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Many-body localization in a quasiperiodic Fibonacci chain

by Nicolas Macé, Nicolas Laflorencie, Fabien Alet

Submission summary

As Contributors: Fabien Alet · Nicolas Macé
Arxiv Link:
Date submitted: 2018-12-05
Submitted by: Macé, Nicolas
Submitted to: SciPost Physics
Domain(s): Theor. & Comp.
Subject area: Quantum Physics


We study the many-body localization (MBL) properties of a chain of interacting fermions subject to a quasiperiodic potential such that the non-interacting chain is always delocalized and displays multifractality. Contrary to naive expectations, adding interactions in this systems does not enhance delocalization, and a MBL transition is observed. Due to the local properties of the quasiperiodic potential, the MBL phase presents specific features. We furthermore investigate the fate of multifractality in the ergodic phase for low potential values. Our analysis is based on exact numerical studies of eigenstates and dynamical properties after a quench.

Current status:
Editor-in-charge assigned

Submission & Refereeing History

Submission 1811.01912v2 (5 December 2018)

Invited Reports on this Submission

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Anonymous Report 1 on 2018-12-21


1 - An interesting study of a model with some unique properties that distinguish it from others typically studied in the context of MBL.
2 - Comprehensive numerical study that covers many different quantities, with convincing arguments backing up the data.
3 - Likely to stimulate further interest in Fibonacci potentials in the MBL community.


1 - As a result of the large number of different quantities computed by the authors, many are introduced briefly and dispensed with very quickly, making it occasionally hard to follow.
2 - Though the results are solid and interesting, the manuscript would benefit from a stronger punchline as to the significance of the results.


This work takes a novel approach to the study of many-body localization by examining a model which is delocalized (and critical) in the non-interacting limit, and becomes localized only when interactions are switched on.

The authors consider a model of interacting fermions in one dimension subject to a quasiperiodic on-site potential of the Fibonacci form. This potential is balanced on a delicate knife-edge, being neither disordered enough to localize the non-interacting system nor smooth enough to entirely delocalize it: this results in the various interesting features of the non-interacting model that are well summarised by the authors.

Using a simple yet persuasive argument, the authors demonstrate that the addition of interactions acts to ‘smooth’ the Fibonacci potential, disrupting this delicate balance. With numerical simulations, the authors go on to demonstrate that the interacting system appears to follow typical MBL phenomenology, i.e. the system is delocalized at small disorder but there is a many-body localization transition at some critical disorder strength. The authors conduct a comprehensive numerical study and present various different quantities which corroborate this claim. As a consequence of the impressive variety of different quantities that the authors consider, some are presented in a rather brief way that relies on readers being already familiar with MBL phenomenology: some small improvements to the discussion would make the manuscript more accessible to a broader audience.

In summary, the authors have given convincing evidence for MBL phenomenology in this model and reviewed the novel features of the Fibonacci potential in a way that serves as an adept introduction of this model to an audience already familiar with MBL. This work likely to stimulate further interest in models with Fibonacci potentials in the MBL community.

I have a variety of minor comments, questions and suggestions which are included in this report. Based on a satisfactory response to these points, I’d be happy to recommend publication in SciPost Physics.



- The perturbative argument given for the ‘smoothing’ of the Fibonacci potential by interactions is plausible and intuitive. This suggests that, in the language of renormalization group, the non-interacting system is controlled by a fixed point unstable to the addition of interactions. The authors cite a variety of RG works which have examined the low-energy properties of the model, but as a future work, it would be interesting to see if RG methods could examine this model in the context of many-body localization.

- In Fig. 10, the bar over the I(t) suggests some form of averaging was used, but this isn’t mentioned anywhere in the text - was this data averaged in some way?

- In Fig. 12, why is the maximum time different for the two largest disorder strengths?

- In future directions, the authors remark that the fate of the transition in the non-interacting limit presents a technical challenge – what precisely is the challenge that this poses?

Requested changes

1 - The current statement in the abstract that ‘the MBL phase presents specific features’ is a little vague. Could the authors say something more definite here, perhaps giving an example of one?

2 - Can the authors clarify what they mean by the sentence “Finally, discrete disorder does not seem to affect the transition as compared to continuous random potential distributions” (p2)?

3 - Can the authors provide some further argument to clarify why there are L+1 Fibonacci words of length L? Currently this is stated without proof (p4).

4 - The authors comment on ‘large statistical errors’ (p4) as a result of the small number of samples available, due to the intrinsic nature of the model. Can the authors provide any further comment on which features of their study may be most strongly affected by these statistical errors?

5 - The existence of secondary peaks in the density (Fig. 5) is interesting and the explanation is ultimately convincing, though this section is a little convoluted to read. The left panel of Fig. 6, however, clearly demonstrates that the secondary peaks are associated with configurations where neighbouring sites have the same sign: I’d suggest leading with this observation before introducing the magic angle states.

6 - Section 5.3 covers a lot of concepts not previously introduced in the paper, and does so very quickly. A slightly expanded discussion of how multifractality can be quantified using the participation entropy would make this section easier to follow, particularly for those unfamiliar with these quantities.

7 - The use of a logarithmic fit to the growth of entanglement entropy in the ETH phase is explained in a confusing way. Although it makes sense after a few reads, can the authors find a clearer way to explain this?

A few typos I spotted:

1) p2: ‘random potentials distributions’ -> ‘random potential distributions’
2) p2: ‘recasted’ -> ‘recast’
3) p3: ‘eigenstates properties’ -> ‘eigenstate properties’
4) p5: ‘free fermions Fibonacci chain’ -> ‘free fermion Fibonacci chain’
5) p10: ‘were’ -> ‘where’
6) p17: ‘quench disordered systems’ -> ‘systems with quenched disorder’ would read better.
7) p7, caption of Fig. 3: the authors say the colour coding is ‘similar’ in both panels, but unless there is some difference, do they mean ‘the same’?
8) p18: ‘disordered strength’ -> ‘disorder strength’

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

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