Tachyonic Instability and Dynamics of Spontaneous Symmetry Breaking

TL;DR

This study reveals that spontaneous symmetry breaking driven by tachyonic (spinodal) instability rapidly transfers the initial potential energy into inhomogeneous classical scalar waves, often completing the process in a single oscillation. Using lattice simulations across a spectrum of potentials, the authors show that long-wavelength fluctuations grow explosively, form domains or topological defects, and drive nonperturbative dynamics that standard perturbative methods fail to capture. The work highlights the central role of the potential’s shape near its maximum and demonstrates that tachyonic preheating is a robust mechanism with broad cosmological implications for reheating in various inflationary and brane-inspired scenarios. These findings emphasize the necessity of nonperturbative, lattice-based approaches to accurately describe symmetry breaking and early-universe dynamics.

Abstract

Spontaneous symmetry breaking usually occurs due to the tachyonic (spinodal) instability of a scalar field near the top of its effective potential at $φ= 0$. Naively, one might expect the field $φ$ to fall from the top of the effective potential and then experience a long stage of oscillations with amplitude O(v) near the minimum of the effective potential at $φ= v$ until it gives its energy to particles produced during these oscillations. However, it was recently found that the tachyonic instability rapidly converts most of the potential energy V(0) into the energy of colliding classical waves of the scalar field. This conversion, which was called "tachyonic preheating," is so efficient that symmetry breaking typically completes within a single oscillation of the field distribution as it rolls towards the minimum of its effective potential. In this paper we give a detailed description of tachyonic preheating and show that the dynamics of this process crucially depend on the shape of the effective potential near its maximum. In the simplest models where $V(φ) \sim -m^2φ^2$ near the maximum, the process occurs solely due to the tachyonic instability, whereas in the theories $-λφ^n$ with n > 2 one encounters a combination of the effects of tunneling, tachyonic instability and bubble wall collisions.

Figures (16)

• Figure 1: Evolution of the occupation numbers for the fluctuations with $k \ll m$ in the model $V(\phi)= - {m^2\over 2}\phi^2 + {\lambda\over 4} \phi^4 + {m^4\over 4\lambda}$
• Figure 2: Same as in Fig 1 for $k = 0.5 m$.
• Figure 3: The process of symmetry breaking in the model (\ref{['aB1']}) for $\lambda = 10^{-4}$. The values of the field are shown in units of $v$, time is shown in units $m^{-1}$. For each moment of time, we also show the occupation numbers $n_k$ (the lower part of each panel), with $k$ measured in units of $m$. At $t = 0$ one has $n_k = 0$, as in the usual quantum field theory vacuum. In the beginning of the process the occupation numbers $n_k$ grow exponentially for $k < m$ ($k < 1$ in the figure), but then this growth spreads to $k > m$ because of domain wall formation and collisions of classical waves of the field $\phi$. Within a single oscillation the occupation numbers for $k \ll m$ grow up to $\sim 10^6$, which is in complete agreement with our estimate $n_k \sim 10^2 \lambda^{-1}$, Eq. (\ref{['occlam']}). The spectrum rapidly stabilizes, but it is not thermal yet, and the occupation numbers remain extremely large. Thermalization takes much more time than spontaneous symmetry breaking.
• Figure 4: Tachyonic growth of quantum fluctuations and the early stages of domain formation in the simplest theory of spontaneous symmetry breaking with $V(\phi)= - {m^2\over 2}\phi^2 + {\lambda\over 4} \phi^4$.
• Figure 5: Formation of domains in the process of symmetry breaking in the model (\ref{['aB1']}).