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Delocalization transition in low energy excitation modes of vector spin glasses

by Silvio Franz, Flavio Nicoletti, Giorgio Parisi and Federico Ricci-Tersenghi

This is not the latest submitted version.

This Submission thread is now published as SciPost Phys. 12, 016 (2022)

Submission summary

As Contributors: Silvio Franz · Flavio Nicoletti · Federico Ricci-Tersenghi
Preprint link: scipost_202109_00008v1
Date submitted: 2021-09-06 15:47
Submitted by: Nicoletti, Flavio
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Statistical and Soft Matter Physics
Approaches: Theoretical, Computational


We study the energy minima of the fully-connected $m$-components vector spin glass model at zero temperature in an external magnetic field for $m\ge 3$. The model has a zero temperature transition from a paramagnetic phase at high field to a spin glass phase at low field. We study the eigenvalues and eigenvectors of the Hessian in the minima of the Hamiltonian. The spectrum is gapless both in the paramagnetic and in the spin glass phase, with a pseudo-gap behaving as $\lambda^{m-1}$ in the paramagnetic phase and as $\sqrt{\lambda}$ in the spin glass phase. Despite the long-range nature of the model, the eigenstates close to the edge of the spectrum display quasi-localization properties. We show that the paramagnetic to spin glass transition corresponds to delocalization of the edge eigenvectors. We solve the model by the cavity method in the thermodynamic limit. We also perform numerical minimization of the Hamiltonian for $N\le 2048$ and compute the spectral properties, that show very strong corrections to the asymptotic scaling approaching the critical point.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2021-10-2 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202109_00008v1, delivered 2021-10-02, doi: 10.21468/SciPost.Report.3609


1) New exact results in the thermodynamic limit about a well known spin-glass model.

2) Accurate numerical results and insights into finite size corrections.


In this paper the fully-connected $m$-component vector spin glass model in an external random field is considered and its zero-temperature properties are studied for $m \ge 3$. These models have been studied for more than forty years and are known to exhibit a zero temperature phase transition from a paramagnetic phase at high field and a spin glass phase at low field. In this paper the paramagnetic phase and the critical point are examined most closely, the energy minima in these cases being unique and isolated. In particular, they investigated the eigenvalues ​​and eigenvectors of the Hessian in the minima of the Hamilton operator. With the cavity method correct in the thermodynamic Limes, the authors obtained several interesting results. In particular they studied the eigenvalues and the eigenvectors of the Hessian in the minima of the Hamiltonian. Using the cavity method, which is correct in the thermodynamic limit, the authors obtained several interesting results. The spectrum of the Hessian is found gapless, with different type of pseudo-gaps in the paramagnetic phase and in the critical point. The eigenstates close to the edge of the spectrum show quasi-localisation behaviour. They also found the distribution of the cavity fields and the distribution of the smallest one. The analytical results are confronted with the results of numerical calculations and very strong finite-size corrections are found close to the critical point.

The subject of the paper is interesting, the results obtained could give new impulses to further clarify the behavior of this type of model. The paper is generally well written. I recommend publishing this work. Below I list a few points that the authors should consider.

1) In the abstract the behaviour of the pseudo-gap in the spin glass phase is explicitly written. As far as I can check this is just a conjecture (see the Discussion). I suggest to change this sentence.

2) The crossover from quasi-localised modes to extended modes is studied in Appendix B. Can this result be illustrated on the numerical results in Fig. 4?

3) The distribution of the smallest cavity field is found the Weibull distribution in Eq.(24), which is valid for independent and identically distributed random variables (iidrv). Could the authors comment on this finding?

4) A related question: for iidrv the finite-size corrections to the asymptotic results are known (see: Phys. Rev. E. 81, 041135 (2010)) and recently applied for interacting random systems (see: Phys. Rev. Res. 3, 033140 (2021)). Would it be possible to perform a similar analysis with the data in Fig.6 ?

5) A minor point, a few typos should be fixed. (ipershere -> hypershere; form -> from; be discuss -> be discussed) Also the abbreviation 1RSB-RFOT should be clarified.

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

Anonymous Report 1 on 2021-9-8 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202109_00008v1, delivered 2021-09-08, doi: 10.21468/SciPost.Report.3507


This is a nice thorough paper. The authors have found interesting results in an already well-studied topic. What is surprising is that quasi-localized states exist despite the long-range form of the interactions in the Sherrington-Kirkpatrick (SK) model. This is most surprising.

I expect that there will now be follow-up studies to ascertain the extent to which the results in the SK model extend to sparse graphs and finite dimensional lattices.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Author:  Flavio Nicoletti  on 2021-09-08  [id 1745]

(in reply to Report 1 on 2021-09-08)

Dear reporter, we plan to extend our studies of quasi-localization properties to the same kind of model (m-component spins with random quenched disorder interactions and random quenched external field) on sparse graphs in the forthcoming year. Many thanks for your report!

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