Entanglement and quench dynamics in the thermally perturbed tricritical fixed point

The authors shall provide more connections to experimental realizations of the corresponding quench dynamics.

The authors study the entanglement dynamics of the $E_7$ particles in the perturbed tricritical Ising field theory, which I would refer to as the quantum $E_7$ integrable field theory (IFT). Although the related techniques, used as building blocks, have been systematically developed in a series of works (mainly Refs. [18], [81], [40], and some more related papers), it is still a non-trivial task to determine the form factors of branch-point twist operators in the quantum $E_7$ IFT. Moreover, the authors also apply the form-factor approach to the quench dynamics of entanglement and show entanglement oscillations of the $E_7$ particles, which compare quite well with the numerics obtained by the iTEBD algorithm. I would suggest that the authors consider more practical experimental realizations of the $E_7$ particles; see Phys. Rev. Lett. 133, 223401 (2024) and related papers for practical realizations and manipulation of tricritical Ising criticality.

Regarding the issues raised by Referee 3 concerning the perturbative framework, it would be better for the authors to emphasize that their calculations are carried out within a post-quench perturbative framework, rather than being obtained by simply replacing the pre-quench interacting-picture results with the post-quench particle masses (even though the expressions may sometimes look similar).

I would like to emphasize the importance of the post-quench perturbative framework. For a general quench-dynamics setup with an arbitrary Hamiltonian H and an arbitrary state |s> that is not an eigenstate of H, the time-dependent physical quantity O can always be expanded as \sum_{nm} <n|s><s|m><m|O|n>\exp{-i(E_n-E_m)t}, where |n> and |m> are eigenstates of H (with |n><n| and |m><m| the corresponding projectors). If one of |n> or |m> is the ground state |0>, this provides the starting point of Eq. (2) in DS2024. The derivation of this formalism is a question at the level of basic quantum mechanics, but in general it is completely impractical. Even in the setup of IFT with small perturbations, in most cases one has no idea how to compute the overlaps "<n|s><s|m>" analytically.

The post-quench perturbative framework provides a faithful method to compute these overlaps within the setup of IFT and small perturbations, as shown in the original paper introducing this method, Ref. [88], where such overlaps are computed up to second order and the results are compared with those from the truncation method. They obtain explicit expressions up to second order in the quench strength and benchmark these analytic formulas against high-precision numerical data from the truncated conformal space approach. Their main conclusion is that the expansion using the post-quench basis works very well, while the alternative expansion in the pre-quench basis suffers from serious limitations in the same model. From a technical point of view, a controlled low-energy expansion around the post-quench Hamiltonian, which actually produces accurate overlaps and reproduces the numerically observed excitation spectrum, is clearly not trivial when compared with a fully general but essentially noncomputable structural result.

I also wish to comment on the limitations of the pre-quench perturbative framework in certain quench problems. Referee 3 repeatedly stresses that a perturbative expansion in the quench strength lambda should be organized around the unperturbed, pre-quench theory. As a matter of textbook perturbation theory this statement is formally correct. However, this does not imply that such an organization is universally useful in practice, even at a formal level, once one considers specific quench protocols.

I would like to provide an extreme example where the pre-quench perturbative framework likely does not work: the quench dynamics of a critical transverse-field Ising chain with a non-critical transverse field (or a non-zero longitudinal field), starting from the ground state of the critical transverse-field Ising chain, where we focus on time-dependent physical quantities. The reason is very simple: In the low-energy region we expect to observe oscillation modes with finite gaps dominated by the post-quench Hamiltonian (a number of exact calculations for non-critical transverse fields support this; e.g., see Phys. Rev. Lett. 106, 227203 (2011), where the oscillations depend only on h, the post-quench transverse field), whereas the pre-quench perturbative framework would require exponentially many excited states in order to build up a finite-gap state, which makes it completely unreasonable to apply it in this problem. However, it still makes sense to apply the post-quench framework to capture part of the remarkable oscillating modes in the low-energy region, in comparison with numerical simulations.

Overall, I do not see any issue with the authors applying the post-quench perturbative framework. It works very well when compared with numerics, providing a reliable analytical approach to understanding such quench dynamics, so I also do not see any necessity to change the method of analysis. Due to the good presentation and detailed derivations in this paper, I shall recommend its publication.

No

Publish (easily meets expectations and criteria for this Journal; among top 50%)