False Vacuum Decay in Quantum Ising Models

• False vacuum decay in quantum Ising models is a quantum tunneling process where a metastable state transitions to a lower-energy true vacuum through bubble nucleation.
• Theoretical methods, including instanton and path integral techniques, quantify critical bubble formation and exponential scaling laws in decay rates.
• Recent advances in tensor network simulations and experimental quantum simulators have enabled direct observation and validation of non-perturbative decay dynamics.

False vacuum decay in quantum Ising models refers to the quantum-mechanical and many-body process in which a metastable state (the “false vacuum”) decays into a lower-energy, stable configuration (the “true vacuum”) via the nucleation and subsequent expansion of domains (or "bubbles") of the true vacuum. This process is of foundational significance in quantum field theory, cosmology, and condensed matter physics, serving as an analogue to first-order phase transitions dominated by quantum tunneling. In the Ising chain and related spin systems, the phenomena are accessible both to theoretical analysis and, increasingly, to direct quantum simulation and experiment, providing an unparalleled platform to paper non-perturbative tunneling, bubble nucleation, and out-of-equilibrium quantum dynamics.

1. Theoretical Foundations and Analogy to Field Theory

The false vacuum decay problem originates in quantum field theory, where it was rigorously formulated in terms of bubble nucleation in a metastable potential landscape (Coleman’s theory of instantons). In the context of quantum Ising models, a close correspondence exists:

• The prototypical Hamiltonian is the transverse-field Ising chain (TFIM) with explicit symmetry-breaking:

$$H = -J \sum_{j=1}^{N} S_j^z S_{j+1}^z - \Gamma \sum_{j=1}^{N} S_j^x - \frac{\epsilon}{2} \sum_{j=1}^{N} S_j^z$$

where $J$ encodes spin-spin interactions, $\Gamma$ acts as a quantum tunneling term, and $\epsilon$ sets the bias between the two nearly degenerate ferromagnetic ground states.

• The two degenerate “vacua” correspond to fully magnetized states, $| \uparrow \uparrow \cdots \uparrow \rangle$ and $| \downarrow \downarrow \cdots \downarrow \rangle$. Introducing a bias ($\epsilon$ or a longitudinal field $h_z$) renders one state globally stable (true vacuum) and the other metastable (false vacuum).
• The decay process is mapped to the nucleation and growth of a “droplet” (or bubble) of true vacuum inside the false vacuum matrix. The discrete spins play the role of a coarse-grained scalar field.
• Path integral formulations and Euclidean action techniques yield an effective droplet action,

$$\frac{S_E(\mathcal{D})}{\hbar} = \int_{\partial D} (2J_1 |dy| + 2J_2 |dx|) - \lambda \int_D dx\,dy$$

with $J_1, J_2$ encoding the effective interfacial tensions in imaginary time and space, and $\lambda$ representing the bulk energy bias (cf. (Moss, 14 Oct 2025)). Optimization leads to a critical droplet with a single negative mode associated with expansion, resulting in a nonzero decay rate via the imaginary part of the lowest energy eigenvalue.

• In one dimension, a fermionic reformulation via the Jordan–Wigner transformation enables direct calculation of tunneling amplitudes and nucleation rates—see, e.g., Rutkevich’s approach and subsequent extensions.

2. Quantum Decay Rates, Bubble Nucleation, and Scaling Laws

The nucleation rate of false vacuum decay in quantum Ising chains, under the thin-wall approximation or via instanton calculations, is governed by a non-perturbative exponential scaling. In canonical form for the 1D chain:

$$\Gamma_{\text{nuc}} = (π/9) \frac{\epsilon}{\hbar} N \exp\left( -\frac{q}{\epsilon} \right)$$

where $q$ is a function of magnetization and transverse coupling, as found in both semiclassical droplet (path integral) and exact fermionic treatments (Moss, 14 Oct 2025). This scaling is analogous to classic field theory results (e.g., Voloshin's for $\varphi^4$ QFT (Szász-Schagrin et al., 2022)) and is characterized by:

• An exponent determined by the balance of the domain wall tension (surface energy) and the energy gain of filling the bubble with true vacuum (bulk bias).
• The critical bubble (or resonant domain) has a typical size inversely proportional to the energy bias $\epsilon$ or $h_z$, which sets the tunneling action's saddle-point.

