Modified hyperscaling relation for phase transitions under random fields

Abstract

A renormalization-group analysis is presented for the internal energies near a second-order phase boundary under random fields, presuming flow to a strong-coupling fixed point. In agreement with the recent result of Bray and Moore, we derive a modified hyperscaling relation, 2-α=(d-$y_{\mathrm{-}}$)ν, where $y_{\mathrm{-}}$ is the strong-coupling runaway eigenvalue exponent. We find the bound $y_{\mathrm{-}}$≤d/2 for Ising or n>1 component spins. For the three-dimensional random-field Ising transition, this result is incompatible with the separately reported exponents α≃0 and ν≃1.5.