Hysteresis Loop Critical Exponents in 6 − ǫ Dimensions

The hysteresis loop in the zero–temperature random–field Ising model exhibits a critical point as the width of the disorder increases. Above six dimensions, the critical exponents of this transition, where the “infinite avalanche” first disappears, are described by mean–field theory. We expand the critical exponents about mean–field theory, in 6 − ǫ dimensions, to first order in ǫ. Despite ǫ = 3, the values obtained agree reasonably well with the numerical values in three dimensions. PACS numbers: 75.60.Ej, 64.60.Ak, 81.30.Kf Typeset using REVTEX 1 In a previous paper [1], we modeled hysteresis in magnetic and martensitic systems using the random–field Ising model at zero temperature. The model exhibited two features characteristic of these systems: the return–point memory effect and avalanche–generated noise. (The noise is called Barkhausen noise in magnetic systems and acoustic emission in martensites.) We also discovered a critical point, separating smooth hysteresis loops at large disorder where all avalanches are finite, from discontinuous hysteresis loops at small disorder where one avalanche turns over a fraction of the whole system. Here we study this critical point in an expansion about mean–field theory. Figure 1(a) shows a schematic of the phase diagram for our model defined by eq. (2) below. The vertical axis H is the external field. The horizontal axis R is the width of the probability distribution of the random fields fi acting on each spin. The bold line represents the location Hc(R) at which the infinite avalanche occurs, when the field H(t) is adiabatically increasing from an initial state where all spins were pointing down. At small disorder, the first spin to flip easily pushes over its neighbors, and the transition happens in one burst (the infinite avalanche). At large enough disorder, the coupling between spins becomes negligible, and most spins flip by themselves: no infinite avalanche occurs. At a special value of the randomness R = Rc the infinite avalanche disappears. We find a critical point with two relevant variables r ≡ (Rc − R)/Rc and h ≡ (H − Hc(Rc)) [1]. At this point we find a universal scaling law for the magnetization m ≡ (M −Mc(Rc)) m ∼ |r|M±(h/|r| ) , (1) where the ± refers to the sign of r. We use a soft-spin version of the random field Ising model, whose energy at a given spin configuration {si} is H = − ∑