False vacuum decay in the ($1+1$)-dimensional ${\phi }^4$ theory

The false vacuum is a metastable state that can occur in quantum field theory, and its decay was first studied semiclassically by Coleman. In this work, we consider the $1+1$ dimensional ${\phi }^4$ theory, which is the simplest model that provides a realization of this problem. We realize the decay as a quantum quench and study the subsequent evolution using a truncated Hamiltonian approach. In the thin wall limit, the decay rate can be described in terms of the mass of the kink interpolating between the vacua in the degenerate limit and the energy density difference between the false and true vacuum once the degeneracy is lifted by a symmetry breaking field, also known as the latent heat. We demonstrate that the numerical simulations agree well with the theoretical prediction for several values of the coupling in a range of the value of the latent heat, apart from a normalization factor, which only depends on the interaction strength.

Figure 1

The kink masses as obtained via THA (markers) together with the semiclassical prediction. The different colors denote different values of the cutoff $n_{\max }=21/2,\dots 37/2$ corresponding to cc. 1000–100000 basis states.

Figure 2

The spectrum of $H_{ϵ}$ for $l=8$ and $\overline{g}=1.1$ as a representative spectrum of the asymmetric theory, with the true vacuum line subtracted. The false vacuum line is denoted by open black markers.

Figure 3

The energy difference between the false and true vacuum as a function of the volume with the linear dependence fitted for some values of the symmetry breaking field $ϵ$ for $\overline{g}=1.1$. The slope of the fitted linear curve is the energy density difference (latent heat) $\mathcal{E}$ for a given $\overline{ϵ}$.

Figure 4

The latent heat $\mathcal{E}$ as a function of $\overline{ϵ}$ for various couplings $\overline{g}$ with the fitted linear dependence. The error bars represent a crude estimate obtained from the uncertainty of the parameter estimation from the fits shown in Fig. 3.

Figure 5

Time evolution of $F(t)$ for $\overline{g}=1.1$ and different values of $ϵ$. The different colors denote different values of the cutoff. Here, results corresponding to $n_{\max }=10$, 12, 14, 16, and 18 are presented, corresponding to cc. 4100–287000 states.

Figure 6

The logarithm of $F(t)$ for $l=20$ and $\overline{g}=1.1$ obtained with the largest cutoff $n_{\max }=18$, plotted together with the linear fits to the apparent tunneling regime. The slope of the linear fits gives the tunneling rate $\mathrm{\Gamma }$.

Figure 7

The logarithm of $\gamma$ obtained from THA (with $n_{\max }=18$) for various couplings in $l=20$ as a function of $1/\overline{ϵ}$ together with the theoretical predictions (8) (dashed lines). The vertical lines correspond to the values of the symmetry breaking field values where the resonant bubble size reaches $a_{*}=l/2$, demonstrating that the difference between the theoretical and numerical results for small values of $\overline{ϵ}$ originate from finite size effects. The error bars represent a crude estimate obtained from the uncertainty of the parameter estimation from the fits, a representative sample of which is shown in Fig. 6.

Figure 8

The (logarithm of the) nucleation rate $\gamma$ obtained from THA for different volumes $l$ at interaction $\overline{g}=1.1$ as a function of $1/\overline{ϵ}$ together with the theoretical predictions (33). The inset shows the fitted values of $\log C$ for different values of the volume, where the (barely visible) error bars show the uncertainty of the parameter estimation from the fits.

Figure 9

THA results (with $n_{\max }=18$) for the logarithm of the prefactor $C$ plotted as a function of the inverse of the interaction strength $1/\overline{g}$ together with a linear fit. The numerical evaluation was performed in volume $l=20$.