Lagnese, Surace, Morampudi, and Wilczek (Phys. Rev. Lett. 133, 240402 (2024)) demonstrated that in the tilted quantum Ising chain a simple global quench reveals, on time scales many orders of magnitude shorter than the decay time, whether the prepared state is a true vacuum or a long-lived false vacuum: the magnitude spectrum of the magnetization differs qualitatively between the two cases, and the exponentially small decay rate can be extrapolated from the amplitudes of subcritical-bubble oscillations. These results were established within a two-fermion truncation and a non-relativistic approximation of the scaling Ising field theory, in one spatial dimension, for unstructured quenches and idealized closed-system dynamics. The open problem is to determine whether and how the diagnostic and the real-time rate extraction survive outside that regime: at small scaling parameter η=m∣h∣−8/15\eta=m|h|^{-8/15} where multiparticle effects are essential; in two and three spatial dimensions where bubble boundaries carry internal shape degrees of freedom; under structured bubble-seeding quenches; on the other experimental platforms where quench experiments are routinely performed; and under cosmological expansion effects. The problem remains open.

Background

In many physical systems the lowest-energy configuration is not the only locally stable one: a metastable "false vacuum" is a local minimum of the energy landscape separated from the true ground state (the "true vacuum") by an energy barrier. In quantum field theory such a state decays by tunneling: fluctuations nucleate a small "bubble" of true vacuum inside the false vacuum, and a bubble larger than a critical size expands because its volume energy gain outweighs its wall cost. Coleman's semiclassical theory expresses the decay rate through imaginary-time ("bounce" or instanton) solutions, and the rate is exponentially small in the inverse of the energy-density bias between the vacua; when the bias is small the false-vacuum lifetime enormously exceeds any accessible observation time, so direct observation of the decay is essentially impossible.

Because of this inaccessibility there is strong interest in quantum simulators - controllable many-body systems that realize the same physics on accessible scales. A convenient setting is the tilted (mixed-field) quantum Ising chain, H=−∑i(σizσi+1z−hzσiz−hxσix)H=-\sum_i(\sigma_i^z\sigma_{i+1}^z-h_z\sigma_i^z-h_x\sigma_i^x): for hz=0h_z=0 and 0<hx<10<h_x<1 it has two degenerate ferromagnetic ground states, and a small longitudinal field hzh_z lifts the degeneracy so that one state becomes the true vacuum and the other a long-lived false vacuum. The elementary excitations of the hz=0h_z=0 chain are free fermions (kinks, i.e. domain walls between the two magnetized states) obtained by a Jordan-Wigner transformation (a standard exact mapping of a spin chain to fermions); a nonzero hzh_z exerts a linear confining potential that binds kink-antikink pairs into discrete bound states ("mesons"). Real-time confinement after quenches in such chains was established by Kormos, Collura, Takács, and Calabrese (2017); the false-vacuum decay of this model was analyzed by Lagnese, Surace, Kormos, and Calabrese (2021), with the analytic decay rate due to Rutkevich (1999).

