Entanglement Entropy with Lifshitz Fermions

In the present manuscript the authors study the entanglement entropy (EE) for a free fermion field with Lifshitz scaling symmetry and Lifshitz integer exponent. They study the single interval case, for both the vacuum and thermal states. The main result of the work is that, for the vacuum state, the EE depends only on the parity of the Lifshitz exponent $z$, obtaining the well-known relativistic result for z even and vanishing EE $z$ for odd. They corroborate their results by numerical lattice computations and using the holographic cMERA approach. For the thermal state, the authors calculate numerically the EE in the small and large temperature regimes. They again find different behaviors for even and odd $z$ in the small temperature limit.

The paper is well written and the theoretical discussion is presented in a clear manner. The presented results are original, and the work set the basis for studying EE for fermion fields with Lifshitz scaling symmetry in more general situations. I therefore strongly recommend publication of the manuscript in SciPost Physics. I would like to address the authors some suggestions for changes, and a question about the scope of the manuscript.

1- Figure 1: "the lattice system (a and b) has" $\rightarrow$ "the lattice system (b and c) has".
2- In eq. (25), it is not clear how the constant $N_{A}$ emerges from eq. (23).
3- Below eq. (26): "This expansion only holds for odd values of $\textit{zero}$" $\rightarrow$ "This expansion only holds for odd values of $z$".
4- Above eq. (37): "where the above $\textit{correlaters}$" $\rightarrow$ "where the above correlators".
5- In figure 3, it is plotted $S(T)$ in terms of $\varepsilon^{z}T$. However, the analytic formulas (26-28) are naturally expressed in terms of $l\beta^{-1/z}$. This point worths to be clarified.
6- I would like to ask the authors why the correlator expressions (equations 17-18, 37, 39) are not expected to be invalid for non-integer $z$.

1 – The paper is clear and concisely written
2 – The main result is derived by two independent methods

1 – The paper is in some places not very self contained (e.g. concerning the derivation of equation (53)).

The paper at hand is very well written and presents results concerning entanglement entropy in theories with Lifshitz scaling, in a clear and concise manner, that are novel (to my best knowledge) and interesting. In my eyes, the paper thus opens a new pathway in an existing research direction, with clear potential for multipronged follow-up work, as required by the acceptance criteria for SciPost Physics. Consequently, I am happy to recommend its publication in SciPost Physics, with only minor revisions listed below.

I have noticed the following minor issues (in order of decreasing importance) which I think can be improved, even though they don’t significantly affect the overall scientific quality of the paper:

1 - In the Conclusion section the authors discuss some possible generalisations of their work, (e.g. interactions and higher dimensions), but one issue that I think is interesting is not directly addressed in the paper, and that is the issue of entanglement entropy of unions of multiple intervals.

For even z, if the ground state is a direct product then the result that entanglement entropy vanishes for all even z will of course also extend to unions of multiple disjoint intervals.

For odd z, however, this is less clear to me. In the CFT case (z=1) for example, it is known that equation (19) is generic, while for the union of two or more intervals there is no generic formula valid for all values of c. (Of course, holography provides results in the large c limit)

So, insofar as this doesn’t require too extensive further calculations, I think it would improve the paper to add a discussion of the case for multiple intervals. Do the authors have reason to expect that for odd z, the entanglement entropy of unions of disjoint intervals in the groundstate will also be independent of z? If this is not the case, it might avoid confusion for some readers if this caveat can be added to the abstract or the Introduction section.

2 - In deriving equation (18) from (16), it should be checked whether there is a factor of i missing. At least Mathematica seems to suggests that:
$\int_{-\infty }^{\infty } \frac{\text{sgn}(k)+1}{4 \pi }e^{-ikx} \, dk = \frac{\delta (x)}{2}-\frac{i}{2 \pi x}$.

3 - The caption of figure 1 reads in part: “… The lattice system (a and b) has N sites and a lattice spacing ε. The interactions are depicted for z = 1 (a) and z = 2 (b). ”. Given the figure which consists of three parts (a,b,c), I believe “a” should be replaced by “b” and “b” by “c” in these sentences.

4 - In figure 2(a), the entanglement entropy is supposed to be plotted for a lattice with N=1000 points, and this is consistent with the maximum value of the curve ($\gtrsim 3.5$), but the markings 20, 40, 60, 80 on the $N_A$ axis would seem to indicate N=100. So this should be corrected.

5 - On the left-hand side of equation (13), there is one more “(“ than “)” bracket.

6 - On page 3, the authors write: “… present the Lagrangian and for free Lifshitz fermions and we determine the two-point correlator. ” I believe one of the “and” should be deleted.

7 - On page 7: “This expansion only holds for odd values of zero, …”. I assume zero should be replaced with z.

8 - On page 15: “The geodesic is the found from the length of the parametric curve”. Something is off with this sentence, and it should be clarified.

9 - On page 16: “In other words, constant of proportionality in question is not manifestly determined in the cMERA framework.” I assume it should be “the constant”.

10 - It appears to me that the hyperlinks that the pdf of the paper includes to ArXiv versions of papers don’t work. Maybe this is an issue with the bibliography style file that is being used.