Random-field Ising and $O(N)$ models: Theoretical description through the functional renormalization group

Abstract

We review the theoretical description of the random field Ising and $O(N)$ models obtained from the functional renormalization group, either in its nonperturbative implementation or, in some limits, in perturbative implementations. The approach solves some of the questions concerning the critical behavior of random-field systems that have stayed pending for many years: What is the mechanism for the breakdown of dimensional reduction and the breaking of the underlying supersymmetry below $d=6$? Can one provide a theoretical computation of the critical exponents, including the exponent ψcharacterizing the activated dynamic scaling? Is it possible to theoretically describe collective phenomena such as avalanches and droplets? Is the critical scaling described by 2 or 3 independent exponents? What is the phase behavior of the random-field $O(N)$ model in the whole ($N$, $d$) plane and what is the lower critical dimension of quasi-long range order for $N=2$? Are the equilibrium and out-of-equilibrium critical points of the RFIM in the same universality class?

Review submitted to the topical issue of EPJB on "Recent advances in the theory of disordered systems"; 19 pages, 7 figures

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• Chaos in the Bose-glass phase of a one-dimensional disordered Bose fluid
• Domain Growth and Aging in the Random Field XY Model: A Monte Carlo Study
• Functional renormalization group for multilinear disordered Langevin dynamics II: Revisiting the $p=2$ spin dynamics for Wigner and Wishart ensembles
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• Wrinkling transition in quenched disordered membranes at two loops
• Dimensional reduction breakdown and correction to scaling in the random-field Ising model
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• Mott-glass phase induced by long-range correlated disorder in a one-dimensional Bose gas
• Energy minima and ordering in ferromagnets with quenched randomness
• On the breakdown of dimensional reduction and supersymmetry in random-field models
• Theory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles
• Crumpled-to-flat transition of quenched disordered membranes at two-loop order
• An Exactly Solvable Model of Randomly Pinned Charge Density Waves in Two Dimensions
• Functional-renormalization-group approach to strongly coupled Bose-Fermi mixtures in two dimensions
• Interplay of Quantum and Thermal Fluctuations in Two-Dimensional Randomly Pinned Charge Density Waves
• Journey from the Wilson exact RG towards the Wegner-Morris Fokker-Planck RG and the Carosso field-coarsening via Langevin stochastic processes
• Emergence of a random field at the yielding transition of a mean-field Elasto-Plastic model
• Low-temperature behavior of the $O(N)$ models below two dimensions
• Renormalization group approach to the elastic properties of graphene bilayers
• The Schwartz-Soffer and more inequalities for random fields
• Critical exponents of the Ising model with quenched structural disorder and long-range interactions at spatial dimension $d=3$