Direct and inverse cascades of spin-wave turbulence in spin-1 ferromagnetic spinor Bose-Einstein condensates

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 $-7\mathrm{/}3$ and $-5\mathrm{/}3$ 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.