for noninteracting phonons and rotons scattering off a strongly pinned core, the $u/\ln u$ and $u/(\ln u)^2$ two-fluid forms around a pinned core, and the superclean-superconductor result. Work that merely re-derives or re-asserts these settled cases does not address the problem; the theoretical items (1)-(3) are jointly needed because they are the pieces of one force law, and each regime supplies boundary conditions and limiting checks for the others. Each of items (1)-(3) is resolved only by an explicit, quantitative force law of the same kind as these settled results - the transverse and dissipative coefficients fixed as functions of the setting's parameters - rather than by a qualitative or order-of-magnitude account that leaves them undetermined.

References

1. D. J. Thouless, Ping Ao, and Qian Niu, 'Transverse force on a quantized vortex in a superfluid', Physical Review Letters 76, 3758-3761 (1996). DOI 10.1103/PhysRevLett.76.3758, https://doi.org/10.1103/PhysRevLett.76.3758 (preprint arXiv:cond-mat/9603196, https://arxiv.org/abs/cond-mat/9603196).
2. G. E. Volovik, 'Comment on "Transverse Force on a Quantized Vortex in a Superfluid"', Physical Review Letters 77, 4687 (1996). DOI 10.1103/PhysRevLett.77.4687, https://doi.org/10.1103/PhysRevLett.77.4687 (preprint arXiv:cond-mat/9606210, https://arxiv.org/abs/cond-mat/9606210).
3. P. Ao and D. J. Thouless, 'Berry's phase and the Magnus force for a vortex line in a superconductor', Physical Review Letters 70, 2158-2161 (1993). DOI 10.1103/PhysRevLett.70.2158, https://doi.org/10.1103/PhysRevLett.70.2158.
4. W. F. Vinen, 'The detection of single quanta of circulation in liquid helium II', Proceedings of the Royal Society of London. Series A 260, 218-236 (1961). DOI 10.1098/rspa.1961.0029, https://doi.org/10.1098/rspa.1961.0029.
5. H. E. Hall and W. F. Vinen, 'The rotation of liquid helium II. II. The theory of mutual friction in uniformly rotating helium II', Proceedings of the Royal Society of London. Series A 238, 215-234 (1956). DOI 10.1098/rspa.1956.0215, https://doi.org/10.1098/rspa.1956.0215.
6. S. V. Iordanskii, 'Mutual friction force in a rotating Bose gas', Soviet Physics JETP 22, 160-167 (1966); Russian original: Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki 49, 225-236 (1965).
7. E. M. Lifshitz and L. P. Pitaevskii, 'Absorption of second sound in rotating helium II', Soviet Physics JETP 6, 418-419 (1958); Russian original: Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki 33, 535-537 (1957).
8. C. F. Barenghi, R. J. Donnelly, and W. F. Vinen, 'Friction on quantized vortices in helium II. A review', Journal of Low Temperature Physics 52, 189-247 (1983). DOI 10.1007/BF00682247, https://doi.org/10.1007/BF00682247.
9. T. D. C. Bevan, A. J. Manninen, J. B. Cook, H. Alles, J. R. Hook, and H. E. Hall, 'Vortex Mutual Friction in Superfluid 3He', Journal of Low Temperature Physics 109, 423-459 (1997). DOI 10.1007/s10909-005-0095-Z, https://doi.org/10.1007/s10909-005-0095-Z.
10. C. Wexler, 'Magnus and Iordanskii Forces in Superfluids', Physical Review Letters 79, 1321-1324 (1997). DOI 10.1103/PhysRevLett.79.1321, https://doi.org/10.1103/PhysRevLett.79.1321 (preprint arXiv:cond-mat/9612111, https://arxiv.org/abs/cond-mat/9612111).
11. E. B. Sonin, 'Magnus force in superfluids and superconductors', Physical Review B 55, 485-501 (1997). DOI 10.1103/PhysRevB.55.485, https://doi.org/10.1103/PhysRevB.55.485.
