Exact Solution for Vortex Dynamics in Temperature Quenches of Two-Dimensional Superfluids

Abstract

An exact analytic solution for the dynamics of vortex pairs is obtained for rapid temperature quenches of a superfluid film starting from the line of critical points below the critical temperature TKT . An approximate solution for quenches at and above above TKT provides insights into the origin of logarithmic transients in the vortex decay, and is in general agreement with recent simulations of the quenched XY model. These results confirm that there is no "creation" of vortices whose density increases with the quench rate as predicted by the Kibble-Zurek theory, but only monotonic decay of the thermal vortices already present at the initial temperature. PACS numbers: 64.60.Ht, 67.25.dj, 67.25.dk, 67.25.dpAlthough the phase-ordering kinetics of temperaturequenched thermodynamic systems have been studied for many decades [1], there are only a few exact results for the dynamics of the recovery to equilibrium [2,3]. For many systems, progress in the field has been made by asserting that dynamic scaling should apply to a quenched system, that for a system with non-conserved order parameter the dynamics will be characterized by a growing length scale ξ(t) = ξ 0 t 1/z , where z ≈ 2 is the dynamical exponent of model A in the classification of Halperin and Hohenberg [4] and t is the time from the quench to low temperature. Scaling holds if solutions involving a length scale r only depend on the ratio r/ξ. The growing length scale characterizes the domain growth of the topological defects of the order parameter as the system becomes completely ordered at long times. A phenomenological argument [1,5] is commonly used to predict the time dependence of the decaying defect density:where n is the number of components of the order parameter. Superfluids are in the n = 2 universality class, where the defects are quantized vortices, so if z = 2 then dynamic scaling predicts a 1/t decay of the vortex density. Computer simulations of spin systems with varying n gave general agreement with Eq. (1), though only at long time and length scales [6]. However, a problematic case for dynamic scaling has been two-dimensional superfluids, where the defects are the vortex pairs of the Kosterlitz-Thouless theory [7] characterizing the equilibrium phase transition occurring at the critical temperature T KT . The above arguments would give a vortex density decaying as t −1 for quenches from well above T KT to very low temperatures. Simulations of the XY model [8][9][10], however, showed the initial vortex decay to be considerably slower than this, and then only at very long times finally approached the predicted exponent of −1. The behavior could be modeled as a ln t/t variation, but this requires altering the dynamic length scale to vary as (t/ ln t) 1/z for initial temperatures above T KT . This change has been cited [11] as a breakdown of dynamic scaling, though others [12] find such a sudden change in the dynamic scale still fully consistent with scaling. To further complicate the issue, numerical solutions of the Fokker-Planck equation f...

Coarsening and thermalization properties of a quenched ferromagnetic spin-1 condensate

“…Following Eqs. (40) and (41) we fit late time numerical data for 1/n k to the functional form a + bk 2 , with a and b as fitting parameters. From a and b we can extract the temperature k B T and the fitted energy gap E gap .…”

Fundamental Connection Between Temperature-Quenched 2D Superfluids and 2D Quantum Turbulence

“…The presence of two different dynamic exponents was unusual, and it was not entirely clear how these came into the vortex dynamics. An exact solution [3] for temperature quenches starting from T KT and below shed light on some of these questions, where the vortex density was found to decay with time t after the quench as t −z scale ∕z . Here, z scale is evaluated at the initial temperature before the quench, and z = 2 is exact.…”

Scintillation yield from electronic and nuclear recoils in superfluid $^4$He

“…Energy deposition in superfluid helium results in a rich variety of excited species, and the relative population of these species contains information about the * Corresponding author: junsonglin@berkeley.edu primary track. Signals in superfluid helium include singlet diatomic molecules (excimers), triplet excimers, rotons, phonons, and perhaps quantum turbulence [18][19][20]. Recent experimental measurements of superfluid helium response signals have focused on scintillation as a function of temperature and applied electric field from α and electron sources [21][22][23][24].…”