# Universal finite-size amplitude and anomalous entangment entropy of $z=2$ quantum Lifshitz criticalities in topological chains

### Submission summary

 As Contributors: Tigran Sedrakyan · Ke Wang Preprint link: scipost_202111_00006v2 Date accepted: 2022-03-29 Date submitted: 2022-03-17 16:33 Submitted by: Wang, Ke Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Quantum Physics 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.

Published as SciPost Phys. 12, 134 (2022)

### Author comments upon resubmission

Dear Editor

Thank you for forwarding us the revision request on Feb. 8, 2022. We are grateful to both Referees for constructive reports and strongly supporting publication. We have carefully considered all the comments and questions raised by Referees.

Our detailed response to the comments and questions raised by the Referees is presented below. The Referee recommendations are primarily targeting the improvement of the presentation and the inclusion of more technical details. We address this point in the present version of the manuscript, where more supporting and technical materials are added. The additions include more detailed explanations in the main part of the manuscript, one additional section on perturbations around the criticality, and two new appendices.

Having met Referee recommendations, we kindly request further consideration of publication in the SciPost Physics.

Sincerely,

The authors.

Response to Referee. II

1. " 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 ground state 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."

Response: We thank the referee for the concise description of the work and suggesting the publication.

1. "As a suggestion (in agreement with the first referee): the manuscript can benefit from language proof-reading and exposing more details of the calculations."

Response: We have introduced several detailed explanations and calculations into the present version of the manuscript. See List of changes.

1. "Authors thought about deriving scaling function, covering deviations from the gapless point would be a welcome addition too"

Response: We thank the reviewer for this suggestion. A new section (Sect. number 5 entitled Velocity perturbation of the $z = 2$ Lifshitz criticality) is added to the manuscript. This section contains the discission of the deviation from the Lifshitz point.

Response to Referee I. 1. " 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 $\sim 1/L^2$ and $L$ is the system size, while the entropy exhibits a non-logarithmic behaviour, 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. "

Response: We thank the referee for the detailed description of the work, finding that the paper is well-written, and suggesting the publication.

2." Indeed, I would suggest to add more details and explanations such that the work can be self-consistent and more pedagogical. "

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

1. "- 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); "

Response: Typos are corrected.

• 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 ?

Response: 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.

• Pag. 6 after Eq. (6): "$a$ is the lattice space": where does $a$ enter in the text?
1. Pag. 6: "entanglment" $\rightarrow$ entanglement; }

Response: 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.

1. "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"

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

1. "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)."

Response: We thank the Referee for the valuable information. We have introduced the corresponding discussion and included the important reference.

1. " 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."

Response: We thank the Referee for pointing to this interesting question. However unfortunately, 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.

### List of changes

1. A comment on the derivation of the boundary condition is added above Eq.~7 in the manuscript.

2. A detailed description of the passage from the correlation function to the entanglement spectrum is added below Eq.~10.

3. A new section titled " 5: Velocity perturbation of the $z = 2$ Lifshitz criticality" is added.

4. A new Appendix titled "A Derivations of the finite-size amplitude" is added.

5. A new Appendix titled "Appendix B Derivations of the correlation function and the entanglement entropy" is added.