Bidirectional universal dynamics in a spinor Bose gas close to a nonthermal fixed point

We numerically study the universal scaling dynamics of an isolated one-dimensional ferromagnetic spin-1 Bose gas. Preparing the system in a far-from-equilibrium initial state, simultaneous coarsening and refining is found to enable and characterize the approach to a nonthermal fixed point. A macroscopic length scale which scales in time according to $L_{\mathrm{\Lambda }}\left(t\right)\sim t^{\beta }$, with $\beta \simeq 1/4$, quantifies the coarsening of the size of spin textures. At the same time kinklike defects populating these textures undergo a refining process measured by a shrinking microscopic length scale $L_{\lambda }\sim t^{{\beta }^{\prime }}$, with ${\beta }^{\prime }\simeq -0.17$. The combination of these scaling evolutions enables particle and energy conservation in the isolated system and constitutes a bidirectional transport in momentum space. The value of the coarsening exponent $\beta$ suggests the dynamics to belong to the universality class of diffusive coarsening of the one-dimensional $XY$ model. However, the universal momentum distribution function exhibiting nonlinear transport marks the distinction between diffusive coarsening and the approach of a nonthermal fixed point in the isolated system considered here. This underlines the importance of the universal scaling function in classifying nonthermal fixed points. Present-day experiments with quantum gases are expected to have access to the predicted bidirectional scaling.

Figure 1

Space-time evolution of the transversal spin $F_{\perp }=\left|F_{\perp }\right|exp\left\{i{\theta }_{F_{\perp }}\right\}$. Quenching the system across the quantum phase transition, here to $q_{\mathrm{f}}=0.9$, introduces exponentially growing unstable modes which, after a few characteristic time scales $t_{\mathrm{s}}$, lead to the formation of a flat background transversal spin, with textures on top. The spin textures, with size given by the distance over which a $2\pi$ phase winding occurs in the phase angle ${\theta }_{F_{\perp }}$, are populated by kinklike defects. The defects are characterized by a dip in the amplitude and a corresponding phase jump as depicted in panels (a) and (b). The solid black lines in panel (a) indicate a sound cone associated with the sound velocity of the spin degree of freedom $c_{\mathrm{s}}=\sqrt{n_0\left|c_1\right|}=1$. The size of the spin textures grows in time which is associated with the dilution of kinklike defects leading to long-range order developing in the phase field [see panel (b)]. The flat background spin length reflects the approximate conservation of the integrated spin structure factor, $\langle |F_{\perp }{|}^2\rangle (k=0)={\sum }_p\langle F_{\perp }^{*}\left(p\right)F_{\perp }\left(p\right)\rangle$, cf. inset of Fig. 2, which governs the scaling evolution towards the nonthermal fixed point. Each panel only shows an excerpt of the total grid of length $\mathcal{L}=554{\xi }_{\mathrm{s}}$.

Figure 2

Two-scale self-similar evolution of the structure factor $S(k,t)$. (a) Overview of the time evolution. The initial polar condensate at $t=0$ shows ground-state fluctuations around zero spin (blue triangles). At $t=0.67t_{\mathrm{s}}$ (orange crosses) the population of the momentum modes is well approximated by the Bogoliubov prediction (dash-dotted line). For times $25t_{\mathrm{s}}\le t\le 200t_{\mathrm{s}}$ the system is in the spatiotemporal scaling regime and evolves in a self-similar manner (dots). Three qualitatively different momentum regimes emerge: a plateau below a characteristic momentum scale $k_{\mathrm{\Lambda }}\left(t\right)$ [which is more clearly seen in panel (b); the dashed line exemplarily marks the scale extracted by means of fitting the scaling function Eq. (5) to the infrared regime of the data at time $t=25t_{\mathrm{s}}]$, a $k^{-\zeta }$ power-law falloff at momenta up to a scale $k_{\lambda }\left(t\right)$ and a steeper power-law decay at large momenta. The inset shows the extracted exponent $\zeta$ (cf. solid line in main frame). (b) Structure factor in the temporal scaling regime rescaled according to Eq. (1) with scaling exponents $\alpha =0.27\pm 0.06$ and $\beta =0.25\pm 0.04$ extracted via a least-square fit. Within the infrared scaling regime, $k\le k_{\mathrm{IR}}^{>}$ (dashed line), all curves collapse onto a single one. The inset shows that, within this momentum regime, the local spin fluctuations are conserved (up to $\sim 2\%$) for times $25t_{\mathrm{s}}\le t\le 200t_{\mathrm{s}}$ (dotted lines). The corresponding upper bound for the integral is set by $k_{\mathrm{u}}\left(t\right)=k_{\mathrm{IR}}^{>}·{(t/t_{\mathrm{ref}})}^{-1/4}$ with $t_{\mathrm{ref}}=25t_{\mathrm{s}}$. Within errors, the scaling exponents are consistent with $\alpha =\beta$. (c) Structure factor in the temporal scaling regime rescaled according to Eq. (1) with ${\alpha }^{\prime }=-0.53\pm 0.08$ and ${\beta }^{\prime }=-0.17\pm 0.04$. Within the ultraviolet scaling regime, $k_{\mathrm{UV}}^{<}\le k\le k_{\mathrm{UV}}^{>}$ (marked by dashed lines), all curves collapse onto a single one.

