Entanglement Entropy in Lifshitz Theories

1) First results in an unexplored domain
2) Well written text
3) Combination of numerical and analytical findings

1) Less clear results (conjectured or numerically suggested) about the EE in the more important linear segment sublattice case
2) No comments about other entropy measures (Rényi entropies, dynamical entropy)
3) Low quality figures

In this paper the entanglement entropy (EE) of (1+1)-dimensional free Lifshitz scalar field theories is computed for different values of the dynamical exponent, z. The authors have considered two geometries for the subsystem: i) the traditional choice of a line segment of consecutive points, and ii) periodic sublattices. In the first case numerical calculations are performed and the qualitative features of the EE as a function of "z", and the length of the subsystem are analysed. Unfortunately no conjectures about the possible functional dependence are obtained through these studies. For periodic sublattices, which are somehow less interesting, there are analytical results, too. They have also studied the behaviour of the EE, when the Lifshitz theory is perturbed with relevant operators. The obtained results are physically motivated and can be considered as plausible. The paper is generally well written, the introduction gives the scientific background, mathematical derivation of the results are well documented. The literature is adequately cited.

1) Phi is not defined in (2.3).
2)The figures are of low quality, they should be replaced.
3) The Conclusions should be extended with some comments. What is expected to happen with other entropy measures, such as the Rényi entropy. What about the behaviour of the dynamical entropy after a sudden change of the parameters (global quench).
4) There are a few typos, such as "prove prove" in p. 15.

1-Studies entanglement entropy in a new class of 1+1d theories.
2-Relatively clean presentation, writing and organization

1-Few substantial analytic results
2-Numerical results are not interpreted carefully.
3-No broad or even unexpected conclusions.

This study explores the entanglement entropy (EE) of Lifshitz theories in 1+1d for arbitrary dynamical exponent $z>1$. This is a previously unexplored area, and one that might be fruitful to understand, e.g. how standard rules of entanglement such as the "area law" may be modified in various situations, or how entanglement should behave in holographic Lifshitz theories. Unfortunately, the authors do not pursue the subject far enough to reach interesting conclusions.

The paper consists of two main parts: the first studies EE on a traditional bipartite subinterval, and the second studies it on a periodic entanglement cut. The first section is entirely numerical whereas the second includes some analytic components. The paper concludes that in Lifshitz theories EE may or may not obey the area law, EE increases linearly with z in some cut geometries, and EE decreases along RG flows in these theories.

There are several crucial areas where the work should be improved. It is unknown how the EE depends on the parameters of the theory --- the dynamic exponent and the mass --- and it would be interesting to learn how they qualitatively change the scaling of the entanglement. In other words, one should try to answer the question: what are the general principles for entanglement with higher $z$? By and large, this work performs numerics that demonstrate the behavior of the entanglement, but does not go on to understand the "physical behavior" responsible. Moreover, the analytical content of the work is relegated to a non-standard and less-useful $p$-periodic entanglement cut.

In conclusion, this work is on an interesting topic and its results are correct, but does not go far enough in providing insight and intuition on its subject matter.

1- Equation (1.2) has a typo. The entire paragraph could be condensed to a sentence, since (1.2) is not needed in the sequel except for in a footnote.

2- Page 3 states "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."

However, the references [16] describes how the replica trick may be applied in any free field theory. Conformal symmetry makes it easier to compute the entanglement entropy, but not impossible.

3-In the "Note added", it would be helpful to refer to an equation number or figure where their results may be seen to conflict.

4-In equation (2.3), $\phi$ should be defined.

5-After Figure One, there is insufficient discussion of the behavior. It is clear that the entanglement increases with $z$, and some arguments are given to say this is reasonable, but the authors miss the opportunity to make a broader point about why this must be or how general this effect should be. It might be good to have a prediction for how the dynamical exponents affects the entanglement and use numerics to confirm it. It also appears there might be a crossover behavior in $S(z)$ at $z \sim N_A = 20$. Above $z = 20$, each site directly interacts with sites outside the entanglement cut, so it is reasonable to expect the entanglement might scale differently in this regime.

6-Similarly after Figure 3, there is insufficient insight into the behavior of the numerics. One should be able to say something about the scaling of the EE.

7-The sentence in the conclusion "While the focus of our paper..." might be better suited for the abstract as it provides a good motivation for studying this topic.

8- Figures: the figures are bitmap images and thus do not render well when printed. The legends in particular become too pixelated to read. Using vector images (e.g. pdf, eps) would improve this. It may be advisable to make the figures colorblind-safe by using different symbols for data, in addition to different colors, when there are multiple sets of data on the same plot, such as in Figure 1.

9-Typos, grammar, notation (non-exhaustive).
In the abstract: "entanglement entropies (EEs)"-> "entanglement entropy (EE)".
Note added: "Dirichlet boundary condition" --- should be "conditions"
In Equation (2.4) and below, it may be easier to read to re-define $J^{-1} \to J$.
Figure 3: "as a function of the size of the subinterval". It may be clearer to say "as a function of the size $N_A$ of the subinterval".
Figure 7: the caption refers to the parameters $\widetilde{J}_0/J_0 = 2^{20}$ but also $\widetilde{J}_0 = 0, 2^{-30},$ and $2^{-40}$. However, the legend refers to $\widetilde{J}_0 = 2^{-20}$ instead. Also, if $\widetilde{J}_0 = 0$, then $\widetilde{J}_0/J_0 = 0 \neq 2^{20}$. Please clarify the parameters.