Notes on false vacuum decay in quantum Ising models

TL;DR

This work adapts Coleman’s false vacuum decay framework to quantum Ising models, using spin chains as concrete analogue systems to explore nonperturbative decay via droplet nucleation. It develops a 1D treatment through imaginary-time path integrals, yielding the nucleation rate $Γ_{nuc}$ and the critical-droplet action, and then leverages exact 2D Ising results to obtain a finite, elliptic-integral expression for the nucleation exponent in the 2D regime. A key contribution is the explicit connection between droplet geometry, action, and decay rate, including a highly speculative conjecture for the 2D rate with a cuboid droplet and a parametrized pre-factor. The paper also discusses alternative field-theoretic and non-relativistic formulations, highlighting technical challenges in discretisation and the continuum limit, and closes with implications for experiments in quantum simulators and potential insights into early-universe phase transitions.

Abstract

This paper aims to gather together some of the basic ideas behind the theory of false vacuum decay in quantum Ising models, focusing on the application of spin chains as analogue systems to false vacuum decay in elementary particle theory. Elementary results on quantum Ising models are rewritten to more closely resemble the original literature on false vacuum decay. A highly speculative conjecture for the false vacuum decay rate in a two dimensional quantum Ising model is also put forward.

Figures (4)

• Figure 1: (a) Rectangular and (b) ellipsoidal droplets with filled circles representing spin up. The ellipsoidal contribution to the action for lengths of the axes that make the action stationary is larger than the rectangular contribution by $4/\pi$.
• Figure 2: Two adjacent rows of spins. The three possibilities, where the edge of the droplet has $dy=1$ and (a) $dx=-1$, (b) $dx=0$ and (c) $dx=1$, all contribute $2J_1|dy|+2J_2|dx|$ to the action.
• Figure 3: The critical droplet for a one dimensional quantum spin chain is shown for various values of $J/\Gamma=2,5,10,20$ moving outwards form the centre, where $J$ is the magnetisation and $\Gamma$ the transverse coupling. The shape becomes more "rectangular" as $J/\Gamma$ increases.
• Figure 4: The nucleation exponent $\Gamma_E$ for the one dimensional quantum spin chain (upper curve) and the logarithmic approximation (lower curve), plotted against $J/\Gamma$, where $J$ is the magnetisation and $\Gamma$ the transverse coupling.