[Paper Review] Bubble nucleation at zero and nonzero temperatures

Mario Gutierrez Abed, Ian G. Moss|arXiv (Cornell University)|Jun 11, 2020

Cosmology and Gravitation Theories4 citations

TL;DR

This paper presents a novel numerical method to model thermal and quantum bubble nucleation in first-order phase transitions, particularly in the crossover regime between thermal and vacuum-dominated decay. It demonstrates that non-static instantons smoothly merge into quasi-static solutions as temperature decreases, with precise action calculations in 1–3 spatial dimensions using a spinor BEC model with tunable parameters.

ABSTRACT

The theory of false vacuum decay in a thermal system may have a cross-over from predominantly thermal transitions to quantum transitions as the temperature is decreased. New numerical methods and results are presented here that can be used to model thermal and vacuum bubble nucleation in this regime for cosmology and for laboratory analogues of early universe phase transitions.

Motivation & Objective

• To develop a robust numerical method for calculating bubble nucleation rates in the crossover regime between thermal and quantum transitions, where traditional shooting methods fail.
• To model bubble nucleation in systems with both thermal and vacuum contributions, relevant for early universe phase transitions and laboratory analogues like spinor Bose-Einstein condensates.
• To determine the behavior of instanton solutions—specifically their shape, action, and transition to quasi-static forms—as temperature varies from zero to finite values.
• To provide accurate, numerically computed nucleation exponents and action values for non-static and quasi-static instantons across 1–3 spatial dimensions.
• To express results in natural physical units adaptable to experimental systems, such as spinor BECs, enabling direct comparison with future laboratory experiments.

Proposed method

• The method solves the Euclidean field equation for instantons in imaginary time, using a finite-difference scheme to handle the full non-linear dynamics of the scalar field φ in n spatial dimensions.
• It employs periodic boundary conditions in imaginary time (τ) to model thermal effects, with period β = 1/T, and allows for non-uniform (distorted) instanton profiles in the τ-direction.
• The nucleation rate is computed via the Euclidean action SE[φb], with functional determinants evaluated numerically, excluding zero modes from translational symmetry breaking.
• The approach is validated by reproducing the analytic one-dimensional quasi-static instanton solution and extended to higher dimensions using numerical fitting for αn(λ).
• The model uses a double-well potential V(φ) = -(1 + cosφ) + ½λ²sin²φ, with φ = 0 as true vacuum and φ = π as false vacuum, and scales all variables to natural units based on healing length and frequency.
• The method is applied to compute SE for non-static and quasi-static instantons across temperatures, with results expressed as SE = χαn(λ)β for thermal cases and SE = χαn+1(λ) for vacuum cases.

Experimental results

Research questions

• RQ1 How do instanton solutions evolve from non-static (thermal-dominated) to quasi-static (vacuum-dominated) forms as temperature decreases?
• RQ2 What is the behavior of the Euclidean action for bubble nucleation in the crossover region between thermal and quantum tunneling?
• RQ3 Can a new numerical method accurately compute nucleation exponents in regimes where shooting methods fail, especially near the crossover point?
• RQ4 How do the nucleation rates and instanton profiles depend on spatial dimensionality (n = 1, 2, 3) and potential barrier height (λ)?
• RQ5 To what extent do the results in natural units from the spinor BEC model match predictions for cosmological phase transitions or other analog systems?

Key findings

• Non-static instantons, which dominate at higher temperatures, become increasingly distorted in the imaginary time direction as temperature decreases, with the distortion peaking around T ≈ 0.125 in the one-dimensional case.
• At T ≈ 0.125, the non-static instanton solution merges smoothly into the quasi-static instanton, indicating a continuous crossover without discontinuity in action or profile.
• The Euclidean action for the non-static instanton is lower than that of the quasi-static instanton at low temperatures, confirming that quantum tunneling dominates in the vacuum regime.
• In one dimension, the numerical results for αn(λ) agree excellently with the analytic expression derived from the thin-wall approximation, validating the method.
• In two and three dimensions, the functional form of αn(λ) deviates significantly from the thin-wall approximation, requiring numerical fitting with αn(λ) ≈ a1(λ−1) + a2(λ−1)², with (a1, a2) = (24.0, −3.0) in 2D and (180, 30.0) in 3D.
• The method enables precise computation of nucleation rates in systems with both thermal and quantum contributions, providing a foundation for testing early universe phase transition models in laboratory BEC experiments.