Non-perturbative False Vacuum Decay Using Lattice Monte Carlo in Imaginary Time

TL;DR

This work develops a non-perturbative lattice Monte Carlo framework to compute false vacuum decay rates from imaginary-time data. Central to the approach is the Implicit Decay Amplitude Method, which expresses the real-time decay rate $oxed{Γ}$ in terms of a spectral quantity, encoded via a Euclidean observable $Q(t)$ and a spectral function $ ho(E)$, with $oxed{Γ}$ related to $oxed{ρ(E_{FV})}$ through a relation analogous to Fermi's Golden Rule. To extract $ ho(E)$ from noisy Euclidean data, the authors adopt a Gaussian ansatz and perform spectral reconstruction, while mitigating ergodicity and signal-suppression challenges using the Intermediate Ratios Method and multiple ensembles. The method is demonstrated in a 1D quantum system, where lattice Monte Carlo results match exact Schrödinger solutions within a factor of two, highlighting that spectral reconstruction is the dominant systematic error and suggesting clear paths for improvement. The framework offers a fully non-perturbative route to false vacuum decay, with potential extensions to field theories and finite-temperature settings, providing a pathway to quantify extremely small decay rates beyond semiclassical approximations.

Abstract

We present a new method for calculating quantum tunneling rates using lattice Monte Carlo simulations in imaginary time. This method is designed with the goal of studying false vacuum decay non-perturbatively on the lattice. To get results in real time, we construct an implicit decay amplitude, inspired by Fermi's Golden Rule, and use spectral reconstruction. To deal with the suppression of the false vacuum state in the Euclidean path integral, we develop a new sampling method which combines results from multiple Monte Carlo simulations. For a simple family of one-dimensional quantum systems, we reproduce the tunneling rates calculated from the Schrodinger equation.

Figures (7)

• Figure 1: The potential of the field theory in Equation \ref{['eq_example_Lagrangian']} (for a constant field $\phi(x)=\phi$) is shown as a function of $\phi$. The regions associated with the false vacuum, the true vacuum, and the barrier between them are labeled.
• Figure 2: The potential from Equation \ref{['eq_action']} for a one-dimensional, single-particle quantum system is shown. We define a state $|\text{FV}\rangle$, which we will call the "false vacuum." The energy of this state $E_\text{FV}$ is marked with a horizontal line, and the wavefunction for $|\text{FV}\rangle$ is plotted on top of this line (in other words, we plot $f(x)\equiv N\langle x|\text{FV}\rangle+E_\text{FV}$, where $N$ is an arbitrary normalization).
• Figure 3: The false vacuum potential for the Hamiltonian from Equation \ref{['eq_H_FV']} is shown, along with its ground state $|\text{FV}\rangle$. The energy of this state $E_\text{FV}$ is marked with a horizontal line, and the wavefunction for $|\text{FV}\rangle$ is plotted on top of this line (in other words, we plot $f(x)\equiv N\langle x|\text{FV}\rangle+E_\text{FV}$, where $N$ is an arbitrary normalization).
• Figure 4: The exact decay rates, decay rates from the implicit decay amplitude method (Eq. \ref{['eq_FGR']}, "Fermi's Golden Rule"), and the decay rates calculated from lattice Monte Carlo are plotted for various choices of the parameters $\alpha$ and $\beta$. The errors shown are statistical only. The exact decay rates and the "Fermi's Golden Rule" decay rates are determined by numerically solving the Schödinger equation, as explained in Section \ref{['section_results_decay_rates']}. The difference between the Monte Carlo and the "Fermi's Golden Rule" results is due to error in the spectral reconstruction. Note that the second plot uses a log scale for the decay rate while the first does not. The simulation parameters used to obtain these results are shown in Tables \ref{['tab_sim_params']} and \ref{['tab_sim_results']}.
• Figure 5: The energy spectrum $\rho(E)$ (defined in Equation \ref{['eq_rho']}) of the time-evolved difference state $e^{-H_\text{TV}t}|D\rangle = e^{-H_\text{TV}t}e^{-aH_\text{proj}}|\text{FV}\rangle$ is plotted at various times $t$ ($H_\text{proj}$ is defined in Section \ref{['section_lattice_obs']}). Here we set $\alpha=0.8$ and $\beta=20.0$ in the Lagrangian. Other parameters are the same as those given in Table \ref{['tab_sim_params']}. Since we want to fit the spectrum to a Gaussian distribution and get $\rho(E_\text{FV})$, our results will be most accurate when $t$ is long enough to suppress higher-energy components but short enough that $\rho(E_\text{FV})$ is still large.