The weak and strong disorder regimes in the continuous random field Ising model

TL;DR

The authors develop a nonperturbative framework based on the distributional zeta-function (DZF) to study quenched disorder in the continuous random-field Ising model (RFIM). By averaging at the level of the effective action, they derive exact $k$-component actions and analyze both weak and strong disorder regimes. In the weak-disorder limit, the infrared structure generates correlated scaling fields with a $1/p^4$ behavior, shifts the upper critical dimension to $d_c^{+}=6$, and yields a minimal infrared action consistent with Cardy decomposition and a supersymmetry-inspired coupling constraint. In the strong-disorder regime, the action diagonalizes to a quadratic form with a discrete spectrum of massive modes, eliminating massless excitations and replacing true criticality with a disorder-driven crossover. Overall, the DZF framework unifies weak and strong disorder descriptions, enabling systematic RG analyses across dimensions and clarifying the infrared degrees of freedom in disordered criticality.

Abstract

We present a nonperturbative analysis of the weak- and strong-disorder regimes of the continuous random-field Ising model using the distributional zeta-function method. By performing the quenched-disorder average at the level of the effective action, we derive exact quadratic and interaction terms. In the weak-disorder limit, we show that the infrared structure of the two-point correlation functions yields a decomposition of the physical field into correlated components with distinct scaling dimensions. This mechanism exhibits the characteristic $1/p^4$ behavior, which shifts the upper critical dimension to $d_c^{+}=6$. The universal critical behavior of the RFIM near this dimension is governed by a minimal infrared effective action. In the strong-disorder regime, we obtain an exact diagonal quadratic action with a discrete spectrum of massive modes. Here, the absence of massless modes implies the absence of conventional criticality. The resulting spectral representation of correlation functions converges rapidly and remains well controlled in the infrared regime.