|As Contributors:||Temple He|
|Submitted by:||He, Temple|
|Submitted to:||SciPost Physics|
|Subject area:||High-Energy Physics - Theory|
We discuss and compute entanglement entropy (EE) in (1+1)-dimensional free Lifshitz scalar field theories with arbitrary dynamical exponents. We consider both the subinterval and periodic sublattices in the discretized theory as subsystems. In both cases, we are able to analytically demonstrate that the EE grows linearly as a function of the dynamical exponent. Furthermore, for the subinterval case, we determine that as the dynamical exponent increases, there is a crossover from an area law to a volume law. Lastly, we deform Lifshitz field theories with certain relevant operators and show that the EE decreases from the ultraviolet to the infrared fixed point, giving evidence for a possible $c$-theorem for deformed Lifshitz theories.
We are grateful for the careful reading of the manuscript, and for the detailed report on it. We have significantly modified the version of the manuscript, and now in particular have an analytic understanding of subinterval EE in massless Lifshitz scalar theories in 1+1 dimensions using cMERA techniques. We have thus reorder section 2, with the new analytical results, which basically describe the scaling of subinterval EE with the dynamical exponent in a very simple way. We compare such analytical results with the previous numerics. In addition, we have also addressed nearly all the comments the referees brought up, and thank the referees for their criticism, which in our opinion made this paper much more complete. The only comments we did not address are the following:
1-One referee suggested comments on Renyi entropy/global quenches in our conclusion. We have not studied those aspects and hence feel we cannot say anything too meaningful in that regard with confidence.
2-One referee suggested us to move a sentence from the conclusion to the introduction/abstract. We believe the paper is a self-contained research within field theories with Lifshitz symmetry. Therefore, we relegated comments on possible applications to other areas and further developments to the conclusion.
3-We are not sure about the reason of the poor resolution of the figures. The referees suggested using pdf format instead of
bitmap format, but the format we are using is indeed pdf. Hopefully this wouldn't be an issue again.
We hope that the present version convinces the editor to accept the manuscript for publication.
With best regards,
1- Fixed typo in v2 equation (1.2) and changed the paragraph around it into a footnote.
2- Clarified what we mean by "Due to the lack of relativistic conformal symmetry for z = 1 [sic], we are unable to directly
apply the replica trick in the calculation of the EE. Thus, we will either resort to numerical
methods, or to very special subsystems in which we can compute the EE analytically."
3- Added relevant equation number in "Note added."
4-In v2 equation (2.3), defined phi.
5-Obtained analytic results for subinterval EE in massless Lifshitz theories using cMERA techniques. Section 2 is now drastically extended to include these analytic results and how they compare with the numerics.
6-We commented on the observed crossover behavior in Fig 3 around $z~N_A$.
1-Studies entanglement entropy in a new class of 1+1d theories.
2-Relatively clean presentation, writing and organization
1-Numerical evidence in support of some claims is perfunctory.
The paper is substantially improved by the changes from the previous version. The use of analytic arguments now provides a qualitative understanding of the effect of the dynamical exponent on entanglement, which is borne out in several of the numerical calculations performed. The main new result, Equation (2.11), is an interesting an straightforward generalization of $z=1$ results, which can be numerically confirmed to very high precision. It is unfortunate that Figure 1 shows somewhat large deviations from the predicted behavior, even at $z=1$ which should be well-understood. It may be appropriate to comment more fully on the non-universal behavior present in the numerical data or, better yet, reduce it (e.g. by going to smaller mass values or larger $z$).
In conclusion, many of the largest problems have been corrected and the paper now provides adequate novelty and understanding of its subject matter.
1 - The quality of the figures is still poor. In particular, Figure 2, 3, and 7 are still bitmapped. They should either be replaced by vector graphics, or raster graphics with a high enough resolution to print well. (PDF is a flexible format that can include both raster and vector images. Modern plotting programs such as matplotlib, mathematica, etc have options for exporting vector graphics. If raster graphics must be employed, one should use a resolution of at least 600DPI so that they print well.)