Abstract
We study the energy minima of the fully-connected mm-components vector spin glass model at zero temperature in an external magnetic field for m\ge 3m≥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}λm−1 in the paramagnetic phase and as \sqrt{\lambda}λ 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 2048N≤2048 and compute the spectral properties, that show very strong corrections to the asymptotic scaling approaching the critical point.
Highlights
B Analytic derivation of the lower band edge spectrum at the critical point and in the spin glass phase
We study the eigenvectors, whose inverse participation ratio turn out to be proportional to 1/N with a prefactor that diverges on the edges of the spectrum as λ−2(m−1), revealing non-trivial localization properties even in the paramagnetic phase of fully-connected mean-field spin glass models, induced by the random external field
We clearly see the deviation from the bulk law for the lowest eigenstates. It remains to understand whether the lowest eigenvectors, i.e. the eigenvectors corresponding to the lowest eigenvalues, are localized and to what extent in the paramagnetic phase, and more importantly what happens approaching the critical point
Summary
We review some well known properties of the fully-connected m-component vector spin glass at zero temperature in an external random field. I< j i where the N spins Si are m-components vectors, normalized to |Si| = 1, and the couplings Ji j (with i < j) are Gaussian independent and identically distributed random variables (iidrv) with Ji j = 0 and Ji2j = 1/N. We choose the external fields to be iidrv with Gaussian distribution of zero mean and variance (biα)2 = ∆2. Eq(2) expresses the fact that in minima the spins are oriented along their local fields This set of equations can be analysed with the cavity method. The variables Ji j are independent from S(ji), so that the cavity fields hi = j Ji jS(ji) are Gaussian random variables with zero mean and covariance matrix hαi hβj 1 m δi j δαβ.
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call