False vacuum decay in long-range interacting quantum systems

We formulate false-vacuum decay in a mixed-field Ising chain with $1/r^\alpha$ interactions as a spatially nonlocal Euclidean $\phi^4$ theory featuring a fractional spatial kinetic term $\sim |q|^\sigma$, where $\sigma=\alpha-1$. The nonlocal bounce is anisotropic in space-time and develops algebraic spatial tails, challenging the standard thin-wall picture of a compact droplet. Combining thin-wall arguments with numerical solutions of the full nonlocal saddle, we show that these tails preserve the leading thin-wall exponents, manifesting instead in subleading corrections. For $0<\sigma<1$, the lifetime exponent scales with the energy bias $h$ of the metastable state as $B\sim h^{-1/\sigma}$; for $1<\sigma<2$, the leading Coleman scaling $B\sim h^{-1}$ is recovered, while long-range effects are retained in the subdominant term $\sim h^{\sigma-2}$. Our results show that tunable long-range interactions fundamentally reshape bubble nucleation and alter false-vacuum decay in quantum simulators.