Long-range random-field models: scaling theory and 1/n expansion

Dimensional reduction breakdown and correction to scaling in the random-field Ising model

We provide a theoretical analysis by means of the nonperturbative functional renormalization group (NP-FRG) of the corrections to scaling in the critical behavior of the random-field Ising model (RFIM) near the dimension d_{DR}≈5.1 that separates a region where the renormalized theory at the fixed point is supersymmetric and critical scaling satisfies the d→d-2 dimensional reduction property (d>d_{DR}) from a region where both supersymmetry and dimensional reduction break down at criticality (d<d_{DR}). We show that the NP-FRG results are in very good agreement with recent large-scale lattice simulations of the RFIM in d=5 and we detail the consequences for the leading correction-to-scaling exponent of the peculiar boundary-layer mechanism by which the dimensional-reduction fixed point disappears and the dimensional-reduction-broken fixed point emerges in d_{DR}.

Phase transitions of the random-bond Potts chain with long-range interactions

We study phase transitions of the ferromagnetic q-state Potts chain with random nearest-neighbor couplings having a variance Δ^{2} and with homogeneous long-range interactions, which decay with distance as a power r^{-(1+σ)}, σ>0. In the large-q limit the free-energy of random samples of length L≤2048 is calculated exactly by a combinatorial optimization algorithm. The phase transition stays first order for σ<σ_{c}(Δ)≤0.5, while the correlation length becomes divergent at the transition point for σ_{c}(Δ)<σ<1. In the latter regime the average magnetization is continuous for small enough Δ, but for larger Δ-according to the numerical results-it becomes discontinuous at the transition point, thus the phase transition is expected of mixed order.