Entanglement and quench dynamics in the thermally perturbed tricritical fixed point

We consider the Blume–Capel model in the scaling limit to realize the thermal perturbation of the tricritical Ising fixed point. We develop a numerical scaling limit extrapolation for one-point functions and Rényi entropies in the ground state. In a mass quench scenario, we found long-lived oscillations despite the absence of explicit spin-flip symmetry breaking or confining potential. We construct form factors of branch-point twist fields in the paramagnetic phase. In the scaling limit of small quenches, we verify form factor predictions for the energy density and leading magnetic field using the dynamics of one-point functions, and branch-point twist fields using the dynamics of Rényi entropies.

💡 Research Summary

This paper investigates the non‑equilibrium dynamics of the thermal perturbation of the tricritical Ising conformal field theory, known as the E₇ integrable quantum field theory, by realizing it as the scaling limit of the quantum Blume–Capel spin‑1 chain. The authors first identify the tricritical point of the lattice model (α≈0.910207, β≈0.415685, γ=0) and introduce a Z₂‑symmetric thermal perturbation H⊥ whose coupling λ drives the system into the high‑temperature (paramagnetic) or low‑temperature (ferromagnetic) phase of the E₇ theory. Using infinite‑time‑evolving block decimation (iTEBD), they compute expectation values of the magnetisation σ and the energy density operator E for a range of λ values, and perform a careful extrapolation to the continuum limit. The scaling of ⟨σ⟩∝a^{-3/40} and ⟨E⟩∝a^{-1/5} is verified, with the measured exponent 0.0419 matching the exact CFT prediction, confirming that the lattice model indeed flows to the E₇ field theory.

In the second part the authors develop the form‑factor bootstrap for branch‑point twist fields 𝒯ₙ, which encode Rényi entropies via the replica trick. Working in the paramagnetic phase, they solve the bootstrap equations for the one‑ and two‑particle form factors of 𝒯ₙ, imposing Z₂ parity, crossing, and the Δ‑theorem to fix normalisation. The resulting expressions are valid for any replica index n and provide the leading contributions to the entanglement dynamics.

The dynamical study focuses on a small mass quench: the system is prepared in the ground state at coupling λ₀ and at t=0 the thermal coupling is changed to λ₀+δλ with |δλ|≪|λ₀|. Post‑quench perturbation theory shows that, because the perturbing operator ε(x) is Z₂‑even, only even‑particle states are excited. Consequently the time evolution of one‑point functions ⟨ε(t)⟩ and ⟨σ(t)⟩ is governed by the two‑particle form factors, leading to undamped oscillations with frequency set by twice the lightest mass. The iTEBD data, after scaling‑limit extrapolation, display precisely these oscillations and match the analytic predictions quantitatively (frequency, amplitude, phase).

For the Rényi entropies Sₙ(t) the leading contribution comes from the two‑particle form factor of the twist field. The authors compute Sₙ(t) analytically and compare with the numerically obtained entropies. Again, excellent agreement is found: the entropies do not exhibit the linear growth typical of thermalising systems but instead show long‑lived oscillations, reflecting the integrable nature of the E₇ model and the restriction to even‑particle excitations. Notably, this behaviour occurs without explicit Z₂ symmetry breaking or confining potentials, distinguishing it from previously studied E₈ quenches.

The paper concludes that (i) the Blume–Capel chain provides a faithful lattice regularisation of the E₇ field theory, (ii) branch‑point twist field form factors can be constructed and successfully applied to entanglement dynamics, and (iii) small thermal quenches in the E₇ model produce long‑lived, undamped oscillations in both local observables and entanglement measures, in quantitative agreement with integrable QFT predictions. The work opens the way to systematic studies of non‑equilibrium entanglement in other perturbed minimal models, to explore larger quenches, and to investigate the role of non‑invertible symmetries that are present at the tricritical point but broken by the thermal deformation.