## SciPost Submission Page

# Renormalization to localization without a small parameter

### by A. G. Kutlin, I. M. Khaymovich

### Submission summary

As Contributors: | Ivan Khaymovich · Anton Kutlin |

Arxiv Link: | https://arxiv.org/abs/2001.06493v1 |

Date submitted: | 2020-01-23 |

Submitted by: | Kutlin, Anton |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | Quantum Physics |

Approaches: | Theoretical, Computational |

### Abstract

We study the wave function localization properties in a d-dimensional model of randomly spaced particles with isotropic hopping potential depending solely on Euclidean interparticle distances. Due to the generality of this model usually called the Euclidean random matrix model, it arises naturally in various physical contexts such as studies of vibrational modes, artificial atomic systems, liquids and glasses, ultracold gases, and photon localization phenomena. We generalize the known Burin-Levitov renormalization group approach, formulate universal conditions sufficient for localization in such models and inspect a striking equivalence of the wave function spatial decay between Euclidean random matrices and translation-invariant long-range lattice models with a diagonal disorder.

###### Current status:

### Submission & Refereeing History

## Reports on this Submission

### Anonymous Report 2 on 2020-2-18 Invited Report

### Strengths

I recommend the publication of this very interesting article.

### Weaknesses

The text refers to many previous works but it is sometimes too allusive.

Two Examples :

a) the 'matrix-inversion trick [19]' is mentioned several times (p3, p10,p16) but is never explained.

b) before Eq 22 : "the generalization of the duality [15,19]" deserves some explanation about what precise duality is meant here.

So I feel that the paper would be much more interesting for a broader audience if it were more self-contained and could be read without having to look at too many previous references. In addition, the authors should state more clearly what is similar and what is different from the previous works.

### Report

Here are some comments that the authors might consider in order to improve their manuscript :

1) The authors use "renormalization time" (p3) , " natural renormalization time variable" (before Eq 19), etc… while it is related to the size R :

I think that the authors should avoid the word 'time' (since nowadays there are actually many works on the dynamics in localized systems) and use some more accurate name like 'RG scale' or equivalent.

2) I think that the arguments given in the numerous long notes [37-38-39-40],[42-43] should be included in the main text instead of being buried among citations.

### Requested changes

English : the last sentence of the Introduction (Part 1) starting with " this significantly reduces …" is not clear and should be rephrased.

### Report 1 by Alexander Burin on 2020-2-4 Invited Report

### Strengths

This is very interesting work resolving the problem of Anderson localization in the presence of the long-range hopping of constant sign. The authors has performed a combination of analytical and numerical studies and characterized localized and delocalized states. Numerics agrees with analytics so the results can be extrapolated to numrically inaccessible regimes

### Weaknesses

Fig. 7 is confusing. I cannot find any demonstration of the predicted dependence of the maximum charge on energy. I think the graph needs major improvement to make it clear for readers.

I would also suggest to add more numerical characterization of ejgen-states including inverse participation ratio and level statistics to give the reader more knowledge about the system behavior.

### Report

As I noticed in my opinion this is very interesting work that deserves publication after the weaknesses will be addressed.

### Requested changes

I would recommend to improve Fig. 7 and add graphs for inverse participation ratio and level statistics.