Silvio Franz, Cosimo Lupo, Flavio Nicoletti, Giorgio Parisi, Federico Ricci-Tersenghi
{"title":"Soft modes in vector spin glass models on sparse random graphs","authors":"Silvio Franz, Cosimo Lupo, Flavio Nicoletti, Giorgio Parisi, Federico Ricci-Tersenghi","doi":"arxiv-2409.10312","DOIUrl":null,"url":null,"abstract":"We study numerically the Hessian of low-lying minima of vector spin glass\nmodels defined on random regular graphs. We consider the two-component (XY) and\nthree-component (Heisenberg) spin glasses at zero temperature, subjected to the\naction of a randomly oriented external field. Varying the intensity of the\nexternal field, these models undergo a zero temperature phase transition from a\nparamagnet at high field to a spin glass at low field. We study how the\nspectral properties of the Hessian depend on the magnetic field. In particular,\nwe study the shape of the spectrum at low frequency and the localization\nproperties of low energy eigenvectors across the transition. We find that in\nboth phases the edge of the spectral density behaves as $\\lambda^{3/2}$: such a\nbehavior rules out the presence of a diverging spin-glass susceptibility\n$\\chi_{SG}=\\langle 1/\\lambda^2 \\rangle$. As to low energy eigenvectors, we find\nthat the softest eigenmodes are always localized in both phases of the two\nmodels. However, by studying in detail the geometry of low energy eigenmodes\nacross different energy scales close to the lower edge of the spectrum, we find\na different behavior for the two models at the transition: in the XY case, low\nenergy modes are typically localized; at variance, in the Heisenberg case\nlow-energy eigenmodes with a multi-modal structure (sort of ``delocalization'')\nappear at an energy scale that vanishes in the infinite size limit. These\ngeometrically non-trivial excitations, which we call Concentrated and\nDelocalised Low Energy Modes (CDLEM), coexist with trivially localised\nexcitations: we interpret their existence as a sign of critical behavior\nrelated to the onset of the spin glass phase.","PeriodicalId":501066,"journal":{"name":"arXiv - PHYS - Disordered Systems and Neural Networks","volume":"31 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - PHYS - Disordered Systems and Neural Networks","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.10312","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We study numerically the Hessian of low-lying minima of vector spin glass
models defined on random regular graphs. We consider the two-component (XY) and
three-component (Heisenberg) spin glasses at zero temperature, subjected to the
action of a randomly oriented external field. Varying the intensity of the
external field, these models undergo a zero temperature phase transition from a
paramagnet at high field to a spin glass at low field. We study how the
spectral properties of the Hessian depend on the magnetic field. In particular,
we study the shape of the spectrum at low frequency and the localization
properties of low energy eigenvectors across the transition. We find that in
both phases the edge of the spectral density behaves as $\lambda^{3/2}$: such a
behavior rules out the presence of a diverging spin-glass susceptibility
$\chi_{SG}=\langle 1/\lambda^2 \rangle$. As to low energy eigenvectors, we find
that the softest eigenmodes are always localized in both phases of the two
models. However, by studying in detail the geometry of low energy eigenmodes
across different energy scales close to the lower edge of the spectrum, we find
a different behavior for the two models at the transition: in the XY case, low
energy modes are typically localized; at variance, in the Heisenberg case
low-energy eigenmodes with a multi-modal structure (sort of ``delocalization'')
appear at an energy scale that vanishes in the infinite size limit. These
geometrically non-trivial excitations, which we call Concentrated and
Delocalised Low Energy Modes (CDLEM), coexist with trivially localised
excitations: we interpret their existence as a sign of critical behavior
related to the onset of the spin glass phase.