Lagnese, Surace, Morampudi, and Wilczek (2024) asked whether the true and false vacua can be distinguished on time scales far shorter than the decay time. Their protocol is a quantum quench, i.e. a sudden change of a Hamiltonian parameter: prepare one of the degenerate ferromagnetic ground states at hz=0h_z=0, then evolve with a small hzh_z of either sign, so the prepared state is either the new ground state (true vacuum) or a metastable state (false vacuum). They track the spatially averaged magnetization ⟨Sz(t)⟩\langle S^z(t)\rangle and its magnitude spectrum ∣Sz(ω)∣|S^z(\omega)|, the modulus of the finite-time Fourier transform, computed both with iTEBD (infinite time-evolving block decimation, a tensor-network method for infinite one-dimensional chains) and analytically. The analytic treatment projects the dynamics onto the zero- and two-fermion sectors - a single meson is a kink-antikink pair, i.e. a small bubble of flipped magnetization - and treats the coupling to higher fermion-number sectors perturbatively. The results are: (i) the magnitude spectra of true- and false-vacuum evolutions differ qualitatively at frequencies comparable to twice the kink mass (the true vacuum shows a gap-driven discontinuity, the false vacuum a smooth low-frequency decay, with the difference largest near the spectrum's maximum amplitude where the signal-to-noise ratio is favorable); (ii) the amplitudes of subcritical-bubble peaks obey γℓ=4πhzEℓ∣Sz(ω=Eℓ)∣\gamma_\ell=4\pi h_z E_\ell|S^z(\omega=E_\ell)|, and an exponential extrapolation Eℓ→0E_\ell\to 0 of γℓ\gamma_\ell yields the exponentially small decay rate (for example log⁡γ=−108\log\gamma=-108 at hz=0.04h_z=0.04, versus −104-104 from Rutkevich's theory); (iii) the lattice spectra converge, at fixed scaling parameter, to those of the Ising field theory - the scaling limit of the chain near criticality, characterized by the single dimensionless parameter η=m∣h∣−8/15\eta=m|h|^{-8/15} with mm the kink mass and hh the magnetic field - computed in a non-relativistic approximation. The Supplemental Material demonstrates the diagnostic's compatibility with a chain of Rydberg atoms (atoms excited to high principal quantum numbers, whose strong tunable interactions make them a leading quantum-simulation platform).

The authors are explicit that these results hold only within a controlled corner of parameter space. The two-fermion truncation and the non-relativistic approximation are reliable at large η\eta and break down at very small η\eta (the strong-field, strongly relativistic regime; note that the thin-wall regime - bubbles whose wall thickness is much smaller than their radius, reached at small vacuum-energy bias - corresponds to η→∞\eta\to\infty, so the failing regime is the opposite of thin-wall). The decay-rate extrapolation inherits the assumptions of Rutkevich's formula, which is known to disagree with other semiclassical estimates by a constant multiplicative prefactor; the authors state that their analysis cannot shed light on that disagreement. The analysis is one-dimensional, with pointlike domain-wall boundaries, unstructured global quenches, and idealized closed-system dynamics. In their Outlook the authors propose that more accurate continuum predictions could come from the Truncated Fermionic Space Approach (TFSA), a Hamiltonian-truncation method that diagonalizes the field theory in a truncated fermionic basis (Rakovszky et al. 2016), and they raise the questions of higher-dimensional bubble boundaries (long-lived strings and membranes carrying low-energy internal shape degrees of freedom), structured "bubble seeding" quenches, other experimental platforms (Rydberg arrays, ultracold atoms, trapped ions, superconducting qubits), and cosmological expansion, which can stabilize otherwise unstable vacua (Coleman and De Luccia 1980).

Problem Statement

Determine whether the short-time quench-based spectral diagnostic for false-vacuum metastability - the qualitative true-versus-false distinction in the magnitude spectrum of the order parameter after a quench, and the extraction of the exponentially small decay rate from subcritical-bubble amplitudes - remains valid and quantitatively accurate outside the controlled regime in which it was demonstrated (a two-fermion truncation of the scaling Ising field theory, non-relativistic kinematics valid at large η=m∣h∣−8/15\eta=m|h|^{-8/15}, one spatial dimension, unstructured global quenches, and idealized closed-system dynamics). The components of this single objective are:

(1) Multiparticle regime: compute the post-quench magnitude spectrum beyond the two-fermion sector and determine how the spectral distinction between true and false vacua and the extrapolated decay rate behave at small η\eta, where the two-particle approximation is invalid.

(2) Higher dimensions: extend the diagnostic to two and three spatial dimensions, where bubble boundaries are extended objects (long-lived strings and membranes) carrying low-energy internal shape degrees of freedom; establish how those boundary modes enter the short-time spectrum, and what the experimentally accessible small bubbles reveal about higher-dimensional near-critical structures.

