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Anisotropy-mediated reentrant localization

by Xiaolong Deng, Alexander L. Burin, Ivan M. Khaymovich

Submission summary

As Contributors: Ivan Khaymovich
Arxiv Link: (pdf)
Date submitted: 2021-07-21 08:08
Submitted by: Khaymovich, Ivan
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
  • Quantum Physics
Approaches: Theoretical, Computational


We consider a 2d dipolar system, $d=2$, with the generalized dipole-dipole interaction $\sim r^{-a}$, with the power $a$ controlled experimentally in trapped-ion or Rydberg-atom systems via their interaction with cavity modes. We focus on the dilute dipolar excitation case when the problem can be effectively considered as single-particle with the interaction providing long-range dipolar-like hopping. % We show that the spatially homogeneous tilt $\beta$ of the dipoles giving rise to the anisotropic dipole exchange leads to the non-trivial reentrant localization beyond the locator expansion, $a<d$, unlike the models with random dipole orientation. The Anderson transitions are found to occur at the finite values of the tilt parameter $\beta = a$, $0<a<d$, and $\beta = a/(a-d/2)$, $d/2<a<d$, showing the robustness of the localization at small and large anisotropy values. % Both extensive numerical calculations and analytical methods show power-law localized eigenstates in the bulk of the spectrum, obeying recently discovered duality $a\leftrightarrow 2d-a$ of their spatial decay rate, on the localized side of the transition, $a>a_{AT}$. This localization emerges due to the presence of the ergodic extended states at either spectral edge, which constitute a zero fraction of states in the thermodynamic limit, decaying though extremely slowly with the system size.

Current status:
Editor-in-charge assigned

Submission & Refereeing History

Reports on this Submission

Anonymous Report 2 on 2021-11-27 (Invited Report)


1. Addresses an interesting topic using appropriate methods.


1. The setup is a little artificial. The angle-dependence makes sense for the regular dipolar interaction, but for the generalized dipolar interaction (realized using ion traps or cavities) there is no reason to expect this angle-dependence.


I think this is a worthwhile addition to the literature. It builds on some previous results of these authors, showing that for very long-range interactions localization of typical states can sometimes be protected essentially by the way low-momentum states 'detach themselves' from the bulk of the spectrum. Long-distance interactions between wavepackets are mediated by these low-momentum states and are therefore suppressed relative to a naive locator expansion. Whether this phenomenon happens or not depends on the sign structure of the long-range interaction, and the dipolar "angle" is a way of tuning this sign structure.

I find this work conceptually quite interesting although from an experimental point of view the setup is highly artificial. But perhaps the authors can comment a little more on what they expect to happen with real dipoles in 3D.

Requested changes

1. Be more explicit about what is meant by the generalized dipolar interaction.
2. Add a little more detail from the appendices into the main text on the RG section.

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

Anonymous Report 1 on 2021-9-25 (Invited Report)


1 - 2D Anderson transition for the effective long-range tunneling model coming from diluted tilted dipoles is studied apparently showing an interesting reentrant transition
2 - Extensive numerical data support the analysis
3 - The numerics is supported by renormalization group analysis


1 - The presentation style and the construction of the paper makes it difficult to follow - see report
2 - Some of numerical results are poorly described with insufficient details so the reader may find it difficult to reproduce the data
3 - Are all the data shown necessary to reach the conclusions of the paper?
4 - Interpretation of the numerics seems controversial


Scientifically the paper seems, at first glance, sound and gives a comprehensive analysis of Anderson localization and spectral properties in an interesting and physically realisable model. The style of the paper makes it, however, extremely difficult to read and understand. Section 2 is entitled "Models and methods", section 3 "methods". What is the difference between methods in 2 and 3? The main body of the paper is rather short and most of the results are hidden in Appendixes which makes the reader jump back and forth unnecessarily. For example Fig.4 in the main text shows the spectrum of fractal dimensions, $f(\alpha)$, (p.9) which is not defined in the main text but is mentioned in A.2 pp. 18-19. Similarly fractal dimension $D_2$ illustrates multifractality in Fig.2 (p.5) with definition again hidden in Appendixes. In the present form the manuscript is unsuitable for publication in Scipost mainly because of the form of the presentation. Also the scientific content which seemingly is sufficiently high requires reconsideration, see requested changes and the comments below.