In higher dimensions (e.g., 2D Ising models), the conjectured form of the decay rate is

$$\Gamma_{\text{nuc}}\propto N_1 N_2 \frac{\epsilon^a}{\hbar} (J_x J_y)^{(1-a)/2} \left[\ln(J_x J_y / \Gamma^2)\right]^{-1/2} \exp\left\{ -\frac{128 J_x J_y}{\epsilon^2} \ln\left( b \sqrt{J_x J_y}/\Gamma \right) \right\}$$

with $a$, $b$ non-universal prefactors and $J_x, J_y$ the anisotropic couplings. This formula mirrors the $-\sigma^3/\epsilon^2$ structure for the bubble action (surface tension $\sigma$) seen in field theory (Moss, 14 Oct 2025).

The main scaling features shared across models:

Model/Regime	Decay Rate Scaling	Reference
1D Ising Chain (thin wall)	$\Gamma \propto (\epsilon/\hbar) N e^{-q/\epsilon}$	(Moss, 14 Oct 2025, Szász-Schagrin et al., 2022)
2D Quantum Ising (conjectured)	$$\Gamma \propto \exp\left(-[J/\epsilon^2]\ln(\cdots)\right)$$	(Moss, 14 Oct 2025)
Field theory ($\varphi^4$)	$$\gamma = (\mathcal{E}/2\pi) e^{-\pi M^2/\mathcal{E}}$$	(Szász-Schagrin et al., 2022)
Lattice/Quantum Simulator (experiment)	Exponential in $1/h_z$ or $1/\Delta_{\text{loc}}$	(Zenesini et al., 2023, Darbha et al., 18 Apr 2024)

3. Nucleation Dynamics and Bubble Expansion

The microscopic mechanism of decay proceeds via three dynamical regimes:

1. Nucleation Initiation: Quantum fluctuations seed small regions (bubbles) of true vacuum via virtual spin flips or coherent tunneling. In the Hamiltonian formalism, the longitudinal field $h_z$ (or staggered detuning) provides the "drive" for nucleation (Maki et al., 2023, Darbha et al., 18 Apr 2024).
2. Growth and Expansion: Having nucleated above a critical size, bubbles expand semi-classically, converting surface energy to kinetic energy for the domain walls. Analytically, this stage is governed by effective Schrödinger-type equations for the bubble wave function (either in coordinate or eigenstate space), capturing the real-time propagation (Maertens et al., 19 Aug 2025, Johansen et al., 19 Aug 2025).
3. Saturation or Bloch Oscillations: On the lattice, domain wall (bubble) expansion is bounded by max group velocity; Bloch oscillations halt indefinite bubble growth, causing oscillatory dynamics and temporary reversals (Pomponio et al., 2021, Maertens et al., 19 Aug 2025).

The role of the "resonant bubble" (with energy $E \sim 0$ in the effective domain wall model) is paramount: it dominates the decay rate and yields Fermi golden rule-like exponential decay within a given window of parameters (Maertens et al., 19 Aug 2025, Johansen et al., 19 Aug 2025).

4. Simulation, Tensor Network Methods, and Experimental Realizations

Recent advances allow both classical and quantum simulation of false vacuum decay in quantum Ising models:

• Matrix Product States (MPS), time-evolving block decimation (TEBD), and truncated conformal space approaches (TCSA) enable real-time and non-equilibrium studies in 1D (or quasi-1D) systems (Lencsés et al., 2022, Maki et al., 2023, Abel et al., 20 Jun 2025, Maertens et al., 19 Aug 2025).
• Large-scale quantum annealers with thousands of qubits have been used to probe bubble nucleation, resonant growth, and bubble–bubble interaction phenomena directly in real time (Vodeb et al., 20 Jun 2024). The observed quantized bubble spectra ($h_z = -2J/n$) and scaling laws (e.g., $h_x^k$ dependence, Landau–Zener collapse) provide direct experimental validation of theoretical predictions.
• Continuous-variable qumode tensor networks now allow simulation of scalar QFT tunneling with highly entangled real-time dynamics, highlighting the necessary bond dimension to capture non-perturbative bubble nucleation (Abel et al., 20 Jun 2025).
• Experimental atomic simulators using cold atoms, Rydberg arrays, or ferromagnetic superfluids have observed exponential decay of order parameters (e.g., the Néel state) attributable to nucleation and expansion of true-vacuum bubbles (Zenesini et al., 2023, Darbha et al., 18 Apr 2024).

