Instanton theory and fluctuation corrections to the thermal nucleation rate of a ferromagnetic superfluid

TL;DR

This work develops a field-theoretical framework for thermal nucleation in a 1D ferromagnetic superfluid, treating the nucleation of ground-state domains as diffusion- and fluctuation-driven dynamics around a critical instanton droplet. It computes both the exponential Arrhenius factor $e^{-eta\Delta E}$ and the fluctuation determinant via Gel'fand–Yaglom methods, yielding a parametric map of the nucleation rate across the phase diagram and a closed-form near the coexistence line. By reformulating domain formation as a Kramers escape problem for the droplet size, the authors connect instanton theory to a simple collective-coordinate description, clarifying two distinct time scales: nucleation and growth. They further extend the analysis to trapped gases using a local-density approach and discuss observable signatures in experiments, providing a rigorous basis for comparing nucleation theory with cold-atom measurements and outlining avenues for future generalizations.

Abstract

We provide a field-theoretical description of thermal nucleation in a one-dimensional ferromagnetic superfluid, a quantum-gas analogue of false-vacuum decay. The rate at which ground-state domains nucleate follows an Arrhenius law, with an exponential factor determined by a saddle-point of the energy functional -- the critical droplet or instanton -- and a magnitude fixed by small fluctuations about this configuration. We evaluate both contributions over the full parameter space, using a Gel'fand-Yaglom approach to reduce the calculation of the fluctuation spectrum to an initial-value problem. In addition, we obtain a closed-form expression for critical droplets close to the coexistence line, and use it to formulate an effective theory of domain nucleation and growth as a Kramers escape problem for the droplet size. Our results determine the parametric dependence of the nucleation rate and predict its signature in experimental images of a nucleating gas, increasing the rigor of comparisons between nucleation theory and experiment.

Figures (8)

• Figure 1: (a) Phase diagram for the mean-field potential $V[\theta]$ [Eq. \ref{['eq:V']}] as a function of dimensionless detuning $a$ and Rabi frequency $c$. The metastability region is shaded in blue, where the hatched portion indicates the parameter range explored in the experiment zenesini2024false. (b-e1) Potential $V[\theta]$ evaluated for the red parameter points in panel (a). They correspond to tilts increasing from almost-degenerate minima (e1), through decreasing potential barriers (d1,c1), until the metastable state vanishes (b). (c2-d2) Critical-droplet configurations associated with the potentials (c1-d1); there is no critical droplet associated with (b) as it lies outside the metastability region. The magnitude of the polarization field $\theta$ is represented as a gradient from red ($-\pi/2$) to blue ($\pi/2$).
• Figure 2: (a) Energy difference $\Delta E$ [Eq. \ref{['eq:deltaE_full']}] between the critical droplet and the metastable state. Dashed and dotted lines show the analytical small-barrier \ref{['eq:cubicE']} and small-tilt \ref{['eq:E_smalla']} approximations, respectively. Transparent vertical lines mark the end points $a_c$ of the metastability region [Eq. \ref{['eq:ac']}]. (b) Relative error of the small-tilt approximation compared to the exact result. Colored lines indicate the parameter range of the lines shown in panel (a).
• Figure 3: Energy of a kink-antikink pair \ref{['eq:kink_ansatz']} as a function of their separation $R$ for $c=0.5$, $a=0$ (blue line) and $a=0.05$ (orange line). Dashed lines show the respective asymptotic behaviors: $\Delta\mathcal{E}_\infty$ and $\Delta\mathcal{E}_\infty - \delta V \,R$. For $a>0$, the critical droplet size $R_c$ (black line) maximizes the energy \ref{['eq:E(R)_asy']}.
• Figure 4: Critical droplet as $a\to0$ (solid lines), obtained by numerically solving Eq. \ref{['eq:instanton_equation']} for $c=0.5$. Dashed lines show the corresponding kink-antikink pairs [Eq.~\ref{['eq:kink_ansatz']}] separated by a distance $R=R_c$ [Eq. \ref{['eq:Rcrit']}].
• Figure 5: (a) Fluctuation determinant $D(a,c)$ evaluated using the Gel'fand-Yaglom method [Eq. \ref{['eq:GY_det']}]. Transparent vertical lines mark the boundary of the metastability region \ref{['eq:ac']}. Dotted lines show the analytical small-tilt (small $a$) result [Eq. \ref{['eq:smallTiltD']}]. (b) Relative error between the analytical small-tilt result compared to the exact result.