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arXiv:1703.09360 · Physical Review Letters

Coarsening Dynamics of an Isotropic Ferromagnetic Superfluid

L. A. Williamson, P. B. Blakie

Published Dec 22, 2017• 28 citations
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Abstract

In zero magnetic field the ground-state manifold of a ferromagnetic spin-1 condensate is SO(3) and exhibits Z_{2} vortices as topological defects. We investigate the phase-ordering dynamics of this system after being quenched into this ferromagnetic phase from a zero-temperature unmagnetized phase. Following the quench, we observe the ordering of both magnetic and gauge domains. We find that these domains grow diffusively, i.e., with domain size L(t)∼t^{1/2}, and exhibit dynamic scale invariance. The coarsening dynamics progresses as Z_{2} vortices annihilate; however, we find that at finite energy a number of these vortices persist in small clumps without influencing magnetic or gauge order. We consider the influence of a small nonzero magnetic field, which reduces the ground-state symmetry, and show that this sets a critical length scale such that when the domains reach this size the system dynamically transitions in order parameter and scaling behavior from an isotropic to an anisotropic ferromagnetic superfluid.

DOI
10.1103/physrevlett.119.255301

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In zero magnetic field the ground-state manifold of a ferromagnetic spin-1 condensate is SO(3) and exhibits Z_{2} vortices as topological defects. We investigate the phase-ordering dynamics of this system after being quenched into this ferromagnetic phase from a zero-temperature unmagnetized phase. Following the quench, we observe the ordering of both magnetic and gauge domains. We find that these domains grow diffusively, i.e., with domain size L(t)∼t^{1/2}, and exhibit dynamic scale invariance. The coarsening dynamics progresses as Z_{2} vortices annihilate; however, we find that at finite energy a number of these vortices persist in small clumps without influencing magnetic or gauge order. We consider the influence of a small nonzero magnetic field, which reduces the ground-state symmetry, and show that this sets a critical length scale such that when the domains reach this size the system dynamically transitions in order parameter and scaling behavior from an isotropic to an anisotropic ferromagnetic superfluid.
Coarsening dynamics, the canonical theory of phase ordering following a quench across a symmetry breaking phase transition, is thought to be driven by the annihilation of topological defects. Here we show that this understanding is incomplete. We simulate the dynamics of an isolated spin-1 condensate quenched into the easy-plane ferromagnetic phase and find that the mutual annihilation of spin vortices does not take the system to the equilibrium state. A nonequilibrium background of long wavelength spin waves remain at the Berezinskii-Kosterlitz-Thouless temperature, an order of magnitude hotter than the equilibrium temperature. The coarsening continues through a second much slower scale invariant process with a length scale that grows with time as t^{1/3} t 1 / 3 . This second regime of coarsening is associated with spin wave energy transport from low to high wavevectors, bringing about the the eventual equilibrium state. The transport displays features of a spin wave energy cascade, providing a potential profitable connection with the emerging field of spin wave turbulence. Strongly coupling the system to a reservoir destroys the second regime of coarsening, allowing the system to thermalise following the annihilation of vortices.
Spinor Bose-Einstein condensates exhibit both superfluid and magnetic order, and accommodate phases with rich symmetry properties and topological defects. Transitions between these phases can be induced by tuning external fields. In this thesis we explore the dynamics of order formation in a quasi-2D spin-1 ferromagnetic condensate following a quench from an unmagnetised phase to one of three ferromagnetic phases. The ferromagnetic phases exhibit distinct symmetry properties (easy-plane, easy-axis or isotropic) and support distinct topological defects. In each phase we observe scale invariant ordering and identify the relevant topological defect affecting the order parameter growth. We find that each phase is characterised by a distinct dynamic critical exponent. In the easy-plane phase we identify a persistent turbulent cascade that affects spin ordering long after all topological defects have annihilated. In addition to our exploration of phase ordering dynamics, we study a microscopic model of spin vortex dynamics in the easy-plane phase. Our work provides a comprehensive theoretical study of phase ordering in a conservative system, provides a thorough foundation for studies of phase ordering in antiferromagnetic and higher spin condensates, and offers prospects for further research into fundamental questions regarding the ordering properties of spin systems. Our work is pertinent to current experiments, which have explored the initial stages of phase ordering in both ferromagnetic and antiferromagnetic spin-1 condensates.
We examine the dynamics of a quasi-two-dimensional spin-1 condensate in which the quadratic Zeeman energy qqq is suddenly quenched to a value where the system has a ferromagnetic ground state. There are two distinct types of ferromagnetic phases, i.e., a range of qqq values where the magnetization prefers to be in the direction of the external field (easy axis) and a range of qqq values where it prefers to be transverse to the field (easy plane). We study the quench dynamics for a variety of qqq values and show that there is a single dynamic critical exponent to characterize the scale-invariant domain growth for each ferromagnetic phase. For both quenches we give simple analytic models that capture the essential scale-invariant dynamics and correctly predict the exponents. Because the order parameter for each phase is different, the natures of the domains and the relevant topological defects in each type of coarsening are also different. To explore these differences we characterize the fractal dimension of the domain walls and the relationship of polar-core spin vortices to the domains in the easy-plane phase. Finally, we consider how the energy liberated from the quench thermalizes in the easy-axis quench. We show that local equilibrium is established in the spin waves on moderate time scales, but continues to evolve as the domains anneal.
We demonstrate that a quasi-two-dimensional spin-1 condensate quenched to a ferromagnetic phase undergoes universal coarsening in its late time dynamics. The quench can be implemented by a sudden change in the applied magnetic field and, depending on the final value, the ferromagnetic phase has easy-axis (Ising) or easy-plane (XY) symmetry, with different dynamical critical exponents. Our results for the easy-plane phase reveal a fractal domain structure and the crucial role of polar-core spin vortices in the coarsening dynamics.
In this paper we study the coarsening dynamics of a spinor condensate quenched into an easy-axis ferromagnetic phase by a sudden change in the quadratic Zeeman energy. We show that applying a spin rotation prior to changing the Zeeman energy accelerates the development of local order and reduces heating. We examine the longitudinal spin ordering and the superfluid ordering of the system and show that the respective order parameter correlation functions exhibit dynamic scaling in the late-time dynamics. Our results also demonstrate that these two types of order grow at different rates, i.e., with different dynamic critical exponents. The spin domain area distribution is calculated and is shown to have power-law scaling behavior expected from percolation theory.
We consider the phase-ordering dynamics of an isolated quasi-two-dimensional spin-1 Bose gas quenched into an easy-plane ferromagnetic phase. Preparing the initial system in an unmagnetized antiferromagnetic state the subsequent ordering involves both polar core and Mermin-Ho spin vortices, with the ratio between the different vortices controllable by the quench parameter. Ferromagnetic domain growth occurs as these vortices annihilate. The distinct dynamics of the two types of vortices means that the domain growth law is determined by two macroscopic length scales, violating the standard dynamic scaling hypothesis. Nevertheless we find that universality of the ordering process manifests in the decay laws for the spin vortices. To provide a fuller picture of the phase ordering dynamics we also present results for the scaling of the order parameter correlation function and the hydrodynamic decomposition of the system kinetic energy.