(3) Bubble seeding: develop and analyze structured quenches - spatially flipped regions or partial magnetization flips that seed large bubbles - and establish systematic protocols for producing controlled populations of large or subcritical bubbles.

(4) Experimental platforms: extend the diagnostic to the other experimental platforms proposed by the authors - ultracold atoms in optical lattices, trapped ions, and superconducting qubits, in addition to the Rydberg-atom chain whose compatibility was demonstrated in the Supplemental Material - and establish whether the true-versus-false spectral distinction and the decay-rate extraction survive under the realistic conditions of quench experiments on such platforms (e.g. experimental noise, finite-size effects, decoherence, and finite observation windows).

(5) Cosmological expansion: determine how the faster expansion of higher-energy-density regions (Coleman-De Luccia-type effects that can stabilize vacua) modifies the short-time, finite-frequency signatures.

These components share one scientific question: which features of the demonstrated signatures are regime-dependent artifacts of the two-particle, one-dimensional, idealized analysis, and which are robust properties of false-vacuum metastability that survive in more general and realistic settings. The five components are independent: fully resolving any one of them constitutes a solution to this task, and the components left unaddressed remain open.

References

  1. G. Lagnese, F. M. Surace, S. Morampudi, F. Wilczek, "Detecting a Long-Lived False Vacuum with Quantum Quenches", Phys. Rev. Lett. 133, 240402 (2024). DOI: 10.1103/PhysRevLett.133.240402, https://doi.org/10.1103/PhysRevLett.133.240402 (arXiv:2308.08340, https://arxiv.org/abs/2308.08340)
  2. S. Coleman, "Fate of the false vacuum: Semiclassical theory", Phys. Rev. D 15, 2929 (1977). DOI: 10.1103/PhysRevD.15.2929, https://doi.org/10.1103/PhysRevD.15.2929
  3. S. Coleman, F. De Luccia, "Gravitational effects on and of vacuum decay", Phys. Rev. D 21, 3305 (1980). DOI: 10.1103/PhysRevD.21.3305, https://doi.org/10.1103/PhysRevD.21.3305
  4. S. B. Rutkevich, "Decay of the metastable phase in d=1 and d=2 Ising models", Phys. Rev. B 60, 14525 (1999). DOI: 10.1103/PhysRevB.60.14525, https://doi.org/10.1103/PhysRevB.60.14525
  5. P. Fonseca, A. Zamolodchikov, "Ising Field Theory in a Magnetic Field: Analytic Properties of the Free Energy", J. Stat. Phys. 110, 527 (2003). DOI: 10.1023/A:1022147532606, https://doi.org/10.1023/A:1022147532606
  6. M. Kormos, M. Collura, G. Takács, P. Calabrese, "Real-time confinement following a quantum quench to a non-integrable model", Nat. Phys. 13, 246 (2017). DOI: 10.1038/nphys3934, https://doi.org/10.1038/nphys3934
  7. G. Lagnese, F. M. Surace, M. Kormos, P. Calabrese, "False vacuum decay in quantum spin chains", Phys. Rev. B 104, L201106 (2021). DOI: 10.1103/PhysRevB.104.L201106, https://doi.org/10.1103/PhysRevB.104.L201106
