Universal scaling at nonthermal fixed points of a two-component Bose gas

Quasistationary far-from-equilibrium critical states of a two-component Bose gas are studied in two spatial dimensions. After the system has undergone an initial dynamical instability it approaches a nonthermal fixed point. At this critical point the structure of the gas is characterized by ensembles of (quasi)topological defects such as vortices, skyrmions, and solitons which give rise to universal power-law behavior of momentum correlation functions. The resulting power-law spectra can be interpreted in terms of strong-wave-turbulence cascades driven by particle transport into long-wavelength excitations. Scaling exponents are determined on both sides of the miscible-immiscible transition controlled by the ratio of the intraspecies to interspecies couplings. Making use of quantum turbulence methods, we explain the specific values of the exponents from the presence of transient (quasi)topological defects.

Figure 1

The figure shows the time evolution of the spin density $S^z\left(\mathbf{x}\right)$ (upper half) and of the incompressible kinetic energy density ${\epsilon }_i=\frac{1}{2}{\left|{\mathbf{w}}_i\left(\mathbf{x}\right)\right|}^2$ (lower half) in the immiscible regime. Numerical parameters are $N_s=1024$, $N_1=N_2=3.2\times {10}^9$, $g=3\times {10}^{-5}$, and $\alpha =2$. Panels (a)–(d) correspond to snapshots of a single run at different grid times $t=500,5000,10000,100000$. Note that only a ${512}^2$ subsection of the computational grid is displayed in order to highlight the emerging structures. Quickly after the start of the simulation domains with oppositely oriented spin emerge as an isotropic pattern, and incompressible energy is created along the domain borders; see panels (a). During the ensuing evolution towards the equilibrium state these domains merge until the system reaches a state in which only two large domains exist; cf. panels (d). On top of the coarse domain structure pointlike domains become visible during the intermediate stage of the evolution which are revealed to be skyrmionic defects by the accompanying pattern of incompressible energy. See the main text for further details.

Figure 2

The single-particle momentum distribution as defined in Eq. (6) at different time steps of the evolution, in dimensionless units defined in the main text. Numerical parameters correspond to Fig. 1 and the spectra are averaged over 100 runs. After the onset of domain formation a scaling regime with $n\sim k^{-3.5}$ forms at intermediate momenta. At a time $t=5000$ this region begins to move towards the IR while a new scaling region with $n\sim k^{-2}$ develops in the UV. The latter signals the onset of thermalization of the UV modes. At late times a stable bimodal power-law behavior has formed with a scaling exponent $\zeta =2$ in the UV and an IR scaling exponent $\zeta \simeq 3.5$.

Figure 3

The decomposition of the occupation spectrum as defined in Eq. (9), in comparison to the full spectrum at grid time $t=5\times {10}^4$, with numerical parameters corresponding to Fig. 1. Averages are taken over 100 runs. The black dots show the full spectrum which is also displayed in Fig. 2 while open colored symbols refer to the different parts of the decomposition, the incompressible and compressible parts of the classical hydrodynamic kinetic energy $n_i$ (red squares) and $n_c$ (blue circles), the quantum pressure part $n_q$ (grey triangles), and the spin pressure part $n_s$ (green stars). The IR scaling behavior $n\sim k^{-\zeta }$ is dominated by incompressible flow generated by vortex excitations with a scaling exponent $\zeta =4$. Spin excitations form domain walls which also give an important contribution in the same momentum regime but with a scaling exponent $\zeta =3$, such that the sum appears to follow a scaling law with $\zeta \simeq 3.5$. See main text for further details.

Figure 4

The time evolution of the spin-density distribution $S^z\left(\mathbf{x}\right)$ (upper half) and of the incompressible kinetic energy density ${\epsilon }_i={\left|{\mathbf{w}}_i\left(\mathbf{x}\right)\right|}^2/2$ (lower half) in the miscible regime. Numerical parameters are $N_s=1024$, $N_1=N_2=3.2\times {10}^9$, $g=1\times {10}^{-5}$, and $\alpha =0.8$, with $v={\xi }^{-1}$. Panels (a)–(d) correspond to snapshots of a single run at different grid times $t=5000,50000,100000,500000$. After the onset of the countersuperflow instability, isotropic $S^z$-polarized spin configurations emerge at intermediate times which are accompanied by uniform distributions of incompressible energy ${\epsilon }_i$ [see panel (a)]. During the thermalization process, the polarization decays in most of the spatial grid, thereby revealing the existence of vortex-antivortex pairs in both components which have $S^z\ne 0$ at the center of their cores. These point defects persist during the whole run time of the simulation and dominate the dynamical features during the late stage [panel (d)]. Consequently, the incompressible energy concentrates around vortical excitations during the later stage of the evolution [panels (b)–(d)], passing from classical to quantum turbulence.

Figure 5

The single-particle momentum distribution as defined in Eq. (6), at different grid times during the onset stage of the countersuperflow instability (CSI). The numerical parameters correspond to Fig. 4, and the spectra are averaged over 100 runs. Initially only modes $k_v$ corresponding to the counterflow velocity are macroscopically occupied (marked by the vertical dashed line). The CSI triggers the growth of certain unstable modes which then act as a source positioned at an intermediate momentum scale. This initiates a typical cascading process in momentum space, causing the transport of predominantly particles to the lower momentum modes and of energy to the higher momentum modes. The inset shows a snapshot of a typical spin configuration $S^z\left(\mathbf{x}\right)$ at $t=500$. One finds that the CSI results in planar spin waves propagating in the direction of the initial counterflow which we chose pointing into the $x$-direction.

