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Universal finite-size 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 | |
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| Preprint Link: | scipost_202111_00006v1 (pdf) |
| Date submitted: | 2021-11-03 18:40 |
| Submitted by: | Wang, Ke |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| 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 symmetry-enriched conformal field theories (CFTs). We report that, at such critical points in one spatial dimension, the finite-size 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 non-logarithmic dependence on $l/L$, where $l$ is the length of the sub-system. In the limit of $l\ll L$, the maximally-entangled ground state has the entropy, $S(l/L)=S_0+(l/L)\log(l/L)$. Here $S_0$ is some non-universal entropy originating from short-range correlations. We show that the novel entanglement originates from the long-range correlation mediated by a zero mode in the low energy sector. The work paves the way to study finite-size effects and entanglement entropy around Lifshitz criticalities and offers an insight into transitions between symmetry-enriched criticalities.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 1) on 2022-2-8 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202111_00006v1, delivered 2022-02-08, 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 low-energy 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 proof-reading 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 2) on 2021-12-8 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202111_00006v1, delivered 2021-12-08, doi: 10.21468/SciPost.Report.4016
Strengths
Interesting and universal results concerning non-conformal 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 non-CFT criticalities, the authors try to investigate some universal features which can emerge in such one-dimensional systems. In particular, they focus on the finite-size 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 non-logarithmic behaviour, $l/L\log(l/L)$, with $l$ the subsystem size, due to the presence of zero-modes 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 Su-Schrieffer–Heeger (SSH) model.
The paper is well-written 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 self-consistent 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 finite-size 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 non-trivial value of the entanglement espectrum, $\epsilon_0$;
- "Zero-modes 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 non-relativistic 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.
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We thank the Referee for the detailed description of the work, finding that the paper is well-written, and suggesting the publication.
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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.
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Typos in the title and around Eqs. 5, 6 are corrected. Some other typos are also corrected.
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We discuss the details of the numerical method to estimate the finite-size effects in the present version (particularly in Sect 3). We also added Ref. 36, discussing some specific details.
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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.
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We have added the details about the steps discussing the calculation of the entanglement spectrum from the correlation function.
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The information on the paper J.Stat.Mech.0512:P12012,2005 is valuable. We have introduced the corresponding discussion and included the important reference.
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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 2022-03-18 [id 2298]
(in reply to Report 2 on 2022-02-08)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