Long-range correlated random magnetic fields in the nonlinear<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>σ</mml:mi></mml:math>model

Abstract

The effect of the long-range correlated random magnetic fields which behave like $〈h(x)h(y)〉$ $\ensuremath{\sim}\frac{1}{{|x\ensuremath{-}y|}^{d\ensuremath{-}\ensuremath{\sigma}}}$ on the critical phenomena is discussed with the use of the nonlinear $\ensuremath{\sigma}$ model. The crossover between the two critical behaviors dominated by the long-range and short-range disorder fixed points is shown to occur at $(m\ensuremath{-}2)\ensuremath{\sigma}\ensuremath{-}d+4=0$ ($m$ is the spin dimensionality). The critical exponents calculated at the long-range disorder fixed point around the lower critical dimensionality $4+\ensuremath{\sigma}$ are not the same as those in $d\ensuremath{-}2$ expansions in the pure system.