Figure 6

The single-particle momentum distribution as defined in Eq. (6) at different grid times, in the miscible regime. Numerical parameters correspond to those in Fig. 4, and the spectra are averaged over 100 runs. With vanishing $S^z$ polarization also the scaling distribution $n\left(k\right)\sim k^{-3}$, which is seen at intermediate times over a wide momentum interval changes towards a bimodal form. While high momentum modes thermalize, forming a Rayleigh-Jeans distribution $n\left(k\right)\sim k^{-2}$, the IR regime is dominated by vortical excitations, implying $n\left(k\right)\sim k^{-4}$.

Figure 7

Decomposition of the momentum spectrum as defined in Eq. (9), in comparison with the full spectrum at grid time $t=500000$. Numerical parameters correspond to those in Fig. 4. Averages are taken over 100 runs. The black dots show the full spectrum also displayed in Fig. 6. Open colored symbols refer to the different parts of the decomposition: the incompressible and compressible parts of the classical hydrodynamic kinetic energy $n_i$ (red squares) and $n_c$ (blue circles), the quantum pressure part $n_q$ (grey triangles) and the spin pressure part $n_s$ (green stars). Other than in the immiscible regime the spin pressure contribution $n_s\left(k\right)$ develops a scaling $n\left(k\right)\sim k^{-4}$ in the IR region which adds up with the incompressible energy distribution $n_i\left(k\right)$ to give an IR scaling exponent $\zeta =4$. See main text for further details.

Figure 8

The spin-pressure distribution $n_s$, divided into its $z$-projection $n_s^z$ and $xy$-projection part $n_s^{xy}$ as defined in Eq. (12), at four different time steps. The time slices as well as the numerical parameters correspond to Fig. 4, and averages are taken over 100 runs. In the intermediate stage both quantities show the scaling behavior $n_s\left(k\right)\sim k^{-3}$ at intermediate momenta [panel (a)] which is expected for kink solutions. During the following time evolution [panels (b)–(d)] the scaling of $n_s^z$ is overtaken by a thermal distribution up to a small intermediate momentum region. In contrast, the distribution of $n_s^{xy}$ develops the bimodal scaling form which is characteristic for vortical flows in two dimensions with an IR scaling exponent $\zeta =4$ [panel (d)]. See the main text for further details.

Figure 9

The time evolution of the spin density $S^z\left(\mathbf{x}\right)$ (upper half) and of the incompressible kinetic energy density ${\epsilon }_i={\left|{\mathbf{w}}_i\left(\mathbf{x}\right)\right|}^2/2$ (lower half). The ratio $\alpha$ of couplings is tuned to the transition point between the miscible and immiscible regimes ($\alpha =1$). Besides this, numerical parameters and depicted times are the same as for Fig. 4. Note that the incompressible energy density has been amplified by a factor of 6 for $t=500000$ in panel (d). Similar to the situation deep in the miscible regime (compare with Fig. 4), after the onset stage of the countersuperflow instability $S^z$-polarized spin configurations emerge at intermediate times which are accompanied by uniform distributions of incompressible energy ${\epsilon }_i$ [panel (a)]. However, here the pseudodomain structure of $S^z$ undergoes a coarse-graining process similar to the situation in the immiscible regime (compare with Fig. 1), instead of decaying in time.

Figure 10

The single-particle momentum distribution as defined in Eq. (6), at different grid times, at the transition point $\alpha =1$. Numerical parameters correspond to those in Fig. 9, and the spectra are averaged over 100 runs. Similar to the miscible regime randomly distributed and orientated pseudokinks at intermediate times imply a domain-wall scaling $n\left(k\right)\sim k^{-3}$ over a wide range of momenta. However, during the following evolution this scaling is retained and the region shifted towards the lower momentum modes while thermalization of high momentum modes sets in. At the latest time step we find a momentum distribution with an almost stationary scaling $n\left(k\right)\sim k^{-3}$ in the IR.

Figure 11

The decomposition of the single-particle spectrum as defined in Eq. (9), in comparison with the full spectrum at grid time $t=500000$. Numerical parameters correspond to those in Fig. 9. Averages are taken over 100 runs. The black dots show the full spectrum which is also displayed in Fig. 10 while open colored symbols refer to the incompressible and compressible parts of the hydrodynamic kinetic energy, $n_i$ (red squares) and $n_c$ (blue circles), respectively, to the quantum pressure part $n_q$ (grey triangles) and the spin-pressure part $n_s$ (green stars). In contrast to the situation deep in the miscible and immiscible regimes the spin-pressure contribution $n_s$ dominates the spectrum of excitations for low momenta with a domain-wall scaling similar to that of $n$, $n_s\left(k\right)\sim k^{-3}$ for $k_d<k<k_{\xi }$. In this regime, incompressible energy excitations are found to be less relevant. See the main text for further details.

Figure 12

The spin-pressure distribution $n_s$ separated into its $z$-projection $n_s^z$ and $xy$-projection $n_s^{xy}$ as defined in Eq. (12), at four different time steps. Those as well as the numerical parameters correspond to Fig. 9, and averages are taken over 100 runs. For the intermediate and late times of evolution depicted here the $xy$-projection follows the form of the $z$-projection, thereby developing the same domain-wall scaling in the IR, $n_s^{xy}\left(k\right)\sim n_s^z\sim k^{-3}$ for $k_d<k<k_{\xi }$. Confer the main text for further details.