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Prove ℓ2-scaling metastable lifetimes for Z2-symmetric perturbations of 2D Ising stripe states

Prove that, in the two-dimensional nearest-neighbor Ising model perturbed by local terms that preserve the global \mathbb{Z}_2 symmetry, stripe-configured metastable states with stripe width ℓ have prethermal lifetimes scaling as t_* = exp(Ω(ℓ^2)), arising from the dominant process of connecting like-polarized stripes via flipping rectangular regions, provided one starts from a genuine metastable state in which each stripe corresponds to a true ground state of the symmetric Hamiltonian.

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Background

The paper identifies stripe configurations in the 2D Ising model as metastable due to domain-wall constraints and discusses lifetime mechanisms arising from distinct nucleation processes.

For generic perturbations, they argue the lifetime should scale as the exponential of the area of the smallest resonant bubble. Under \mathbb{Z}_2-symmetric perturbations, they assert the dominant channel yields an exp(Ω(ℓ2)) lifetime but state that they have not supplied a proof.

References

Furthermore, if we require that the perturbation preserves the Ising \mathbb{Z}2 symmetry, there is no longer a false vacuum decay as in Section \ref{sec:falsevacuum}, so we expect (but did not prove) that $t* = \exp{\Omega(\ell2)}$ coming from the latter channel alone, so long as one starts from the genuine metastable state where each stripe corresponds to a ``true ground state'' of the symmetric Hamiltonian.

— Theory of metastable states in many-body quantum systems (2408.05261 - Yin et al., 9 Aug 2024) in Section “Two-dimensional Ising model,” final paragraph