# Delocalization transition in low energy excitation modes of vector spin glasses

### Submission summary

 As Contributors: Silvio Franz · Flavio Nicoletti · Federico Ricci-Tersenghi Preprint link: scipost_202109_00008v2 Date accepted: 2021-11-10 Date submitted: 2021-10-11 16:01 Submitted by: Nicoletti, Flavio Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Statistical and Soft Matter Physics Approaches: Theoretical, Computational

### Abstract

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 $\sqrt{\lambda}$ at criticality and 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.

Published as SciPost Phys. 12, 016 (2022)

Dear Editor,

Please find below the resubmittal letter of our paper “Delocalization transition in low energy excitation modes of vector spin glasses”

We thank both referees for the appreciation of our work. We have modified the paper according to the suggestions of the second referee:

"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."

We added details, in the main text and in the appendix, to our arguments to determine the behavior of the pseudo-gap in the spin glass phase. We hope that the referee finds these arguments convincing and allows us to keep the sentence in the introduction.

"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?"

We have marked the computed cross-over point on three of the curves in figure 4 where the cross-over is more visible. We changed the caption to explain that.

"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?"

The cavity fields are indeed iidv with algebraic distribution close to zero, this is the origin of the Weibull distribution of their minimum.

"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?"

We added a sentence to mention that one could analyse the finite size corrections of the distribution of the minimum field along the lines of Phys. Rev. E. 81, 041135 (2010) (the analysis in the case of the min eigenvalue would be more complicated).

"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."

We corrected the typos, hopefully all of them. We defined the acronymous RFOT → Random First Order transition, added a reference and avoided 1RSB.

In addition we included a comment about the similarity of Eigenvector Localization in our model and the Einstein condensation in the Bose gas.

Sincerely,
Silvio Franz, Flavio Nicoletti, Giorgio Parisi, Federico Ricci-Tersenghi

### List of changes

Abstract:

"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."

We added details, in the main text and in the appendix, to our arguments to determine the behavior of the pseudo-gap in the spin glass phase. We hope that the referee finds these arguments convincing and allows us to keep the sentence in the introduction.

Page 8, fig. 4:

"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?"

We have marked the computed cross-over point on three of the curves in figure 4 where the cross-over is more visible. We changed the caption to explain that.

Page 8, fig. 6:

"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?"

We added a sentence to mention that one could analyse the finite size corrections of the distribution of the minimum field along the lines of Phys. Rev. E. 81, 041135 (2010) (the analysis in the case of the min eigenvalue would be more complicated).

Page 4:

In addition we included a comment about the similarity of Eigenvector Localization in our model and the Einstein condensation in the Bose gas.

Page 15:

We included the expression for the crossover at m=3.