Figure 3

Time evolution of the structure factor $S(k,t)$ on time scales up to $t=4000t_s$ indicating the departure from the scaling regime. Bidirectional self-similar dynamics can be observed even up to $t\simeq 400t_{\mathrm{s}}$ as the scaling function remains the same as in Fig. 2. For $t\ge 800t_{\mathrm{s}}$ a thermal tail given by an approximate $k^{-2}$ power law (see black dashed line) is present in the UV regime of momenta. The mean kinetic energy in the tail gradually increases as time evolves up to $t=4000t_{\mathrm{s}}$. Note that the final equilibration process is beyond the time scales considered in our numerical simulations.

Figure 4

Time evolution of the characteristic IR scale $k_{\mathrm{\Lambda }}\left(t\right)$ [panel (a)], as well as the UV scale $k_{\lambda }\left(t\right)$ [panel (b)]. While the IR scale decreases algebraically as $k_{\mathrm{\Lambda }}\left(t\right)\sim L_{\mathrm{\Lambda }}{\left(t\right)}^{-1}\sim t^{-0.25}$ over the whole range of evolution times considered [see solid line in (a)], the UV scale starts to deviate from $k_{\lambda }\left(t\right)\sim L_{\lambda }{\left(t\right)}^{-1}\sim t^{0.17}$ [see solid line in (b)] at $t\simeq 500t_{\mathrm{s}}$. The deviation arises as the corresponding UV length scale approaches the spin healing length, i.e., as $L_{\lambda }\left(t\right)=2\pi /k_{\lambda }\left(t\right)\to 1$. While this causes the system to leave the regime of two-scale universal dynamics, it stays close to the nonthermal fixed point as the IR scaling behavior remains unchanged. The characteristic scales $k_{\mathrm{\Lambda }}\left(t\right)$ and $k_{\lambda }\left(t\right)$ are obtained by means of fitting the scaling form (5) to the structure factor up to a maximum momentum given by $k_{\mathrm{UV}}^{>}$ [see Fig. 2]. We remark that the scaling function, Eq. (5), does not appropriately capture the data in the UV regime of momenta anymore for evolution times $t\gtrsim 2000t_{\mathrm{s}}$, which strongly influences the extraction of the UV scale $k_{\lambda }\left(t\right)$. Hence we restrict our analysis to the time window $25t_{\mathrm{s}}\le t\le 2000t_{\mathrm{s}}$. For all times considered $k_{\mathrm{\Lambda }}\left(t\right)$ is comfortably larger than the momentum scale $k_{\mathcal{L}}=2\pi /\mathcal{L}=0.011$ associated with the system size $\mathcal{L}$. Errors bars correspond to the fit error of the extracted scales. Note the double-log scale.

Figure 5

Spatial first-order coherence function $C(r,t)$ within the temporal scaling regime. $C(r,t)$ is calculated by applying a Fourier transform to the structure factor $S(k,t)$. At larger distances the correlation function decays exponentially $C(r,t)\sim exp(-r/L_{\mathrm{\Lambda }})$, with time-evolving correlation length $L_{\mathrm{\Lambda }}\left(t\right)\sim t^{\beta }$. The inset shows $C(r,t)$ rescaled with $\tilde{\alpha }=0.02$ and $\beta =0.25$ for reference time $t_{\mathrm{ref}}=25t_{\mathrm{s}}$ such that the data collapses at large distances. This enables one to observe the shrinking of the characteristic UV length scale $L_{\lambda }\left(t\right)$ occurring at distances below $r\simeq 0.5$. Note the semilog scale of the inset.

Figure 6

(a) Scaling exponents $\alpha$ (blue stars) and $\beta$ (orange circles) obtained from least-square rescaling fits to $S(k,t)$ within the time window $[t_{\mathrm{ref}},t_{\mathrm{ref}}+\mathrm{\Delta }t]$ with $\mathrm{\Delta }t=120t_{\mathrm{s}}$. The scaling exponents are independent of the reference time $t_{\mathrm{ref}}$. A constant fit to the data reveals $\alpha =0.284\pm 0.013$ (dashed line) and $\beta =0.261\pm 0.008$ (solid line). The error is given by the standard deviation of all data points as they are not statistically independent. (b) Scaling exponents ${\alpha }^{\prime }/3$ (blue stars) and ${\beta }^{\prime }$ (orange circles) obtained by the same method and within the same time window as in (a). A constant fit to the data yields ${\alpha }^{\prime }/3=-0.183\pm 0.002$ (dashed line) and ${\beta }^{\prime }=-0.168\pm 0.004$ (solid line). Error computed as in (a).