Modified scaling relation for the random-field Ising model

We investigate the low-temperature critical behavior of the three-dimensional random-field Ising ferromagnet. By a scaling analysis we find that in the limit of temperature T→0 the usual scaling relations have to be modified as far as the exponent α of the specific heat is concerned. At zero temperature, the Rushbrooke equation is modified to α+2β+γ=1, an equation which we expect to be valid also for other systems with similar critical behavior. We test the scaling theory numerically for the three-dimensional random-field Ising system with Gaussian probability distribution of the random fields by a combination of calculations of exact ground states with an integer optimization algorithm and Monte Carlo methods. By a finite-size scaling analysis we calculate the critical exponents ν≈1.0, β≈0.05, γ̄≈2.9, γ≈1.5 and α≈−0.55.