Microscopic Dynamics of False Vacuum Decay in the $2+1$D Quantum Ising Model

TL;DR

The paper addresses false vacuum decay through bubble nucleation in the $2+1$D quantum Ising model after a global quench. It uses tree tensor networks (TTN) combined with the time-dependent variational principle (TDVP) to simulate real-time dynamics on a $16\times16$ lattice with transverse and longitudinal fields $h_{\perp}$ and $h_{\parallel}$. The main finding is that bubble fate—expansion or contraction—depends sensitively on geometry and microscopic parameters, with a detailed analysis based on bond perimeter $P_b$ and site perimeter $P_s$, and a proposal for realizing these dynamics in Rydberg-array quantum simulators. This work demonstrates nonperturbative real-time dynamics in $d>1$, provides a path toward experimental observation, and suggests extensions to lattice gauge theories and related quantum simulations.

Abstract

False vacuum decay, which is understood to happen through bubble nucleation, is a prominent phenomenon relevant to elementary particle physics and early-universe cosmology. Understanding its microscopic dynamics in higher spatial dimensions is currently a major challenge and research thrust. Recent advances in numerical techniques allow for the extraction of related signatures in tractable systems in two spatial dimensions over intermediate timescales. Here, we focus on the $2+1$D quantum Ising model, where a longitudinal field is used to energetically separate the two $\mathbb{Z}_2$ symmetry-broken ferromagnetic ground states, turning them into a ``true'' and ``false'' vacuum. Using tree tensor networks, we simulate the microscopic dynamics of a spin-down domain in a spin-up background after a homogeneous quench, with parameters chosen so that the domain corresponds to a bubble of the true vacuum in a false-vacuum background. Our study identifies how the ultimate fate of the bubble -- indefinite expansion or collapse -- depends on its geometrical features and on the microscopic parameters of the Ising Hamiltonian. We further provide a realistic quantum-simulation scheme, aimed at probing bubble dynamics on atomic Rydberg arrays.

Figures (8)

• Figure 1: Sketch of the dynamical phase diagram corresponding to a homogeneous quench, in the ordered phase, for an initial state $|\Psi(t=0)\rangle$ consisting of a domain of spin-downs (true vacuum) in a spin-up background (false vacuum). The quench is performed with the Ising Hamiltonian \ref{['eq:H_TLFI']} with parameters $h_{\perp}$ and $h_{\parallel}$. The transverse field $h_{\perp}$ determines the speed of the dynamics. Even at small $h_{\perp}$ shapes with the same perimeter can freely fluctuate on short timescales within the square in which they are inscribed, as there is no surface-energy cost to this. This is purely a lattice effect. The longitudinal field $h_{\parallel}$, on the other hand, sets the energy difference between the two vacua and favors bubble expansion.
• Figure 2: Average value of the magnetization over time after a quench where initial states consisting of square spin-down domains of side $L$ are evolved under the Hamiltonian \ref{['eq:H_TLFI']}. The plots clearly reveal a critical size of the bubble, which, for a fixed value of $h_{\perp}$, depends on $h_{\parallel}$. Above this critical size, the bubble expands until it occupies the whole system.
• Figure 3: Snapshots of the local magnetization on the $16 \times 16$ lattice at different times for $h_{\parallel}=-0.15$ (upper panel) and $h_{\parallel}=-0.05$ (lower panel), for an initial $8 \times 8$ square bubble. The change in the longitudinal field drastically affects the bubble dynamics, leading to vacuum decay in the first case and shrinking in the latter. When the bubble expands the time evolution produces diamond-like shapes, which are the square-lattice equivalent of circles.
• Figure 4: (a) Snapshots of the magnetization along the path shown in panel (c) for different times, $h_{\parallel}=-0.25$, and increasing values of the transverse field $h_{\perp}$. (b) Local magnetization at sites inside the square patch (green) and outside of it (blue) for increasing values of $h_{\perp}=0.5, 0.8, 1.1$ (lighter to darker shades). The results show that for moderate values of the transverse field, over intermediate timescales, the dynamics occurs entirely within the patch, in agreement with the perturbative analysis outlined in the main text. (c) Sketch of the system, showing the initial "true vacuum" bubble (blue) and the patch covering a domain with the same perimeter as the bubble and maximal area (light red).
• Figure 5: Data points corresponding to initial true vacuum domains of different shape and size, classified exclusively by their bond and site perimeters $P_\text{b}$ and $P_\text{s}$, for longitudinal fields $h_{\parallel}={-}0.05,\,{-}0.1,\,\text{and}\,{-}0.15$. Green markers correspond to expanding bubbles.