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arXiv:1304.4674 · Physical Review A

Spin-glass-like behavior in the spin turbulence of spinor Bose-Einstein condensates

Makoto Tsubota, Yusuke Aoki, Kazuya Fujimoto

Published Dec 6, 2013• 14 citations
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Abstract

We study numerically the spin turbulence (ST) in spin-1 ferromagnetic spinor Bose-Einstein condensates (BECs). ST is characterized by a $\ensuremath{-}7/3$ power law in the spectrum of the spin-dependent interaction energy. The direction of the spin density vector is spatially disordered but temporally frozen in ST, showing an analogy with the spin-glass state. Thus, we introduce the order parameter of spin glass into ST in spinor BECs. When ST develops through some instability, the order parameter grows with a $\ensuremath{-}7/3$ power law, thus succeeding in describing ST well.

DOI
10.1103/physreva.88.061601

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We study numerically the spin turbulence (ST) in spin-1 ferromagnetic spinor Bose-Einstein condensates (BECs). ST is characterized by a \ensuremath−7/3\ensuremath{-}7/3\ensuremath−7/3 power law in the spectrum of the spin-dependent interaction energy. The direction of the spin density vector is spatially disordered but temporally frozen in ST, showing an analogy with the spin-glass state. Thus, we introduce the order parameter of spin glass into ST in spinor BECs. When ST develops through some instability, the order parameter grows with a \ensuremath−7/3\ensuremath{-}7/3\ensuremath−7/3 power law, thus succeeding in describing ST well.
We summarize the recent theoretical and numerical works on spin turbulence (ST) in spin-1 spinor Bose-Einstein condensates. When the system is excited from the ground state, it goes through hydrodynamic instability to ST in which the spin density vector has various disordered direction. The properties of ST depend on whether the spin-dependent interaction is ferromagnetic or antiferromagnetic. ST has some characteristics different from other kinds of turbulence in quantum fluids. Firstly, the spectrum of the spin-dependent interaction energy exhibits the characteristic power law different from the usual Kolmogorov -5/3 law. Secondly, ST can show the spin-glass-like behavior; the spin density vectors are spatially random but temporally frozen.
We theoretically and numerically study spin turbulence (ST) with small spin magnitude in spin-1 spinor Bose-Einstein condensates by using the spin-1 spinor Gross-Pitaevskii (GP) equations. This kind of ST is realized in two cases: (i) with antiferromagnetic interaction and (ii) with ferromagnetic interaction under a static magnetic field. The ST with small spin magnitude can exhibit two characteristic power laws in the spectrum of the spin-dependent interaction energy: $\ensuremath{-}1$ and $\ensuremath{-}7/3$ power laws in the low- and high-wave-number regions, respectively. These power laws are derived from a Kolmogorov-type dimensional scaling analysis for the equations of motion of the spin vector and nematic tensor. To confirm these power laws, we perform a numerical calculation of the spin-1 spinor GP equations in a two-dimensional uniform system. In case (i), the $\ensuremath{-}7/3$ power law tends to appear in the high-wave-number region, but the spectrum in the low-wave-number region deviates from the $\ensuremath{-}1$ power law. In contrast, both the $\ensuremath{-}1$ and $\ensuremath{-}7/3$ power laws tend to appear in case (ii).
We perform a numerical study of spin turbulence in a two-dimensional trapped spin-1 spinor Bose-Einstein condensate, focusing on the energy spectrum. The spin turbulence in the trapped system is generated by instability of the helical structure of the spin density vector in the initial state. Our numerical calculation finds that in the trapped system, the spectrum of the spin-dependent interaction energy in the ferromagnetic case exhibits a $\ensuremath{-}7/3$ power law, which was confirmed in a uniform system by our previous study. The relation between the $\ensuremath{-}7/3$ power law and the motion of the spin density vector is discussed by investigating the orbits of dynamical variables in the spin space.
We theoretically study counterflow instability and turbulence in a spin-1 spinor Bose-Einstein condensate by the Gross-Pitaevskii equation and the Bogoliubov-de Gennes equation. Our study considers (i) the dynamics induced by the counterflow of two components with different magnetic quantum numbers, which leads to turbulence with spin degrees of freedom, and (ii) the properties of the turbulence. For (i), the behavior of the condensate induced by the counterflow strongly depends on whether the spin-dependent interaction is ferromagnetic or antiferromagnetic, leading to different behaviors for the dispersion relation and the spin density vector, etc. For (ii), we numerically calculate the spectrum of the spin-dependent interaction energy, which also depends on the spin-dependent interaction. The spectrum of the spin-dependent interaction energy in the ferromagnetic case clearly exhibits a $\ensuremath{-}7/3$ power law, which can be explained by scaling analysis. The spectrum in the antiferromagnetic case seems to show some power law too, but the power exponent cannot be estimated as definitely as the ferromagnetic case.
We theoretically and numerically study spin wave turbulence in spin-1 ferromagnetic spinor Bose-Einstein condensates, finding direct and inverse cascades with power-law behavior. To derive these power exponents analytically, the conventional weak wave turbulence theory is applied to the spin-1 spinor Gross-Pitaevskii equation. Thus we obtain the $\ensuremath{-}7/3$ and $\ensuremath{-}5/3$ power laws in the transverse spin correlation function for the direct and inverse cascades, respectively. To confirm these power laws, numerical calculations are performed that obtain results consistent with these power laws.