Figure 1

Distribution of spin density vectors in ST at $t/\tau =4000$ obtained by the counterflow instability with $V_R/c_{\text{sound}}=0.78$, where $V_R$, $c_{\text{sound}}$, and $\tau$ are the relative velocity of counterflow, the sound velocity, and the characteristic time in a uniform system [13]. The system size is $128\xi \times 128\xi$ with the coherence length $\xi$. This numerical calculation starts from the initial state of the counterflow between the $m=\pm 1$ components, which induces the spin modulation, finally leading to ST [13].Reuse & Permissions

Figure 2

Spectrum of the spin-dependent interaction energy in three cases: (a) counterflow instability in a uniform system, (b) instability of the initial helical structure of the spin density vector in a trapped system, and (c) application of an oscillating magnetic field in a uniform ferromagnetic system. Each spectrum is calculated at (a) $t/\tau =4000$, (b) $t\omega =200,$ and (c) $t/\tau =9000$, when ST is fully developed. In the case of (c), the magnetic field is tuned off at $t/\tau =5000$. The parameters $\omega$, $a_h$, $R_{\text{TF}}$, $n_0$, $\xi$, and $L$ are, respectively, the trapping frequency, the harmonic oscillator length, the Thomas-Fermi radius, the total density in the uniform system, the coherence length, and the system size. The details of these parameters are described in [13, 15, 23], respectively. The blue broken lines show the $-7/3$ power law. The spectrum in all cases exhibits the $-7/3$ power law in the region [(a) $2\pi /L<k<2\pi /{\xi }_s$, (b) $2\pi /R_{\text{TF}}<k<2\pi /{\xi }_s$, (c) $2\pi /L<k<2\pi /{\xi }_s$] [25], where ${\xi }_s$ is the spin coherence length.Reuse & Permissions

Figure 3

Time development of the exponent $n_p$ and the spin-glass order parameter $\sqrt{q\left(t\right)}$ for (a) the counterflow instability in a uniform system, (b) the instability of the initial helical structure of the spin density vector in a trapped system, and (c) the application of the oscillating magnetic field in a uniform ferromagnetic system. In (b), the squares and triangles show that the calculation of $\sqrt{q}$ is performed in the regions $r<R_{\text{TF}}/2$ and $r<R_{\text{TF}}$, respectively. The exponents $n_p$ in (a)–(c) are evaluated by the least-squares method in the region [(a) $4\pi /L<k<2\pi /{\xi }_s$, (b) $2\pi /R_{\text{TF}}<k<2\pi /{\xi }_s$, (c) $4\pi /L<k<2\pi /{\xi }_s$]. Then, in the uniform systems of (a) and (b), we omit the point $2\pi /L$, where the spectrum can be affected by the finite-size effect. The insets of (a) and (b) are the enlarged graphs of $n_p$.Reuse & Permissions