12. H. E. Hall and J. R. Hook, 'Comment on "Magnus and Iordanskii Forces in Superfluids"', Physical Review Letters 80, 4356 (1998). DOI 10.1103/PhysRevLett.80.4356, https://doi.org/10.1103/PhysRevLett.80.4356.
13. C. Wexler, D. J. Thouless, P. Ao, and Q. Niu, 'Wexler et al. Reply', Physical Review Letters 80, 4357 (1998). DOI 10.1103/PhysRevLett.80.4357, https://doi.org/10.1103/PhysRevLett.80.4357.
14. E. B. Sonin, 'Comment on "Berry's Phase and the Magnus Force for a Vortex Line in a Superconductor", "Transverse Force on a Quantized Vortex in a Superfluid", and "Magnus and Iordanskii Forces in Superfluids"', Physical Review Letters 81, 4276 (1998). DOI 10.1103/PhysRevLett.81.4276, https://doi.org/10.1103/PhysRevLett.81.4276.
15. C. Wexler, D. J. Thouless, P. Ao, and Q. Niu, 'Wexler et al. Reply', Physical Review Letters 81, 4277 (1998). DOI 10.1103/PhysRevLett.81.4277, https://doi.org/10.1103/PhysRevLett.81.4277.
16. C. Wexler and D. J. Thouless, 'Scattering of Phonons by a Vortex in a Superfluid', Physical Review B 58, R8897-R8900 (1998). DOI 10.1103/PhysRevB.58.R8897, https://doi.org/10.1103/PhysRevB.58.R8897 (preprint arXiv:cond-mat/9804118, https://arxiv.org/abs/cond-mat/9804118).
17. G. E. Volovik, 'Vortex versus spinning string: Iordanskii force and gravitational Aharonov-Bohm effect', JETP Letters 67, 881-887 (1998). DOI 10.1134/1.567781, https://doi.org/10.1134/1.567781 (preprint arXiv:cond-mat/9804308, https://arxiv.org/abs/cond-mat/9804308).
18. Michael R. Geller, Carlos Wexler, and David J. Thouless, 'Transverse force on a quantized vortex in a superconductor', Physical Review B 57, R8119-R8122 (1998). DOI 10.1103/PhysRevB.57.R8119, https://doi.org/10.1103/PhysRevB.57.R8119 (preprint arXiv:cond-mat/9712173, https://arxiv.org/abs/cond-mat/9712173).
19. Michael Stone, 'Iordanskii Force and the Gravitational Aharonov-Bohm effect for a Moving Vortex', Physical Review B 61, 11780-11786 (2000). DOI 10.1103/PhysRevB.61.11780, https://doi.org/10.1103/PhysRevB.61.11780 (preprint arXiv:cond-mat/9909313, https://arxiv.org/abs/cond-mat/9909313).
20. Michael R. Geller, David J. Thouless, Sung-Wu Rhee, and W. F. Vinen, 'Iordanskii and Lifshitz-Pitaevskii Forces in the Two-Fluid Model', Journal of Low Temperature Physics 121, 411-416 (2000). DOI 10.1023/A:1017553814235, https://doi.org/10.1023/A:1017553814235 (preprint arXiv:cond-mat/0005438, https://arxiv.org/abs/cond-mat/0005438).
21. D. J. Thouless, M. R. Geller, W. F. Vinen, J.-Y. Fortin, and S. W. Rhee, 'Vortex Dynamics in the Two-Fluid Model', Physical Review B 63, 224504 (2001). DOI 10.1103/PhysRevB.63.224504, https://doi.org/10.1103/PhysRevB.63.224504 (preprint arXiv:cond-mat/0101297, https://arxiv.org/abs/cond-mat/0101297).
22. Jean-Yves Fortin, 'Dynamics of vortices in neutral superfluids with noninteracting phonons', Physical Review B 63, 174502 (2001). DOI 10.1103/PhysRevB.63.174502, https://doi.org/10.1103/PhysRevB.63.174502.
