Confinement and false vacuum decay on the Potts quantum spin chain

The authors study non-equilibrium physics in the mixed-filed 3-state Potts model in 1+1D. The authors observe rich dynamics compare to the Ising model. In particular, the authors claim that in addition to Wannier-Start localization and dynamical confinement, they observe baryonic excitations.

I find the subject interesting and timely, though the explanations are in many places are too lengthy and full of technical details making it hard to follow the general idea. For instance, the definition of the Potts model (very standard and well known in the field) takes two full pages. And then it is repeated again but with the longitudinal field in Eq.(3.3), and again in Eq.(3.5), and again in Eq.(4.47)

I'd recommend to significantly shorten the main part, perhaps, moving technical details into appendices (obviously, this makes the paper very pedagogical and contributes to the reproducibility of the results, but the way it is written now, the amount of technical information included distracts the attention from the main results)

I do not see any conceptual difference between model of Eq. (3.3)/sketches 3.1 and the model in Eq. (3.5)/sketches 3.2. By applying an overall rotation on one model, one man make the two of the identical. Do I miss something?

The captions are minimalistic. I strongly recommend the authors to include more details in the description of their results and to guide readers on what they could (or could not) see in the Figures.

I appreciate the details on TEBD in the appendix, still minimal information on the used truncation error/bond dimension can be mentioned in the beginning of the section 3.2.

­In fig.4.8, 4.10, I guess, symbols mean exact diagonalization results and lines are from semiclasics? This should be mentioned.

Fourrier spectrum is not properly defined in the main text (one Eq. In the main text and all technical details in the appendix would be great, I think)

In Fig.5.5/5.6 the correlations are non-zero even outside the light-cone. The reason for this should be briefly mentioned in the text.

I find the last sentence “Finally, it would be interesting to see whether the
various phenomena explored here can be realized in experimental settings, e.g., by ultra-cold atomic
simulators.” too vague. The authors should either remove it or support it with some concrete ideas.

I do recommend the publication of the present study in SciPostPhysics once the authors have considered the comments above.

Publish (meets expectations and criteria for this Journal)

In this work the authors study the non-equilibrium dynamics of the three-state Potts quantum chain in the presence of mixed fields. As in the previously studied quantum Ising chain, a rich physical phenomenology including confinement is expected, and indeed observed, in this case. Furthermore, the 3-state Potts chain offers a wider scope of quench protocols, which result in a richer physics (which, borrowing from the QCD terminology, are associated with "baryonic excitations").
Starting from a purely ferromagnetic initial state and with an external magnetic field pointing in one of the three directions there are four possible quench protocols (depending on the relative orientation of the initial magnetization with the external longitudinal magnetic field, and of the sign of the latter), each of which comes with a different phenomenology (light-cone propagation, standard or partial confinement or anticonfinement). Numerical simulations by iTEBD are presented for each case, and the numerical data for the entanglement entropy, average magnetizations or correlation functions, are confronted with a semi-classical analysis.
Importantly, the model is not integrable on the lattice, therefore requiring a numerical evaluation of the various scattering phases. Combined with a semi-classical analysis, this results in a very precise estimation of (a significant part of) the low-lying energy spectrum in the various regimes, which is used to provide a compelling interpretation of the non-equilibrium spectral data obtained from numerics.

Altogether this is a very complete study of a new and interesting problem, completing the state-of the art (the case of the Ising model) with a richer phenomenology (even though the analogy to QCD may be somewhat artificial).
I recommend its publication in SciPost physics once the comments below have been considered.

1-Eq. (2.2) : in the definition of the Hamiltonian, I suggest that the authors recall which term break integrability, and how this compares with the Ising case.

2-A significant part of the interesting physics (relying on Bloch oscillations and Wannier-Stark localisation) goes as in the previously studied case of the Ising model, and therefore very little background is given. I suggest to add a few lines of explanation of the Wannier-Stark localisation phenomenology, and why it is expected to hold in the present model, as opposed to other quantum spin chains.

3- Appendix A.4 : since the underlying CFT is known, I find surprising that no exact estimation of $\alpha$ can be found. Could the authors comment on that ?

4- Fig A.3: Could the authors confront their phase shift with those known from field theory ?

5- Eq (4.37) and following : notation $\mathcal{E}_n$ for the energy is unexplained. Is it the same as $E_n$ ? I so notations should be harmonized

Ask for minor revision

The manuscript meets the SciPost Physics acceptance criteria.
I recommend its publication in SciPost Physics with minor revision in response to my two comments.

1- I propose to add vertical gridlines in Figures 5.1 -5.4 indicating the boundaries of the continuous two-kink spectrum in the infinite unperturbed chain.
2 - I think, that the interpretation of the computer simulations in the oblique regimes should be modified, as it is explained in the report.

Publish (surpasses expectations and criteria for this Journal; among top 10%)