Finite-frequency dynamics of vortex loops at the ${}^4$He superfluid phase transition

Abstract

The finite-frequency dynamics of the ${}^4$He superfluid phase transition can be formulated in terms of the response of thermally excited vortex loops to an oscillating flow field. The key parameter is the Hausdorff fractal dimension $d_H$ of the loops, which affects the dynamics because the frictional force on a loop is proportional to the total perimeter $P$ of the loop, which varies as $P\sim a^{d_H}$ where $a$ is the loop diameter. Solving the 3D Fokker-Planck equation for the loop response at frequency $\omega$ yields a superfluid density which varies at $T_{\lambda }$ as ${\omega }^{1/(d_H-1)}$. This power-law variation with $\omega$ agrees with the scaling form found by Fisher, Fisher, and Huse, since the dynamic exponent $z$ is identified as $z=d_H-1$. Flory scaling for the self-avoiding loops gives a fractal dimension in terms of the space dimension $d$ as $d_H=(d+2)/2$, yielding $z=d/2=3/2$ for d = 3, in complete agreement with dynamic scaling.

4 pages, submitted to Journal of Physics: Conference Series (Proceedings of LT25)

References in corpus (7)

• The critical exponents of the superfluid transition in He4
• Vortex Loop Phase Transitions in Liquid Helium, Cosmic Strings, and High-T_c Superconductors
• High Precision Measurement of the Thermal Exponent for the Three-Dimensional XY Universality Class
• Hausdorff dimension of critical fluctuations in abelian gauge theories
• Vortex Fluctuations in the Critical Casimir Effect of Superfluid and Superconducting Films
• Methods to determine the Hausdorff dimension of vortex loops in the three-dimensional XY model
• Critical exponent η_ϕof the Lattice London Superconductor and vortex loops in the 3D XY model