Kibble-Zurek scalings of continuous magnetic phase transitions in spin-1 spin-orbit-coupled Bose-Einstein condensates

We investigate the universal dynamics of a continuous magnetic phase transition from a unmagnetized phase to a ferromagnetic phase in a spin-1 spin-orbit-coupled Bose-Einstein condensate. When the system approaches the critical point, the Landau critical velocity gradually decreases to zero and so the correlation length diverges. Therefore, during the slow passage through the critical point, ferromagnetic domains are spontaneously created according to the Kibble-Zurek mechanism. Through calculating the Bogoliubov excitations, we give the Landau critical velocity and the correlation length, from which we derive the Kibble-Zurek scalings. Furthermore, we numerically simulate the critical spatial-temporal dynamics of the bifurcation delay and spontaneous domain formation, and extract the universal scaling exponents. The numerical scalings extracted from the critical spatial-temporal dynamics are very consistent with the scalings derived from the correlation length.

Figure 1

Ground-state phase transition of a spin-1 spin-orbit-coupled BEC. (a),(c) Dispersion relations of the lowest band with $\mathrm{\Omega }/E_r=7.737$ ($\mathrm{\Omega }/E_r=3.0609$). The white solid and red dashed lines represent minima in the dispersion spectra, with the former representing the lowest-energy states. (b) Ground-state phase diagram [33, 34, 35]. The dash-dotted (solid) line represents the first-order (second-order) phase transition and the dotted lines define crossover boundaries. The insets show schematic dispersion relations of the lowest band in different regions, where the spectrum has, respectively, one, two, and three minima. Particularly, the three minima are degenerate along the first-order transition curve. The tricritical point is indicated by the symbol $*$, where the single-minimum, double-minima, and triple-minima states merge. In our calculations, $g_0\overline{n}/E_r=1$ and $g_2\overline{n}=0$.

Figure 2

(a) Schematic diagram of the Kibble-Zurek mechanism. As a system is driven across a critical point with a linear ramp, the evolution becomes impulse when the transition time is less than the reaction time. The freeze-out time is then given as ${\tau }_r\left(\widehat{t}\right)={\tau }_t\left(\widehat{t}\right)=\widehat{t}$, which marks crossovers between the impulse and adiabatic stages of the time evolution. (b) The lowest band of the excitation spectrum for $\mathrm{\Lambda }=0.8{\mathrm{\Lambda }}_c$ (triangles), $\mathrm{\Lambda }={\mathrm{\Lambda }}_c$ (solid line), and $\mathrm{\Lambda }=1.2{\mathrm{\Lambda }}_c$ (circles). Here we chose $\mathrm{\Omega }/E_r=7.737,{\mathrm{\Lambda }}_c/E_r=-2,g_0\overline{n}/E_r=1$, and $g_2\overline{n}/E_r=0$.

Figure 3

(a) The Landau critical velocity $v_L$ vs $\mathrm{\Lambda }/E_r$ for different spin-dependent interaction strength $g_2$. (b)–(d) The Landau critical velocity $v_L$ vs $\left|\epsilon \right|$ for (b) $g_2=0$, (c) $g_2=-0.1g_0$, and (d) $g_2=0.1g_0$. The open triangles and circles correspond to the PW phase and the ZM phase, respectively. All parameters are chosen the same as those for Fig. 2. We can see that in the presence of spin-dependent interactions, the transition between the PW phase and ZM phase is slightly affected as the critical point shifts to larger (smaller) $\mathrm{\Lambda }$ for $g_2<0$ ($g_2>0$) but the scaling laws stand.

Figure 4

The universal scaling of the averaged bifurcation delay $\overline{b_d}$. The error bars denote the standard deviations. The inset shows fluctuations of the spin polarization $\mathrm{\Delta }{J}_z$ vs $\mathrm{\Lambda }\left(t\right)/{\mathrm{\Lambda }}_c$ for different ${\tau }_Q$. After the impulse-adiabatic transition, fluctuations quickly increase and $\mathrm{\Delta }{J}_z$ starts to develop, from which we determine the bifurcation delay $b_d$. All parameters are chosen the same as those in Fig. 2.

Figure 5

The universal scaling of the averaged domain numbers $\overline{N_d}$ counted at the freeze-out time. The error bars denote the standard deviations. The inset shows the time evolutions of $J_z\left(x\right)$ in a single run for different quench time ${\tau }_Q$. All parameters are chosen the same as those for Fig. 2.

Figure 6

The universal scaling of the averaged domain size $\overline{L_d}$. The error bars denote the standard deviations. The insets show the correlation functions $G\left(x\right)$ for different quench time ${\tau }_Q$. The typical domain size $L_d$ is illustrated for ${\tau }_Q={10}^3$ by the arrow in the left inset. The right inset shows that the correlation functions $G\left(x\right)$ collapse to a single cure when the distance is rescaled by the domain size $L_d$. All parameters are chosen the same as those for Fig. 2.