Microscopic Description of Critical Bubbles

TL;DR

This paper presents a fully microscopic holographic description of critical bubbles in a strongly coupled 4D gauge theory at finite temperature by constructing static, inhomogeneous, unstable black-brane solutions that are dual to $O(3)$-symmetric bubbles. It computes the bubble profile, surface tension $\sigma$, and nucleation rate $\mathcal{P}(T)$ across the metastable branch and demonstrates remarkable agreement with a two-derivative effective action for the order parameter when the EFT is derived from the microscopic theory, while large discrepancies arise when the EFT is constrained only by the equation of state and dimensional analysis; these discrepancies can be resolved by imposing the correct surface-tension constraint, highlighting the critical role of $\sigma$ in EFT validity. The study reveals a substantial suppression of the kinetic term relative to naive estimates and identifies two distinct regimes for bubble profiles: near $T_c$ the wall is well described by a hyperbolic tangent with a finite thickness, and near $T_0$ the profile becomes Gaussian, with a nucleation-action scaling $\Delta F\propto(T-T_c)^{-2}$ near the critical temperature and $\Delta F\propto(T-T_0)^{0.75}$ near the spinodal. These results establish a controlled framework for testing EFTs of FOPTs in strongly coupled systems and provide benchmarks for lattice and phenomenological models, with potential implications for cosmology and high-density QCD phenomena.

Abstract

First-order phase transitions occur through the nucleation of critical bubbles of the stable phase within the metastable phase. Using holography, we present a fully microscopic description of these bubbles in a strongly coupled, four-dimensional gauge theory at finite temperature. In the gravitational dual, these bubbles correspond to static, inhomogeneous and unstable black-brane solutions with a localized deformation on the horizon. We construct these solutions across the entire metastable branch and compute the surface tension and the nucleation rate. We then compare these microscopic results with those obtained from a two-derivative effective action for the order parameter in two different scenarios. When the effective action is derived from the microscopic theory via holography, we find remarkable agreement. However, when the effective action is constrained only by the equation of state and dimensional analysis, significant discrepancies emerge. These discrepancies can be resolved if an additional constraint related to the surface tension is imposed.

Figures (16)

• Figure 1: A critical bubble (top part of the figure) is dual to an inhomogeneous static black brane geometry (bottom part).
• Figure 2: Energy density as a function of temperature for the theory with $\lambda_4 = -1/4$, $\lambda_6 = 1/10$. The thick solid curves correspond to the two branches of stable configurations. The two vertical, thin, dashed black lines indicate the location of the turning point at $T = T_0$, where the spinodal branch begins, and the first-order phase transition at $T = T_{\text{c}}$. The two dashed curves that extend between the critical temperature and the corresponding turning point are metastable branches. They are connected by the spinodal branch (intermediate dotted curve), where the system is both thermodynamically and dynamically unstable. The gray solid curve that interpolates between $T_0$ and $T_{\text{c}}$ indicates the energy density at the center of the critical bubbles. At the top, we present the density plot for the relative energy density of four representative bubble solutions, indicating the value of their relative temperature difference $\mathcal{T}$, defined in Eq. \ref{['eq:rel_temperature_difference']}. Their energy profiles expand from the metastable branch (dots) to the corresponding value at the center of the bubble (squares), as indicated by the arrows. In the legend, $e_+$ and $e_-$ refer to the energy densities in the metastable, high-energy phase and in the stable, low-energy phase, respectively.
• Figure 3: Free energy density defined through (\ref{['freedensity']}) (top) and pressure (bottom) of the homogeneous solutions as functions of the temperature. The conventions are the same as in Fig. \ref{['fig:phase_diagram']}, with the thickest curves representing homogeneous, stable states. The critical temperature, marked by a thick black dot, is the point where these curves cross each other. For the same four bubbles shown in Fig. \ref{['fig:phase_diagram']}, we have drawn bars indicating the range probed by each bubble (see also Fig. \ref{['fig:results_bubbles']}), extending from the origin (marked by a square) to infinity (on the metastable dashed branch). Close to $T_c$, the state at the bubble center lies near the stable homogeneous state, although the bubble wall itself remains non-trivial. Note that $p=-f$ for homogeneous states but not for inhomogeneous ones, as expected on general grounds.
• Figure 4: (Top) Energy density of a bubble as a function of the radial direction, normalized to the difference between the metastable ($+$) and stable ($-$) branches, for the four different choices of $\mathcal{T}$ indicated on each column, where $\mathcal{T}$ is defined in Eq. \ref{['eq:rel_temperature_difference']}. Near $T_0$ (left, $\mathcal{T} = 0$) the profile of the energy density is Gaussian and its characteristic size grows as $T_0$ is approached. On the other side, near $T_c$ (right, $\mathcal{T} = 1$), the profile becomes closer and closer to a hyperbolic tangent. The profiles of the entropy density and the expectation value of $\mathcal{O}$ exhibit qualitatively similar behavior. (Middle) Transverse (solid curves) and longitudinal (dashed curves) pressures as a function of the radial direction for the same values of the temperature. These coincide at the origin, where isotropicity is restored, and at infinity, where the asymptotic, metastable, homogeneous state is recovered. (Bottom) Free energy density difference, $\Delta f=f(\rho)-f_+$, as a function of the radial direction for the same values of the temperature. These are the profiles that we integrate to obtain the bubble action.
• Figure 5: Best Gaussian (thin, solid) and hyperbolic tangent (dashed, thick) fits on top of the energy density profile from Fig. \ref{['fig:results_bubbles']} (top). The hyperbolic tangent appears to approximate the profiles well in all the range of temperatures, but it fails to give the correct Newman boundary condition at $\rho = 0$, specially close to $T_0$. The Gaussian profile Eq. \ref{['eq:Gaussian_profile']}, in contrast, has a minimum at the origin, but fails to reproduce the profile away from $T_0$.