On the symmetry of a Preisach map

At the very heart of the successful phenomenological model of magnetic hysteresis there is the so called Preisach distribution. In the existing literature it is implicitly assumed, that this distribution has a mirror symmetry. We show, by simple and convincing example, that this common assumption is plainly wrong. Dropping it, we gain the ability to model not only the usual hysteresis loops (major and minor) more accurately than ever before, but also those displaying the exchange bias effect, what is impossible within the framework of the symmetrical Preisach model. It is hoped, that our observation paves the way towards the unified description of all hysteretic systems, including, but not necessarily limited to, superconductors, (multi)layered structures, nanocrystalline materials, patterned media, and – perhaps – the other non-magnetic hysteretic phenomena. Introduction The major hysteresis loop, observed in the sizable samples of homogeneous ferromagnetic materials, exhibits well known symmetry: Mlb(H) = −Mub(−H), (1) where Mlb (Mub) denotes the lower (upper) branch of the sample’s magnetization, M vs. exciting field H . The hysteresis curve, not necessarily the major loop, but also the response to the arbitrary sequence of exciting fields as well, can be described, or modelled, in many ways. One of them, the Classical Preisach Model (CPM), was first proposed by Ferenc Preisach [1] and then subsequently developed, generalized and tested by many researchers. In this model, the change of the sample’s magnetization is expressed by the double integral: ∆M = M(Hf)−M(Hi) = 2Ms ∫ ∫ H↑≥H↓ Hi≤H↑≤Hf dH↑ dH↓ ̺ (H↑,H↓) , (2) for the monotonously increasing field (‘i’ – the initial state, ‘f’ – final) and by ∆M = M(Hf)−M(Hi) = −2Ms ∫ ∫ H↑≥H↓ H f ≤H↓≤Hi dH↑ dH↓ ̺ (H↑,H↓) , (3) if the field is monotonously decreasing. Ms is the saturation magnetization, while the distribution ̺ (H↑,H↓) ≥ 0, supported over the domain H↑ ≥ H↓ is called the Preisach density. With the additional condition: