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Real-time false-vacuum decay in the quantum Ising chain reproduces the Fonseca–Zamolodchikov rate

Dylan Z. Liu Liu

Published Oct 27, 2026• 0 citations
Publisher

Abstract

Recent real-time simulations of false-vacuum decay in the quantum Ising chain report decay-rate prefactors that disagree with each other and with field theory by factors up to three. We prepare the metastable state by a real-time ramp of the longitudinal field, measure the survival rate per site of an infinite matrix-product state, and compare it with the decay rate that Fonseca and Zamolodchikov obtained from the free energy of the Ising field theory continued to negative field. At two lattice spacings and scaling variables y = 2.2–2.8 the ratio is 1.008 with a standard deviation of 0.015 over 6 runs, with no adjustable parameter, once the first correction to the thin-wall formula is kept. The two-kink coefficient π∆E/18 used in recent comparisons is too large by π2/9, as anticipated by Fonseca and Zamolodchikov, and exponential fits to the order parameter return rates 147–390 times larger than the nucleation rate. The survival rate, or equivalently the growth rate of the bubble density, is the observable that quantum simulators should compare with theory.

DOI
10.5281/zenodo.23707742

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Recent real-time simulations of false-vacuum decay in the quantum Ising chain report decay-rate prefactors that disagree with each other and with field theory by factors up to three. We prepare the metastable state by a real-time ramp of the longitudinal field, measure the survival rate per site of an infinite matrix-product state, and compare it with the decay rate that Fonseca and Zamolodchikov obtained from the free energy of the Ising field theory continued to negative field. At two lattice spacings and scaling variables y = 2.2–2.8 the ratio is 1.008 with a standard deviation of 0.015 over 6 runs, with no adjustable parameter, once the first correction to the thin-wall formula is kept. The two-kink coefficient π∆E/18 used in recent comparisons is too large by π2/9, as anticipated by Fonseca and Zamolodchikov, and exponential fits to the order parameter return rates 147–390 times larger than the nucleation rate. The survival rate, or equivalently the growth rate of the bubble density, is the observable that quantum simulators should compare with theory.
Extracting a thermodynamic nucleation rate from finite-time quantum dynamics requires separating observable decay from estimator and finite-size validity. We develop a multilevel identification framework for real-time tensor-network simulations of false-vacuum decay in the one-dimensional quantum Ising chain. Across twelve parameter points, the same coherent two-kink amplitudes semi-quantitatively predict both infinite-chain survival and magnetization dynamics: the survival coefficient has a median lattice-to-theory ratio of 0.896, while the magnetization-area slope ratios span 0.767--0.931. By contrast, the microscopic nearest-neighbour bond response is coherence dominated: vacuum--pair coherence contributes 60.0--81.5% across seven points with matched bond-dimension control, while substantial late-window slope discrepancies remain that cannot be removed by a scalar normalization. The analysis establishes finite-time survival and magnetization benchmarks and identifies the additional finite-size and branch-validation requirements for a bulk thermodynamic rate interpretation. Within the two-kink model and under the adopted common normalization, the lattice-resolved WKB action gives a median fixed-prefactor rate discrepancy of 4.13% from the coherent-bubble spectral calculation. These results distinguish finite-time lattice--theory consistency from the additional observable and finite-size evidence required to identify a thermodynamic nucleation rate.
We revisit quantum false vacuum decay for the one-dimensional Ising model, focusing on the real-time nucleation and growth of true vacuum bubbles. Via matrix product state simulations, we demonstrate that for a wide range of parameters, the full time-dependent quantum state is well described by a Gaussian ansatz in terms of domain wall operators, with the associated vacuum bubble wave function evolving according to the linearized time-dependent variational principle. The emerging picture shows three different stages of evolution: an initial nucleation of small bubbles, followed by semi-classical bubble growth, which in turn is halted by the lattice phenomenon of Bloch oscillations. Furthermore, we find that the resonant bubble only plays a significant role in a certain region of parameter-space. However, when significant, it does lead to an approximately constant decay rate during the intermediate stage. Moreover, this rate is in quantitative agreement with the analytical result of Rutkevich (Phys. Rev. B 60, 14525) for which we provide an independent derivation based on the Gaussian ansatz.
