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Universal finitesize amplitude and anomalous entangment entropy of $z=2$ quantum Lifshitz criticalities in topological chains
by Ke Wang, T. A. Sedrakyan
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Submission summary
Authors (as registered SciPost users):  Tigran Sedrakyan · Ke Wang 
Submission information  

Preprint Link:  scipost_202111_00006v1 (pdf) 
Date submitted:  20211103 18:40 
Submitted by:  Wang, Ke 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We consider Lifshitz criticalities with dynamical exponent $z=2$ that emerge in a class of topological chains. There, such a criticality plays a fundamental role in describing transitions between symmetryenriched conformal field theories (CFTs). We report that, at such critical points in one spatial dimension, the finitesize correction to the energy scales with system size, $L$, as $\sim L^{2}$, with universal and anomalously large coefficient. The behavior originates from the specific dispersion around the Fermi surface, $\epsilon \propto \pm k^2$. We also show that the entanglement entropy exhibits at the criticality a nonlogarithmic dependence on $l/L$, where $l$ is the length of the subsystem. In the limit of $l\ll L$, the maximallyentangled ground state has the entropy, $S(l/L)=S_0+(l/L)\log(l/L)$. Here $S_0$ is some nonuniversal entropy originating from shortrange correlations. We show that the novel entanglement originates from the longrange correlation mediated by a zero mode in the low energy sector. The work paves the way to study finitesize effects and entanglement entropy around Lifshitz criticalities and offers an insight into transitions between symmetryenriched criticalities.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 202228 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202111_00006v1, delivered 20220208, doi: 10.21468/SciPost.Report.4358
Report
The paper addresses finite size scaling at a topological phase transition with a degenerate dispersion relation \epsilon \sim k^2. There is a body of well established literature, starting from the universal finite size scaling at a conformal critical point, where the correction is known to be proportional to the central charge of the theory. These finding were subsequently generalized to an entire universal scaling function, covering small deviation from the conformal critical point. All these findings, however, are restricted to the linear Dirac dispersion relation. While most prominent, the Dirac dispersion is not unique. There are instances (e.g. multicrtical points) where the lowenergy dispersion relation degenerates into k^2. The present paper presents results for finite size scaling of grounsntate energy and entanglement entropy in such situation. This
is a welcome contribution to the field and a valuable addition to the existing body of knowledge. I support its publication.
As a suggestion (in agreement with the first referee): the manuscript can benefit from language proofreading and exposing more details of the calculations. Authors thought about deriving scaling function, covering deviations from the gapless point would be a welcome addition too.
Report #1 by Anonymous (Referee 1) on 2021128 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202111_00006v1, delivered 20211208, doi: 10.21468/SciPost.Report.4016
Strengths
Interesting and universal results concerning nonconformal invariant criticalities.
Weaknesses
Lack of details and explanations about the derivation of the results.
Report
This paper deals with theories that at criticality are not conformal invariant, but they have a dynamical exponent $z=2$, rather than $z=1$ as occurs in conformal field theories (CFTs). Starting from these nonCFT criticalities, the authors try to investigate some universal features which can emerge in such onedimensional systems. In particular, they focus on the finitesize corrections to the energy and on the entanglement entropy of the ground state. For the energy, they find a universal correction $\sim L^{2}$, with $L$ the system size, while the entropy exhibits a nonlogarithmic behaviour, $l/L\log(l/L)$, with $l$ the subsystem size, due to the presence of zeromodes at the Fermi surface of the considered models.
They benchmark their results against lattice computations which involve a generalisation of the Majorana chain and of the SuSchriefferâ€“Heeger (SSH) model.
The paper is wellwritten and it contains some nontrivial results. Therefore, I would recommend it for publication once a minor revision work has been done. Indeed, I would suggest to add more details and explanations such that the work can be selfconsistent and more pedagogical.
Here is a short list of comments/questions/typos:
 Typo in the title: entangment $\rightarrow$ entanglement;
 Pag. 4 before Eq. (5) and after Eq. (6): Hamilotnian $\rightarrow$ Hamiltonian (and hamiltonian $\rightarrow$ Hamiltonian pag. 2);
 Pag. 5: "the computation of the finitesize amplitude of the ground state is similar to the method used in references [10,32]": could the author explain in more details the numerical method used to obtain the universal coefficient $A$?
 Pag. 6 after Eq. (6): "a is the lattice space": where does $a$ enter in the text?
 Pag. 6: "entanglment" $\rightarrow$ entanglement;
 Pag. 7: it would be more clear if you could comment the steps from the correlation function to the nontrivial value of the entanglement espectrum, $\epsilon_0$;
 "Zeromodes are present and influencing entanglement entropy in other contexts, including CFTs": the authors could be interested into another scenario where the presence of a zero mode at the conformal point of a free scalar theory affects the behaviour of the entanglement entropy (J.Stat.Mech.0512:P12012,2005).\\
 Do the authors have any insights about what happens in higher dimensional systems? For example, free massless nonrelativistic fermions show logarithmic violations of the area law.
Here we respond to comments made by the Referee in the same order they appear in the report.

We thank the Referee for the detailed description of the work, finding that the paper is wellwritten, and suggesting the publication.

We appreciate the Referee's helpful suggestions. The manuscript has been updated with further information. Now it includes more explanations and two new appendices with details of calculations.

Typos in the title and around Eqs. 5, 6 are corrected. Some other typos are also corrected.

We discuss the details of the numerical method to estimate the finitesize effects in the present version (particularly in Sect 3). We also added Ref. 36, discussing some specific details.

Again, we thank the Referee for pointing to these and other typos. The sentence "$a$ is the lattice space" is deleted (which was a leftover from an older version). We have corrected this and many other typos present in the text.

We have added the details about the steps discussing the calculation of the entanglement spectrum from the correlation function.

The information on the paper J.Stat.Mech.0512:P12012,2005 is valuable. We have introduced the corresponding discussion and included the important reference.

We thank the Referee for pointing out this interesting question of higher dimension. Although the generalization to higher dimensions is of great interest, we at this moment do not have a good understanding of such generalization. For that reason, we prefer not to speculate about it in the present paper.
Bests The authors
Author: Ke Wang on 20220318 [id 2298]
(in reply to Report 2 on 20220208)Here we respond to comments made by the Referee in the same order they appear in the report.
We thank the Referee for the concise description of the work and for suggesting the publication.
The suggestion of exposing more details is valuable. We have introduced several detailed explanations and calculations into the present version of the manuscript. The summary of changes is contained in the List of Changes.
Also, the suggestion about exploring the deviations from the gapless point is important. We have added a new section (Sect. 5 titled Velocity perturbation of the $z = 2$ Lifshitz criticality) to the manuscript. This section contains a discussion of the deviation from the Lifshitz point.
Bests The authors