Critical Exponents of the Random Field Hierarchical Model

Abstract

We have studied the one dimensional Dyson hierarchical model in presence of a random field. This is a long range model where the interactions scale with the distance with a power law-like form J(r) ~ r^{-ρ} and we can explore mean field and non-mean field behavior by changing ρ. Thus, it can be used to approach the phase transitions in finite-dimensional disordered models. We studied the model at T=0 and we numerically computed its critical exponents in the non-mean field region for Gaussian disorder. We then computed an analytic expression for the critical exponent δ, that holds in the non-mean field region, and we noted an interesting relation between the critical exponents of the disordered model and the ones of the pure model, that seems to break down in the non-mean field region. We finally compare our results for the critical exponents with the expected ones in D-dimensional short range models and with the ones of the straightforward one dimensional long range model.

9 pages, 3 figures

References in corpus (13)

• Study of the de Almeida-Thouless line using power-law diluted one-dimensional Ising spin glasses
• Diluted one-dimensional spin glasses with power law decaying interactions
• Monte Carlo studies of the one-dimensional Ising spin glass with power-law interactions
• The correspondence between long-range and short-range spin glasses
• Finite size scaling in Ising-like systems with quenched random fields: Evidence of hyperscaling violation
• Overlap Interfaces in Hierarchical Spin-Glass models
• Nonlinear Aspects of the Renormalization Group Flows of Dyson's Hierarchical Model
• Colloid-polymer mixtures in random porous media: Finite size scaling and connected versus disconnected susceptibilities
• No-passing Rule in the Ground State Evolution of the Random-Field Ising Model
• A critical Dyson hierarchical model for the Anderson localization transition
• Random field Ising model : statistical properties of low-energy excitations and of equilibrium avalanches
• Ensemble renormalization group for the random field hierarchical model
• The renormalization flow of the hierarchical Anderson model at weak disorder

Cited by in corpus (6)

• Many-Body Localization : construction of the emergent local conserved operators via block real-space renormalization
• Meta-stable states in the hierarchical Dyson model drive parallel processing in the hierarchical Hopfield network
• Critical behavior of the random-field Ising model with long-range interactions in one dimension
• Dyson Hierarchical Long-Ranged Quantum Spin-Glass via real-space renormalization
• Existence of long-range order in random-field Ising model on Dyson hierarchical lattice
• Statistical mechanics of the spherical hierarchical model with random fields