False vacuum decay describes the relaxation of a metastable state through the nucleation and growth of bubbles of the stable phase. Despite describing a broad variety of phenomena across different fields, the quantum version of the nucleation theory has little experimental or numerical support. Testing its predictions is particularly important in two or more spatial dimensions, where bubble nucleation acquires its true geometrical nature. Here, we study false vacuum decay in the quantum Ising model in two dimensions. Through tree tensor network simulations we extract the decay rate, the effective interface tension and the critical bubble size. We compare them to new semi-classical field theory calculations, and find excellent agreement. These results provide numerical evidence that the critical-bubble picture survives in an interacting quantum spin system in 2+1 dimensions.
The false vacuum decay has been a central theme in physics for half a century with applications to cosmology and to the theory of fundamental interactions. This fascinating phenomenon is even more intriguing when combined with the confinement of elementary particles. Due to the astronomical timescales involved, the research has so far focused on theoretical aspects of this decay. The purpose of this Letter is to show that the false vacuum decay is accessible to current optical experiments as quantum analog simulators of spin chains with confinement of the elementary excitations, which mimic the high energy phenomenology but in one spatial dimension. We study the nonequilibrium dynamics of the false vacuum in a quantum Ising chain and in an XXZ ladder. The false vacuum is the metastable state that arises in the ferromagnetic phase of the model when the symmetry is explicitly broken by a longitudinal field. This state decays through the formation of ``bubbles'' of true vacuum. Using infinite volume time evolving block decimation (iTEBD) simulations, we are able to study the real-time evolution in the thermodynamic limit and measure the decay rate of local observables. We find that the numerical results agree with the theoretical prediction that the decay rate is exponentially small in the inverse of the longitudinal field.
We formulate false-vacuum decay in a mixed-field Ising chain with 1/rα1/r^α1/rα interactions as a spatially nonlocal Euclidean φ4φ^4φ4 theory featuring a fractional spatial kinetic term ∼∣q∣σ\sim |q|^σ∼∣q∣σ, where σ=α−1σ=α-1σ=α−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<σ<10<σ<10<σ<1, the lifetime exponent scales with the energy bias hhh of the metastable state as B∼h−1/σB\sim h^{-1/σ}B∼h−1/σ; for 1<σ<21<σ<21<σ<2, the leading Coleman scaling B∼h−1B\sim h^{-1}B∼h−1 is recovered, while long-range effects are retained in the subdominant term ∼hσ−2\sim h^{σ-2}∼hσ−2. Our results show that tunable long-range interactions fundamentally reshape bubble nucleation and alter false-vacuum decay in quantum simulators.
In this work we characterize the false vacuum decay in the ferromagnetic quantum Ising chain with a weak longitudinal field subject to continuous monitoring of the local magnetization. Initializing the system in a metastable state, the false vacuum, we study the competition between coherent dynamics, which tends to create resonant bubbles of the true vacuum, and measurements which induce heating and reduce the amount of quantum correlations. To this end we exploit a numerical approach based on the combination of matrix product states with stochastic quantum trajectories which allows for the simulation of the trajectory-resolved non-equilibrium dynamics of interacting many-body systems in the presence of continuous measurements. We show how the presence of measurements affects the false vacuum decay: At short times the departure from the local minimum is accelerated while at long times the system thermalizes to an infinite-temperature incoherent mixture. For large measurement rates the system enters a quantum Zeno regime. The false vacuum decay and the thermalization physics are characterized in terms of the magnetization, connected correlation function, and the trajectory-resolved entanglement entropy.