5. Extensions, Generalizations, and Theoretical Frontiers

Several important extensions and speculative developments frame the ongoing paper of false vacuum decay in quantum Ising models:

• Higher Dimensions: Direct analytic treatment remains elusive. Conjectured decay rates based on saddle point/droplet theory indicate scaling in the exponential with $J_x J_y / \epsilon^2$ and logarithmic corrections (Moss, 14 Oct 2025). Recent tensor network simulations in 2D, notably using tree tensor networks (TTN), directly observe that energetic (magnon) collisions can trigger bubble nucleation and drive the violent decay of the false vacuum in a symmetry-broken regime (Pavešić et al., 2 Sep 2025).
• Coherent Dynamics and Superradiant Scaling: Under resonance conditions ($h \approx 2J/n$), coherent two-state oscillations between the false vacuum and a resonant single-bubble state can dominate, with the Rabi frequency exhibiting a $\sqrt{L}$ enhancement akin to superradiance. Bubble-size blockade ($n \gtrsim L/2$) and long-range interactions further stabilize these coherent oscillations against conventional many-bubble decay (Ge et al., 4 Sep 2025).
• Quantum Measurements and Decoherence: Continuous measurement protocols (e.g., local magnetization monitoring) accelerate initial decay by activating spin flips, but for strong measurement rates, the system enters a quantum Zeno regime suppressing bubble nucleation and eventually causing full thermalization to an infinite-temperature state (Maki et al., 2023).
• Field-Theoretic and Path Integral Connections: Advanced approaches using self-consistent Green’s function techniques provide a framework for evaluating quantum corrections to the decay rates, incorporating domain wall inhomogeneity and inhomogeneous fluctuation determinants (Garbrecht et al., 2017). Extensions to finite temperature reveal nontrivial interplay between shifted bounces and "shot" solutions, modifying the decay prefactor and action (Harada et al., 25 Oct 2024).
• Analytical Framework and Effective Theories: Multi-bubble dynamics and bubble–bubble interactions are encapsulated in effective bosonic theories ("coherent bubble theory") by second-quantizing the bubble eigenstates. Observable quantities such as magnetization, bubble density, and correlation fronts are directly expressed in terms of these bosonic modes, allowing for analytic characterization and comparison to matrix product state (MPS) numerics (Johansen et al., 19 Aug 2025).

Phenomenon / Regime	Key Dynamical Feature	Supporting References
Exponential false vacuum decay (1D)	Fermi golden rule regime (expandable bubbles)	(Lagnese et al., 2021, Szász-Schagrin et al., 2022, Johansen et al., 19 Aug 2025)
Coherent oscillations (resonant tunneling)	Two-state Rabi-like dynamics, $\sqrt{L}$ scaling	(Ge et al., 4 Sep 2025)
Bubble growth and Bloch oscillations	Lattice-limited bubble expansion, oscillatory	(Pomponio et al., 2021, Maertens et al., 19 Aug 2025)
Nucleation under measurement/decoherence	Accelerated or suppressed decay via Zeno effect	(Maki et al., 2023)
Nucleation in 2D and scattering-induced decay	Ballistic expansion of true vacuum bubble	(Pavešić et al., 2 Sep 2025)
Experimental bubble nucleation	Exponential decay of observable, bubble imaging	(Zenesini et al., 2023, Darbha et al., 18 Apr 2024)

6. Outlook, Open Questions, and Implications

The paper of false vacuum decay in quantum Ising models continues to illuminate both foundational quantum physics and practical experimental frontiers.

• Validation of Field-Theoretic Predictions: Discrete spin models—via real-time numerics, quantum simulation, and cold-atom experiments—now robustly reproduce the exponential scaling of tunneling rates, bubble nucleation, and associated correlation spreading that underpin cosmological and particle physics analogues.
• Universal versus Non-universal Behavior: Although the exponential dependence and critical droplet structure are well established, the prefactor (“model-dependent constant” $\mathcal{C}$ in nucleation rates) remains non-universal and is a focus for further numerical and analytical refinement (Lencsés et al., 2022).
• Extensions to Other Symmetry Classes and Degeneracy Structures: Studies involving tricritical Ising theories (Lencsés et al., 2022) and antiferromagnetic (staggered) spin models (Darbha et al., 18 Apr 2024) reveal new decay channels, intermediate metastable plateaus, and discrete "cascading" transitions tied to richer vacuum structures.
• Non-Perturbative Dynamics and Many-Body Coherence: The existence of long-lived coherent oscillations and effective two-state regimes in finite and large systems suggests a broader spectrum of dynamical possibilities—transcending standard tunneling decay.
• Experimental Realizability and Control: State-of-the-art quantum hardware (annealers, Rydberg platforms, cold atom arrays) allow for direct preparation, quenching, and measurement in regimes where non-perturbative decay occurs, enabling direct tests of theoretical conjectures.

By reformulating field-theoretical and cosmological tunneling dynamics in the precise, experimentally tractable framework of quantum Ising models, these studies provide a fertile ground for advancing both fundamental understanding and experimental quantum many-body physics.