Super-heated first order phase transitions

Abstract

We study first order phase transitions that occur when the temperature of the system increases and we identify the conditions that lead to super-heating, a phase where the system can heat up arbitrarily. First order phase transitions with super-heating behave as inverse transitions. We quantify these claims by studying a prototypical example of a dark sector with a large number of interacting light bosons at finite temperature. Depending upon thermalisation, a super-heated phase transition in cosmology is often associated with another transition when the system is eventually cooling down, enriching the spectrum of gravitational waves from bubble collisions.

Figures (7)

• Figure 1: Finite--temperature effective potential $V(\phi,T)$, normalised to $|V(\phi_\ast,T_0)|$, shown as a function of $\phi/\phi_\ast(T_0)$ at the characteristic temperatures $T_0$ (blue), $T_c$ (black dashed), and $T_{\rm SP}$ (red) (see sec. \ref{['sec:tree-level']}). The plot illustrates the presence of a metastable minimum away from the origin, its degeneracy with the symmetric phase at $T_c$, and its disappearance at the spinodal temperature $T_{\rm SP}$. The orange and green curves correspond to intermediate nucleation temperatures, $T_n^\leftarrow$ and $T_n^\rightarrow$ respectively (see sec. \ref{['sec:nucleation']}), while arrows indicate the inverse (orange) and direct (green) transition paths. Dotted light red curves denote the potential for higher temperatures in the case when superheating does not last forever.
• Figure 2: Numerical sample of $B(\kappa)$ for a 3d bounce solution, with boundary conditions as in eq. \ref{['eq:B']}.
• Figure 3: Bounce action $S_{3}/T$ as a function of $a_{2}$ and $a_{3}$, with $a_{4}=0.1$. Only the scale-invariant region in which the potential admits a metastable vacuum is shown. The solid red line denotes where the bounce action vanishes. The blue‑shaded area indicates the domain in which, upon adding an instability term at the origin, the spinodal temperature, eq. \ref{['eq:T spinodal']}, is well defined. The grey‑shaded region corresponds to $\phi_{b}<\pi T$, where resummation is important.
• Figure 4: Bounce actions for heating (solid) and cooling (dashed) phase transitions, colour-coded by critical temperature $T_c/T_0$. Each solid curve at a given colour corresponds to its matching dashed curve upon cooling. Solid lines stop at the spinodal temperature, eq. \ref{['eq:T spinodal']}, when it exists, while dashed ones stop at $T_0$. The colorbar shows $T_c/T_0$. We highlight that the red‐shaded region ($T_c/T_0>3$), above the black dotted line, marks the maximal superheating phase where the barrier persists to arbitrarily high $T$ and it is disjoined from the rest of the parameter space. In this phase, the action, at large temperature, converges to the scale‐invariant value.
• Figure 5: Illustration of the model with two phase transitions: while heating (top row) and cooling (bottom row). Heating is represented in red, while cooling is in blue. Left column: Schematic trajectories of the order‐parameter value. Solid curves show the actual evolution, arrows on the lines indicate the direction of time, and stars mark the nucleation temperatures for the inverse ($\leftarrow$) and direct ($\rightarrow$) transitions. Dashed extensions depict the hypothetical path if nucleation did not occur. The red shaded area at the bottom indicates the thermalisation of the system below $T_0$. Right column: Corresponding bounce action $S_3/T$ (black curve) with the nucleation criterion from Eq. \ref{['eq:nucleation']} overlaid (coloured lines for different Hubble rates). Solid segments represent the realised evolution. In both heating and cooling cycles, once the inverse transition occurs, the direct transition inevitably follows.