  8. T. Rakovszky, M. Mestyán, M. Collura, M. Kormos, G. Takács, "Hamiltonian truncation approach to quenches in the Ising field theory", Nucl. Phys. B 911, 805 (2016). DOI: 10.1016/j.nuclphysb.2016.08.024, https://doi.org/10.1016/j.nuclphysb.2016.08.024
  9. D. Maertens, J. Haegeman, K. Van Acoleyen, "Real-time bubble nucleation and growth for false vacuum decay on the lattice", arXiv:2508.13645 (2025), https://arxiv.org/abs/2508.13645
  10. C. Johansen, A. Recati, I. Carusotto, A. Biella, "Many-body theory of false vacuum decay in quantum spin chains", Phys. Rev. B (2026). DOI: 10.1103/kskv-9mhr, https://doi.org/10.1103/kskv-9mhr (arXiv:2508.13780, https://arxiv.org/abs/2508.13780)
  11. I. G. Moss, "Notes on false vacuum decay in quantum Ising models", arXiv:2510.12592 (2025), https://arxiv.org/abs/2510.12592
  12. B. Yu, R. Zhou, H. Su, "Observable dependence and rate identifiability in false-vacuum decay of the quantum Ising chain", arXiv:2608.11339 (2026), https://arxiv.org/abs/2608.11339
  13. U. Borla, A. Lazarides, C. Groß, J. C. Halimeh, "Microscopic Dynamics of False Vacuum Decay in the 2+1D Quantum Ising Model", arXiv:2601.04305 (2026), https://arxiv.org/abs/2601.04305
  14. L. Pavešić, M. Di Liberto, S. Montangero, "Scattering and induced false vacuum decay in the two-dimensional quantum Ising model", Nat. Commun. 17, 6576 (2026). DOI: 10.1038/s41467-026-73056-6, https://doi.org/10.1038/s41467-026-73056-6 (arXiv:2509.02702, https://arxiv.org/abs/2509.02702)
  15. Z.-H. Zhu, Y. Liu, G. Lagnese, F. M. Surace, W.-Y. Zhang, M.-G. He, J. C. Halimeh, M. Dalmonte, S. C. Morampudi, F. Wilczek, Z.-S. Yuan, J.-W. Pan, "Probing false vacuum decay on a cold-atom gauge-theory quantum simulator", Phys. Rev. Lett. (2026). DOI: 10.1103/vhnm-j9bw, https://doi.org/10.1103/vhnm-j9bw (arXiv:2411.12565, https://arxiv.org/abs/2411.12565)
  16. Y.-X. Chao, P. Ge, Z.-X. Hua, C. Jia, X. Wang, X. Liang, Z. Yue, R. Lu, M. K. Tey, X. Wang, L. You, "Probing False Vacuum Decay and Bubble Nucleation in a Rydberg Atom Array", Phys. Rev. Lett. 136, 120407 (2026). DOI: 10.1103/kqzq-fnr4, https://doi.org/10.1103/kqzq-fnr4 (arXiv:2512.04637, https://arxiv.org/abs/2512.04637)
  17. D. Luo, F. M. Surace, A. De, A. Lerose, E. R. Bennewitz, B. Ware, A. Schuckert, Z. Davoudi, A. V. Gorshkov, O. Katz, C. Monroe, "Quantum simulation of bubble nucleation across a quantum phase transition", arXiv:2505.09607 (2025), https://arxiv.org/abs/2505.09607
  18. P. Osterholz, F. Bensch, S. Tang, S. Baburaj Sheela, B. Sbierski, I. Lesanovsky, C. Groß, "Collective cluster nucleation dynamics in quantum magnets", arXiv:2512.04656 (2025), https://arxiv.org/abs/2512.04656
  19. C. Yin, F. M. Surace, A. Lucas, "Theory of Metastable States in Many-Body Quantum Systems", Phys. Rev. X 15, 011064 (2025). DOI: 10.1103/PhysRevX.15.011064, https://doi.org/10.1103/PhysRevX.15.011064
  20. S. Darbha, M. Kornjača, F. Liu, J. Balewski, M. R. Hirsbrunner, P. L. S. Lopes, S.-T. Wang, R. Van Beeumen, K. Klymko, D. Camps, "Long-lived oscillations of metastable states in neutral atom systems", Phys. Rev. B 110, 155114 (2024). DOI: 10.1103/PhysRevB.110.155114, https://doi.org/10.1103/PhysRevB.110.155114