Finite-frequency dynamics of vortex loops at the He superfluid phase transition

The finite-frequency dynamics of the 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 dH 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 ∼ aH where a is the loop diameter. Solving the 3D Fokker-Planck equation for the loop response at frequency ω yields a superfluid density which varies at Tλ as ωH. This power-law variation with ω agrees with the scaling form found by Fisher, Fisher, and Huse, since the dynamic exponent z is identified as z = dH −1. Flory scaling for the self-avoiding loops gives a fractal dimension in terms of the space dimension d as dH = (d+2)/2, yielding z = d/2 = 3/2 for d = 3, in complete agreement with dynamic scaling.