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Topological Hall Angles and Lifetime Hierarchies of Knotted Vortex Filaments

Omar Iskandarani

Published Feb 1, 2026• 0 citations
Publisher

Abstract

Recent measurements of mutual friction in strongly interacting Fermi superfluids establish a direct link between the transverse dynamics of quantized vortices and the relaxation of localized core states. Motivated by these results, we investigate the dynamics of knotted vortex filaments within an effective hydrodynamic framework. By extending the vortex Hall angle formalism to non-planar filament geometries, we derive a phenomenologically motivated scaling relation between filament topology and internal relaxation times. We show that a hierarchy of decay times emerges naturally from a topology-dependent Reynolds-like barrier, with lifetimes controlled by writhe-dependent geometric phase lags. While no claim is made of microscopic particle identity, the resulting hierarchy closely parallels the ordering observed in several physical systems exhibiting metastable topological excitations.

DOI
10.5281/zenodo.18388701

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Recent measurements of mutual friction in strongly interacting Fermi superfluids establish a direct link between the transverse dynamics of quantized vortices and the relaxation of localized core states. Motivated by these results, we investigate the dynamics of knotted vortex filaments within an effective hydrodynamic framework. By extending the vortex Hall angle formalism to non-planar filament geometries, we derive a phenomenologically motivated scaling relation between filament topology and internal relaxation times. We show that a hierarchy of decay times emerges naturally from a topology-dependent Reynolds-like barrier, with lifetimes controlled by writhe-dependent geometric phase lags. While no claim is made of microscopic particle identity, the resulting hierarchy closely parallels the ordering observed in several physical systems exhibiting metastable topological excitations.
We investigate the two-dimensional motion of a single vortex orbiting a pinned anti-vortex in a unitary Fermi superfluid. By analyzing its trajectory, we measure the yet-unknown longitudinal and transverse mutual friction coefficients, which quantify the vortex-mediated coupling between the normal and superfluid components. Both coefficients increase while approaching the superfluid transition. They provide access to the vortex Hall angle, which is linked to the relaxation time of the localized quasiparticles occupying Andreev bound states within the vortex core, and to the vortex Reynolds number Reα associated with the transition from laminar to quantum turbulent flows. We compare our results with numerical simulations and an analytic model originally formulated for superfluid 3He, finding good agreement. Our work suggests that vortex dynamics in unitary Fermi superfluids is essentially affected by the interplay between delocalized thermal excitations and vortex-bound quasiparticles. Further, it provides a novel testbed for studying vortex dynamics at finite temperatures. Vortex dynamics and mutual friction in quantum fluids are intimately connected to the fundamental properties of superfluids. Here, the authors reveal previously unexplored mechanisms underlying the mutual friction coefficients in ultracold Fermi superfluids in the unitary limit, suggesting bound quasiparticles within the vortex core play a significant role.
We extend the emergent hydrodynamic framework for fermionic statistics in a 3+1D quantized superfluid to include non-Abelian topological excitations.Single framed vortex loops reproduce the familiar Abelian π Berry phase corresponding to fermionic exchange. Multi-component vortex networks exhibit matrix-valuednon-Abelian holonomies associated with internal symmetry groups G. Linked and knotted vortices provide a mechanical realization of both fermionic statistics andinternal quantum numbers. We discuss how effective gauge interactions emerge from topological constraints, provide energy scale estimates, and present a roadmapconnecting vortex network structures to Standard Model-like particles and forces.
AbstractWe present an emergent hydrodynamic framework for fermionic statistics within a 3 + 1-dimensional quantized superfluid effective field theory (EFT). While classical vortices in irrotational fluids are bosonic, we demonstrate that quantized vortex loops generically acquire an effective internal framing arising from the anisotropic geometry and phase-gradient structure of their cores. These framed vortex loops inhabit a configuration space C ∼= (T^3 × SO(3))N /S^N. We define the Berry connection on the Hilbert bundle of condensate states and show that the adiabatic exchange of two such defects corresponds to a non-contractible loop in SO(3). Due to π1(SO(3)) ∼= Z2, the resulting holonomy yields a topological phase π, producing the fermionic exchange sign e^iπ = −1. This provides a concrete mechanical realization of the Finkelstein–Rubinstein constraint and suggests that half-integer spin and fermionic statistics emerge as topological properties of a superfluid vacuum.
The dynamics of torus vortex configurations V n, p, q in a superfluid liquid at zero temperature ( n is the number of quantum vortices, p is the number of turns of each filament around the symmetry axis of the torus, and q is the number of turns of the filament around its central circle; radii R 0 and r 0 of the torus at the initial instant are much larger than vortex core width ξ) has been simulated numerically based on the regularized Biot–Savart law. The lifetime of vortex systems till the instant of their substantial deformation has been calculated with a small step in parameter B 0 = r 0 /R 0 for various values of parameter Λ = ln( R 0 /ξ). It turns out that for certain values of n, p , and q , there exist quasi-stability regions in the plane of parameters ( B 0 , Λ), in which the vortices remain almost invariable during dozens and even hundreds of characteristic lengths.
In a recent article, Kwon et al. [Nature (London) 600, 64 (2021)NATUAS0028-083610.1038/s41586-021-04047-4] revealed nonuniversal dissipative dynamics of quantum vortices in a fermionic superfluid. The enhancement of the dissipative process is pronounced for the Bardeen-Cooper-Schrieffer interaction regime, and it was suggested that the effect is due to the presence of quasiparticles localized inside the vortex core. We test this hypothesis through numerical simulations with time-dependent density-functional theory: a fully microscopic framework with fermionic degrees of freedom. The results of fully microscopic calculations expose the impact of the vortex-bound states on dissipative dynamics in a fermionic superfluid. Their contribution is too weak to explain the experimental measurements, and we identify that thermal effects, giving rise to mutual friction between superfluid and the normal component, dominate the observed dynamics.
Excitations in vortex cores in superconductors and other Fermi superfluids are single-particle excitations with a peculiar energy spectrum. These excitations are responsible for many important thermodynamic properties such as specific heat, London penetration length, etc. They also determine the dynamic characteristics of superconductors and superfluids through their interaction with vortices. Flux flow resistance, the Hall effect in type II superconductors and the mutual friction in superfluids are the most important phenomena which strongly depend on vortex core excitations. These phenomena determine the electromagnetic responses of type II superconductors and the hydrodynamic behaviour of superfluids and are of great significance for practical applications of superconducting devices and for understanding the most fundamental properties of correlated electrons and other Fermi particles. In this review we consider the dynamic properties of superconductors and superfluids and outline the basic ideas and results on the vortex dynamics in clean superfluid Fermi systems. The forces acting on moving vortices are discussed including the problem of the transverse force which was a matter of confusion for quite some time. We formulate the equations of the vortex dynamics, which include all the forces and the inertial term associated with excitations bound to the moving vortex.