Anomalous non-thermal fixed point in a quasi-two-dimensional dipolar Bose gas

TL;DR

The paper investigates anomalous non-thermal fixed-point dynamics in a quasi-two-dimensional dipolar Bose gas by analyzing self-similar space-time scaling of momentum spectra and coarsening of vortex patterns. Using truncated-Wigner simulations of the Gross-Pitaevskii equation with both contact and dipolar interactions, the authors identify a universal sub-diffusive coarsening regime with $eta \\approx 0.2$ that is robust across ultradilute and quantum parameter sets, followed by a late-time diffusion-type scaling with $eta \\approx 0.5$ driven by vortex–sound friction. They connect this anomalous fixed point to vortex annihilation dynamics while revealing that dipolar interactions and roton excitations qualitatively alter vortex patterns (anti-clustering and roton-induced density ripples) without changing the universal scaling exponent. The results support the existence of a partial attractor near non-thermal fixed points in dipolar systems and highlight how long-range, anisotropic interactions shape defect configurations while preserving key scaling properties. These findings advance the classification of universality classes for non-thermal fixed points and inform future analytic characterizations of anomalous scaling in low-dimensional quantum fluids.

Abstract

The emergence of distinctly sub-diffusive scaling in the vicinity of an anomalous non-thermal fixed point is discussed in a quasi-two-dimensional dipolar Bose gas in the superfluid phase, carrying ensembles of vortices and antivortices with zero net angular momentum. The observed scaling behavior reflects coarsening dynamics driven by the mutual annihilation of vortices and antivortices, with the mean inter-defect distance growing algebraically over time as $\ell_\text{v}(t)\sim t^{\,β}$. A sub-diffusive ($β<1/2$) exponent $β\approx0.2$ is extracted for various parameter regimes, initial conditions, and dipolar configurations from both scaling occupation-number spectra and the evolution of inter-defect distances as well as the corresponding total vortex densities. As vortex-antivortex annihilation progresses, excitations of the background condensate increase. This gives rise to a transition in the scaling behavior at late times, toward a non-thermal fixed point governed by diffusion-type scaling with $β\approx1/2$ as expected for the mutual annihilation of well-separated vortex-antivortex dipoles. While the temporal scaling with $β$ does not depend significantly on the strength and anisotropy of the dipolar interactions and thus underlines the universality of the anomalous as well as diffusion-type non-thermal fixed points, we find distinctly different vortex patterns resulting in the dipolar case. While in the superfluid with contact interactions only, same-sign vortices tend to cluster and form large-scale eddies, in the dipolar and tilted cases, roton excitations appear to prevent such motion, giving rather rise to a maximisation of distances between vortices of either sign.

Figures (11)

