Soft modes in vector spin glass models on sparse random graphs

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 ${\chi }_{SG}=⟨1/{\lambda }^2⟩$. 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.

15 pages, 8 figures

References in corpus (53)

• Localization of interacting fermions at high temperature
• The distribution of the ratio of consecutive level spacings in random matrix ensembles
• The Bethe lattice spin glass revisited
• Anharmonicity, vibrational instability and Boson peak in glasses
• Continuum limit of the vibrational properties of amorphous solids
• Statistics and properties of low-frequency vibrational modes in structural glasses
• Universal Spectrum of Normal Modes in Low-Temperature Glasses: an Exact Solution
• Low-frequency vibrational modes of stable glasses
• On the stationary points of the TAP free energy
• Bosonic Excitations in Random Media
• Universality of the nonphononic vibrational spectrum across different classes of computer glasses
• Two critical localization lengths in the Anderson transition on random graphs
• Anomalous vibrational properties in the continuum limit of glasses
• Effect of instantaneous and continuous quenches on the density of vibrational modes in model glasses
• Probing the non-Debye low frequency excitations in glasses through random pinning
• Sound attenuation in stable glasses
• Soft Modes, Localization and Two-Level Systems in Spin Glasses
• Critical properties of the Anderson transition in random graphs: two-parameter scaling theory, Kosterlitz-Thouless type flow and many-body localization
• Cavity and replica methods for the spectral density of sparse symmetric random matrices
• Low-frequency vibrational spectrum of mean-field disordered systems
• Random Matrix Approach to Glassy Physics --- Low Temperatures and Beyond
• Potts Glass on Random Graphs
• Universal low-frequency vibrational modes in silica glasses
• Finitely connected vector spin systems with random matrix interactions
• Theory for the density of interacting quasi-localised modes in amorphous solids
• The de Almeida-Thouless line in vector spin glasses
• High q-State Clock Spin Glasses in Three Dimensions and the Lyapunov Exponents of Chaotic Phases and Chaotic Phase Boundaries
• Diluted Mean-Field Spin-Glass Models at Criticality
• Mean-field model of interacting quasilocalized excitations in glasses
• Marginal stability of soft anharmonic mean field spin glasses
• Finitely coordinated models for low-temperature phases of amorphous systems
• Hessian spectrum at the global minimum of high-dimensional random landscapes
• Odd q-State Clock Spin-Glass Models in Three Dimensions, Asymmetric Phase Diagrams, and Multiple Algebraically Ordered Phases
• Approximating the XY model on a random graph with a $q$-state clock model
• Absence of Mobility Edge in Short-range Uncorrelated Disordered Model: Coexistence of Localized and Extended States
• Thermal origin of quasi-localised excitations in glasses
• Universality in Glassy Low-Temperature Physics
• The nature of non-phononic excitations in disordered systems
• The random field XY model on sparse random graphs shows replica symmetry breaking and marginally stable ferromagnetism
• The marginally stable Bethe lattice spin glass revisited
• Fully localized and partially delocalized states in the tails of Erdös-Rényi graphs in the critical regime
• Mean-field description for the architecture of low-energy excitations in glasses
• Delocalization transition in low energy excitation modes of vector spin glasses
• The metastable minima of the Heisenberg spin glass in a random magnetic field
• Bose-Einstein-Like condensation of deformed random matrix: A replica approach
• The most uniform distribution of points on the sphere
• Comparison of Gabay-Toulouse and de Almeida-Thouless instabilities for the spin glass XY model in a field on sparse random graphs
• The mean and variance of the distribution of shortest path lengths of random regular graphs
• The full replica symmetry breaking in the Ising spin glass on random regular graph
• Low energy excitations of mean-field glasses
• Multifractality and statistical localization in highly heterogeneous random networks
• The Ising spin glass on random graphs at zero temperature: not all spins are glassy in the glassy phase
• Analytical solution to Heisenberg spin glass models on sparse random graphs and their de Almeida-Thouless line

Cited by in corpus (2)

• Further testing the validity of generalized heterogeneous-elasticity theory for low-frequency excitations in structural glasses
• Statistical mechanics of vector Hopfield network near and above saturation