Abstract
We numerically study spin turbulence (ST) in spin-1 spinor Bose-Einstein condensates with ferromagnetic and antiferromagnetic interaction by solving the Gross-Pitaevskii equation. In the previous study for ST with ferromagnetic interaction, the directions of the spin density vectors were found to be spatially disordered but temporally frozen. This behavior of ST is similar to that of spin glass. In this study, to characterize the “spin-glass-like” behavior, we calculate the order parameter of the spin glass. In ST with ferromagnetic interaction, we confirm the growth of the order parameter, which indicates that ST behaves like spin glass. On the other hand, in ST with antiferromagnetic interaction, the order parameter does not grow. This means that spin density vectors temporally fluctuate, not being frozen.
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Notes
The spectrum of the spin-dependent interaction energy is obtained like this. Substituting \({\bf F}({\bf r})=\sum_{\bf k}\tilde{\bf F}({\bf k})e^{i{\bf k}\cdot{\bf r}}\) into the spin-dependent interaction energy \(E_{s}=\frac{c_{1}}{2}\int|{\bf F}|^{2}d{\bf r}\), we obtain \(E_{s}=\frac{c_{1}L^{2}}{2}\sum_{\bf k}|\tilde {\bf F}({\bf k})|^{2}\). Therefore, the spectrum of the spin-dependent interaction energy is given by \(E_{s}(k)=\frac{c_{1}L^{2}}{2\Delta k}\sum_{k\leq|{\bf k}|<k+\Delta k}|\tilde{\bf F}({\bf k})|^{2}\) with Δk=2π/L.
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Aoki, Y., Fujimoto, K. & Tsubota, M. Calculation of Spin Glass Order Parameter in Spin Turbulence of Spin-1 Spinor Bose-Einstein Condensate. J Low Temp Phys 175, 216–221 (2014). https://doi.org/10.1007/s10909-013-0966-7
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DOI: https://doi.org/10.1007/s10909-013-0966-7