# Asymmetric butterfly velocities in 2-local Hamiltonians

### Submission summary

 As Contributors: Yong-Liang Zhang Preprint link: scipost_201912_00045v1 Date submitted: 2019-12-29 Submitted by: Zhang, Yong-Liang Submitted to: SciPost Physics Discipline: Physics Subject area: Quantum Physics Approaches: Theoretical, Computational

### Abstract

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.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission scipost_201912_00045v1 on 29 December 2019

## Reports on this Submission

### Strengths

1) Well written
2) Relevant topic

### Weaknesses

1) Results are a bit elementary

### Report

In this work, the authors demonstrate the asymmetry in the speed of local operator spreading (or butterfly velocity, vB) in a particular 1D theory. By mapping this theory to a free Hamiltonian they compute the OTOC, hence vB, exactly. Using three different complementary methods they obtain similar conclusions in a related non-integrable theory.

The paper is clearly written and presents original results. Therefore, I recommend the paper for publication after the following comments are addressed:

1) A subjective comment is — at times the paper reads like a follow up of past works of some of the authors. May be a more self-contained discussion of the various (borrowed) aspects might improve the readability of this paper.

2) lambda = 0, 1 case : For the lambda=0 case the vB is symmetric (Fig 1) despite the lack of inversion symmetry, is it obvious why?

3) It might be interesting to understand the role (and physics) of the constants h_ij or h_k in controlling the butterfly velocity.

4) Could the authors briefly comment on whether any of the features they discuss are in any sense restricted to the (a) dimensionality (1D), or (b) zero temperature solutions of the problem.

### Requested changes

There are few small typos that I came across: ‘Jordan Wigner’ after eq. 2 has missing hyphen (also in App. A); ‘about center’ in the same paragraph, probably has missing article; Figure 2 uses ‘upper panel’ which is absent; In the conclusion ‘may also interesting..’ has missing verb; same paragraph has ‘subbalistic’, missing hyphen.

• validity: good
• significance: good
• originality: high
• clarity: high
• formatting: excellent
• grammar: good

### Strengths

1- Construction of a simple solvable model with asymmetric light cones
2- Comparison of various numerical methods
3- Clear presentation

### Weaknesses

1- Lack of a more careful assessment of the strengths/weaknesses of each method
2- Somewhat limited scope of the results

### Report

The authors of this manuscript study the transport of quantum information, as characterized by the spreading of local operators, in one-dimensional systems where inversion symmetry is explicitly broken. In this case, operator spreading can be asymmetric between left and right, characterized by two different 'butterfly' velocities.. They first construct a model that is exactly solvable by a Jordan-Wigner mapping to free fermions, where the two velocities can be analytically calculated, and are shown to be different generically. They then go on to numerically study a non-integrable extension of the model, extracting both velocities using three separate methods (velocity-dependent Lyapunov exponents, operator weights and operator entanglement).

The paper is clearly written and the results presented are sound, albeit of somewhat limited interest - asymmetric light cones were studied previously in the literature, as the authors note in their introduction. I believe it would eventually be suitable for publication in SciPost, but only after some modifications and extensions. In particular, the three numerical methods used to extract v_l,r give significantly different results, making it difficult to discern what the actual butterfly velocities are. Since the comparison of these various methods is a major part of the paper, a more careful assessment of their particular merits seems necessary. In particular, since the authors present a solvable model, it would be quite natural to benchmark all three methods on this test case, to evaluate the accuracy of each. I believe, doing so would significantly strengthen the paper.

- I think it would be useful to write explicitly the real-space free fermionic version of the Hamiltonian Eq. (2) in the appendix, it would help with developing some intuition for this model.
- The bond dimensions used for TEBD simulations should be stated somewhere. Especially, since the maximal times obtained (t=20) are in fact rather large, and it seems like at this point there is already a quite significant amount of operator entanglement (as shown by Fig. 8)
- In their evaluation of v_l,r from operator weights, presented in Fig 7, it is unclear to me why the authors use the half of the peak, rather than the peak itself, as their reference point. Since the wavefront broadens in time, as they note in the introduction, it seems like this could affect the velocity they extract?
- In the caption of Fig. 1, when stating the parameters of the model used, say "h_z = 0.5". I believe this is a typo, as these results are supposed to come from the integrable case, which has h_z = 0 (indeed, the Hamiltonian Eq. (2) refered to in the same caption does not have h_z among its parameters)
- In the 3rd sentence of the caption of Fig. 2, the first plot is referred to as the 'upper' panel: it should be 'left'. Also, in the same figure, I would suggest making the stars that denote the points corresponding to v_l,r larger to be more visible.
- Some typos: "spacial and "one can get the growth function is" (should be "as", presumably).

### Requested changes

1- Test methods on solvable model
2- Provide more detail on TEBD simulations (e.g. bond dimensions used, truncation errors, etc)
3- Explain why half peak is used instead of the peak in Fig. 7

• validity: high
• significance: ok
• originality: ok
• clarity: high
• formatting: excellent
• grammar: excellent