• Figure 1: (a,b) Initial density $\rho_2/ \langle \rho_2 \rangle$ and (c,d) phase $\varphi=\arg \psi$ for both (a,c) lattice and (b,d) random sampling after a phase imprint and a brief evolution in imaginary time for the vortex cores to form. Panels (e--h) show the normalized, absolute current $|\vb{j}| = |\rho_2 \nabla \varphi|$ in the ultradilute case for $\epsilon_\mathrm{dd}=0.5$ and $\theta=0$, starting from the lattice initial condition. The gray streamlines indicate the local strength and direction of the current field $\vb{j}$ and the dots and crosses for $t>0$ show the positions of the vortices and antivortices, respectively. The temporal evolution is shown for the initial configuration at (e) $t=0$, as well as at (f) $t=10^4 (2\pi)/\omega_z$, (g) $t=10^5 (2\pi)/\omega_z$, and (h) $t=10^6 (2\pi)/\omega_z$. In particular at early times we observe high flows encircling clusters of vortices which coarsen over time leading to a weaker flow and smaller vortex clusters.
• Figure 2: Rescaled occupation number $n(k,t)$ in the (a) ultradilute regime for $\epsilon_\mathrm{dd}=0.5$ and (b) quantum regime for $\epsilon_\mathrm{dd}=1.47$, both for $\theta=0$, lattice sampling and over two, respectively one, order of magnitude in time. The gray, ultraviolet (UV) region is excluded from the rescaling procedure in order to identify the IR scaling behavior, such that the exponents are extracted by fitting the scaling collapsed distributions within the momentum window $k\in[0,k_\text{max}(t)]$ only, see main text for details. We obtain $\alpha=0.40\pm0.02$ and $\beta=0.21\pm0.01$ in the ultradilute and $\alpha=0.37\pm0.06$ and $\beta=0.19\pm0.02$ in the quantum case. The non-rescaled density spectra are shown in the lower-left inset for comparison. In (a) at $t=10^5 (2\pi)/\omega_z$ we extract the spatial exponent $\zeta=5.44\pm0.09$ and in (b) at $t=10^2 (2\pi)/\omega_z$ the exponent $\zeta=4.39\pm0.10$, cf. (\ref{['eq:occupation_number_scaling']}), \ref{['eq:scalingfunction']}. The inverse of the deviation $\chi^2$ of the least-squares fit for the respective scaling exponents is shown in the upper-right inset and indicates the optimal scaling exponents.
• Figure 3: Time-varying scaling exponents $\alpha(t)$ (solid) and $\beta(t)$ (dashed) are shown for different tilting angles in the ultradilute case at $\epsilon_\mathrm{dd}=1.5$ for (a) lattice and (b) random sampling and (c) in the quantum case for $\epsilon_\mathrm{dd}=1.47$ and random sampling. Panels (d--e) show the respective evolutions of the exponent $\zeta(t)$. The exponents $\alpha$ and $\beta$ are obtained by collapsing $10$ momentum spectra, logarithmically spaced in time within $[t,10t]$, onto each other. As in Fig. \ref{['fig:rescale_occ_number']}, the exponents are extracted by fitting the scaling collapsed distributions within the momentum window $k\in[0,k_\text{max}(t)]$, where the initial cutoff $k_\text{max}(t_0)$ is fixed for the smallest and largest initial time and interpolated to intermediate initial times $t_0$ using a power law. The gray shadow displays the error obtained for the scaling exponents of the non-dipolar case; its magnitude is of similar order for the other settings in the ultradilute regime (not shown). For the strongly tilted case in (c) the error is indicated as a red shadow, which demonstrates its increase in the quantum regime. In (a) and (b) the extracted exponents are consistent with $\beta=1/5$ and $\alpha=d\beta$, with a trend to higher values at late times, which is more pronounced for randomly sampled initial vortex configurations, cf. the main text. For (c) we also find consistent exponents within errors, we observe however stronger trends in the variations of the exponents. The exponent $\zeta$ is extracted by fitting the scaling function \ref{['eq:scalingfunction']} to the momentum spectra. The UV cutoff is fixed both at the earliest and latest times and interpolated by a power law for intermediate times. The extracted non-dipolar exponents are consistent with $\zeta=5.7\pm0.3$ found in Karl2017b for most times. In general, a constant exponent $\zeta$ is found in all ultradilute scenarios until $t\approx 3\cdot10^5$ before a tendency of decreasing exponents indicates the transition of the scaling function towards the diffusion-like scaling, cf. the main text. In the quantum regime $\zeta$ is constant for most times, besides for the weakly-tilted $\theta=\pi/8$ case. The insets show the characteristic momentum scale, which exhibits power-law scaling $k_\Lambda(t)\sim t^{-1/5}$ in the universal regime of constant $\zeta$.
• Figure 4: In panels (a--c) the average inter-defect distance $\ell_\mathrm{v}$ and in (d--f) the vortex number $N_\mathrm{v}$ are shown for (a,d) lattice and (b,e) random sampling in the ultradilute regime at $\epsilon_\mathrm{dd}=1.5$ and (c,f) in the quantum regime for random sampling, all for different tilting angles. For comparison, the corresponding non-dipolar $\epsilon_\mathrm{dd}=0$ (gray) case is shown in all panels, respectively. The scaling in (a--c) with $\beta=1/5$ and in (d--f) with $-2\beta=-2/5$ of the anomalous non-thermal fixed point is indicated by the black-dashed lines and qualitatively follows the observed scaling at late times. In the insets the dynamically fitted scaling exponent $\beta$ is plotted where the shaded area in the insets indicates the error of the fitting procedure.
• Figure 5: (a) The average inter-defect distance for the non-dipolar ($\epsilon_\mathrm{dd}=0$) and the isotropic dipolar ($\epsilon_\mathrm{dd}=0.5$) case in the ultradilute regime starting from random vortex sampling of either $1000$ or $2400$ vortices. The evolution time is extended by an order of magnitude in time until almost no vortices are left in the system. Starting form $2400$ vortices, we observe the transition towards the diffusion-type scaling with $\beta\approx1/2$ at $t\approx10^6 (2\pi)/\omega_z$ which is shifted to $t\approx4\cdot10^6 (2\pi)/\omega_z$ for an initial vortex number $N_\mathrm{v,in}=1000$. The two black-dashed lines serve as a guide to the eye for power-law scaling with $\beta=1/5$ and $\beta=1/2$. (b) The corresponding average number of vortices is shown for the same scenarios and exhibits the same transition from sub-diffusive towards diffusive coarsening behavior. The black-dashed lines indicate the power-law scalings with exponents $-2/5$ and $-1$. The insets show the dynamically fitted scaling exponent $\beta$ extracted from (a) the average inter-defect distance and (b) the average vortex number. They exhibit a transition between scaling exponents of $\beta\approx0.2$ towards $\beta\approx0.5$ at late times.