Entanglement and quench dynamics in the thermally perturbed tricritical fixed point

TL;DR

The paper investigates entanglement and quench dynamics near the thermal perturbation of the tricritical Ising fixed point by realizing the $E_7$ model in the Blume--Capel spin chain. It develops a numerical scaling-limit extrapolation for ground-state one-point functions and Rényi entropies, and demonstrates long-lived, undamped oscillations after mass quenches in the paramagnetic phase, even without explicit spin-flip symmetry breaking or confinement. A central theoretical advance is the extension of the form-factor bootstrap to branch-point twist fields within the $E_7$ theory, yielding two-particle twist-field form factors and enabling analytic predictions for energy density, leading magnetic field, and entropy dynamics that are then validated against scaling-limit iTEBD simulations and quench spectroscopy. The results substantiate the scaling relations, mass-coupling relations, and spectrum-consistency of the $E_7$ description and highlight the utility of twist-field FFs for entanglement observables, with potential experimental relevance for quantum simulators of tricritical Ising physics. The work also opens questions about duality, non-invertible symmetries, and entanglement structure in both phases, suggesting further extensions to the odd sector and kink-related phenomena.

Abstract

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.

Figures (10)

• Figure 1: The scaling of the expectation value of the magnetic field. We found the expected scaling behavior, with the measured exponent $0.0419$ in an excellent agreement to its theoretical value.
• Figure 2: Expectation value of the perturbing operator in the low- and high-temperature phases, with $\lambda<0$ and $\lambda>0$ respectively, with scaling limit fits separately in both phases. One can see that it exponentially converges to approximately the same constant and with the same exponent in both phases.
• Figure 3: Scaling limit of the Rényi entropies. Unusual scaling ansatz is important to extract the conformal weights of the twist fields, which in turn are in excellent agreement with the theory.
• Figure 4: Rescaling of time/energy scale. In the left panel we show the data produced by iTEBD simulations for various quenches with $5$% change in the coupling. The post-quench couplings are indicated in the middle panel, where we show the rescaled data. On the rescaled data, one can perform a time-dependent extrapolation to the scaling limit, as we show in the right panel.
• Figure 5: The power spectrum of the time evoultion of the magnetization after the mass quench with $\lambda_0=0.001 \rightarrow 0.00105$. The uppermost horizontal line is fitted by hand, the other two were computed using one-particle form factors from Acerbietal1996Cortesetal2021. Locations of peaks and the height of the particle peaks are compatible with the integrability results for even particles.