Let me exemplify the problem with the analysis of level spacing ratio (gap ratio) statistics. For systems with energy dependent mobility edge the safe, standard approach is to find the mean ratio, say $r$, in narrow energy intervals (see e.g. Luitz et al. 2015, PRB) averaging over different disorder realisations. It seems to be indeed done, somehow, in Fig.2b) where $r$ is plotted as a function of energy. While for $\beta=1,2$ $r$ seems not to be dependent on energy and shows signs of localized and extended phases, respectively, the intermediate $\beta=1.5$ case seems to show significant energy dependence. While in the caption we can read that data are for $L=250$, one cannot find out the width of energy window used for this plot (each yellow point for $\langle r\rangle$ corresponds, I believe, to some tiny energy interval). It is written that vertical dashed lines provide "the position of FSME extracted from finite-size data for r-statistics". But in fact the dashed lines point to the edge of the spectrum where, at low energies, density of states (DOS a) panel) significantly drops indicating the edge of the system. This cannot be the position of the mobility edge!.
It is not written in the caption which "r-statistics" is being considered. If this is the one shown in Fig.3 then it is performed badly. As written, energies from $[-W/2,W/2]$ interval, i.e. "95% of all the states" are taken for analysis. But this is simply wrong, as mean $r$ depends on energy, see Fig.2b) - the authors are again referred to e.g. Luitz et al. seminal paper.
The proper way of doing the analysis, let me repeat, would be to devide the energy interval $[-W/2,W/2]$ in which the bulk of the spectrum is contained into say 20 or 30 intervals and make the finite size analysis ala Fig.3 in each of the interval separately getting the critical $\beta(E)$ value depending on energy.
Authors are aware of the strong energy dependence of eigenstates, in particular in Fig.8 they study $f(\alpha)$ around energy $E=5$.
By the way, a similar analysis could be done for different disorder amplitude $W$ values as the manuscript considers a single choice $W=20$.
Panel 2c) shows the dimension $D_2$ which is both strongly energy dependent for $\beta=1.5$ (again - how yellow curve is obtained?). Its noninteger value suggests a fractal character. Multifractality requires, however, that $D_q$ change with $q$. The statement in the caption that panel b) and c) "show localized, multifractal and ergodic eigenstate properties in the spectral bulk" is thus wrong. In particular panel b) shows mean $r$ which is the eigenvalues property and is not linked with multifractality.
Appendices contain a number of results that are not used or referred to in the main text. I believe a selection should be made validating the claims of the paper (or showing its limitations). They are described sometimes not precisely. While, as mentioned above $f(\alpha)$ is analysed around $ E=5$ what about the following $D_q$ analysis? In Fig.9 a comparison with power-law random banded matrices is shown but is the power law assumed similar to that of the dipole model (1)? In the caption we read about 2d power-law random banded model - how 2d model is constructed? The referee (and [26] cited) knows only a standard PLRBM model where no dimensional structure is present. The precise definition of PLRBM used should be given in 2 lines explaining e.g. what is the sense of $W$ in PLRBM? Is energy dependence included in the analysis of IPR in Fig.10? Is energy dependence included in the higher order gap ratio statistics that appears in Sec.C of the Appendix and fig.15? Let me stop here asking the questions about the wealth of the material scattered throughout appendices.

Requested changes

1- I strongly recommend rewriting the paper is a style suitable for Scipost Physics. All notions and observables used in the main text must be properly defined before their use in the text and/or figures.
More detailed changes:
a/In the caption of Fig.1 it is written that the transition occurs at $r\simeq 0.47$ with $r$ defined in Eq.(4) while in Fig.2 or Fig.3 $\langle r\rangle$ is used to describe the vertical axis; please unify the notation. Strictly the level statistics is given as a set of $r_{n,1}$ values with the average $r$. This should be stated around (4).
b/it is not frequent to call an earlier paper of one of coauthors as a ``pioneering paper'' (p.4). Maybe the use of ``early'' instead of ``pioneering'' would sound better?
c/The level statistics ratio analysis should be performed in energy dependent style as described in the report.
d/Similar comments affect other measures used in the paper/appendix.
e/Authors should limit the numerical results presented in the appendices to a subset needed for the confirmation of the work claims and its conclusions.
f/For example what is the relevance of the "spectrum of hopping" (appendix D) as well as Ioffe-Regel criterion (F) for the content of the paper? Either discuss this issues and their relevance to the main body of the paper or remove them for clarity.

  • validity: poor
  • significance: good
  • originality: good
  • clarity: low
  • formatting: mediocre
  • grammar: excellent

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