23. N. B. Kopnin, 'Vortex dynamics and mutual friction in superconductors and Fermi superfluids', Reports on Progress in Physics 65, 1633-1678 (2002). DOI 10.1088/0034-4885/65/11/202, https://doi.org/10.1088/0034-4885/65/11/202.
24. E. B. Sonin, 'Magnus Force and Aharonov-Bohm Effect in Superfluids', in R. P. Huebener, N. Schopohl, and G. E. Volovik (editors), 'Vortices in Unconventional Superconductors and Superfluids', Springer Series in Solid-State Sciences, Springer, pp. 119-145 (2002). DOI 10.1007/978-3-662-04665-4_8, https://doi.org/10.1007/978-3-662-04665-4_8.
25. Markus Flaig and Uwe R. Fischer, 'Global singularity-free solution of the Iordanskii force problem', Physical Review B 74, 224503 (2006). DOI 10.1103/PhysRevB.74.224503, https://doi.org/10.1103/PhysRevB.74.224503 (preprint arXiv:cond-mat/0608053, https://arxiv.org/abs/cond-mat/0608053).
26. Assa Auerbach, Daniel P. Arovas, and Sankalpa Ghosh, 'Quantum tunneling of vortices in two-dimensional condensates', Physical Review B 74, 064511 (2006). DOI 10.1103/PhysRevB.74.064511, https://doi.org/10.1103/PhysRevB.74.064511.
27. E. B. Sonin, 'The Aharonov-Bohm effect in neutral liquids', Journal of Physics A: Mathematical and Theoretical 43, 354003 (2010). DOI 10.1088/1751-8113/43/35/354003, https://doi.org/10.1088/1751-8113/43/35/354003.
28. L. Thompson and P. C. E. Stamp, 'Quantum Dynamics of a Bose Superfluid Vortex', Physical Review Letters 108, 184501 (2012). DOI 10.1103/PhysRevLett.108.184501, https://doi.org/10.1103/PhysRevLett.108.184501.
29. L. Thompson and P. C. E. Stamp, 'Vortex Dynamics: Quantum Versus Classical Regimes', Journal of Low Temperature Physics 171, 526-538 (2013). DOI 10.1007/s10909-012-0727-z, https://doi.org/10.1007/s10909-012-0727-z.
30. E. B. Sonin, 'Transverse forces on a vortex in lattice models of superfluids', Physical Review B 88, 214513 (2013). DOI 10.1103/PhysRevB.88.214513, https://doi.org/10.1103/PhysRevB.88.214513.
31. E. B. Sonin, 'Dynamics of Quantised Vortices in Superfluids', Cambridge University Press (2016). DOI 10.1017/cbo9781139047616, https://doi.org/10.1017/cbo9781139047616.
32. U. Leonhardt and P. Öhberg, 'Optical analog of the Iordanskii force in a Bose-Einstein condensate', Physical Review A 67, 053616 (2003). DOI 10.1103/PhysRevA.67.053616, https://doi.org/10.1103/PhysRevA.67.053616 (preprint arXiv:cond-mat/0110514, https://arxiv.org/abs/cond-mat/0110514).
33. Lorenzo Gavassino, Marco Antonelli, and Brynmor Haskell, 'Superfluid dynamics in neutron star crusts: the Iordanskii force and chemical gauge covariance', Universe 7, 28 (2021). DOI 10.3390/universe7020028, https://doi.org/10.3390/universe7020028 (preprint arXiv:2012.10288, https://arxiv.org/abs/2012.10288).
34. Yuri A. Sergeev, 'Mutual Friction in Bosonic Superfluids: A Review', Journal of Low Temperature Physics 212, 251-305 (2023). DOI 10.1007/s10909-023-02972-4, https://doi.org/10.1007/s10909-023-02972-4.
35. Edouard Sonin, 'Joe Vinen and the Transverse Force on Vortex', Journal of Low Temperature Physics 212, 306-320 (2023). DOI 10.1007/s10909-023-02977-z, https://doi.org/10.1007/s10909-023-02977-z (preprint arXiv:2308.06514, https://arxiv.org/abs/2308.06514).