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Asymmetric butterfly velocities in 2-local Hamiltonians

by Yong-Liang Zhang, Vedika Khemani

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Submission summary

Authors (as registered SciPost users): Yongliang Zhang
Submission information
Preprint Link: scipost_201912_00045v4  (pdf)
Date accepted: 2020-07-29
Date submitted: 2020-07-13 02:00
Submitted by: Zhang, Yongliang
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approaches: Theoretical, Computational


The speed of information propagation is finite in quantum systems with local interactions. In many such systems, local operators spread ballistically in time and can be characterized by a ``butterfly velocity", which can be measured via out-of-time-ordered correlation functions. In general, the butterfly velocity can depend asymmetrically on the direction of information propagation. In this work, we construct a family of simple 2-local Hamiltonians for understanding the asymmetric hydrodynamics of operator spreading. Our models live on a one dimensional lattice and exhibit asymmetric butterfly velocities between the left and right spatial directions. This asymmetry is transparently understood in a free (non-interacting) limit of our model Hamiltonians, where the butterfly speed can be understood in terms of quasiparticle velocities.

Author comments upon resubmission

Dear Editor,

Thank you and the referees for reviewing our manuscript "Asymmetric butterfly velocities in 2-local Hamiltonians" (scipost_201912_00045v1). We are grateful for the referees’ comments and suggestions to improve our manuscript. We have replied to the referee reports using SciPost comments, and would like to resubmit our improved manuscript to SciPost Physics as a regular article. Thank you very much for your consideration.

Sincerely yours,
Yong-Liang Zhang and Vedika Khemani

List of changes

(1) We have slightly edited the introduction and conclusion to better emphasize the main results of the work and how it adds to previous results on asymmetric speeds in the literature, including correcting the misconception that asymmetric speeds require anyonic statistics.

(2) We previously presented three different methods for computing butterfly speeds, leading the first referee to conclude that this was a central part of our paper. In fact, these different methods are not essential for illustrating our central result of presenting simple and solvable models with asymmetric butterfly speeds, and we agree with the referee that the large amount of space devoted to the different methods distracts from our central message. To address this, we have removed the last method which was subject to the largest amount of numerical uncertainty and did not add any qualitatively new insight.

(3) In the caption of Fig. (1), we have removed “h_z=0.5”. Also, we have fixed the typos that the referees pointed out. In Fig. (2), we have enlarged the stars to increase visibility.

(4) Below Eq. (3) and (5)(6), we have added sentences to explain that for the lambda=0 case, the v_B is symmetric despite the lack of inversion symmetry.

(5) Under Fig. (4), we have provided parameters for TEBD simulations.

(6) In Fig. (7), we have changed our figures and the fitting results using the peaks because of the broadening of the wavefront. Corresponding estimation results have been changed, and show better agreement between the two methods of computing butterfly speeds.

(7) In Appendix A, we have explicitly written the real-space free-fermionic version of the Hamiltonian Eq. (2). In addition, we have plotted a figure to demonstrate the nearest and next-nearest neighbor hopping terms.

Published as SciPost Phys. 9, 024 (2020)

Reports on this Submission

Anonymous Report 2 on 2020-7-22 (Invited Report)


The authors have addressed all the questions raised in my previous report. In particular, I am glad to see that the two methods of estimating the butterfly velocities now agree to a good precision.

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Anonymous Report 1 on 2020-7-13 (Invited Report)


The authors have adequately addressed all my comments.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
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