Phase transitions in ferromagnets and singularities (Q1077027)

The paper is devoted to the phenomenological investigation of the critical phenomena in ferromagnets by means of the modern theory of singularities. In order to take into account the hysteresis phenomena the authors postulate that the set of thermodynamical parameters (M,T) (magnetization and temperature) corresponding to \(T<t_ c\) \((t_ c\) is the critical temperature) consists of two copies of a halfplane (M\(\in {\mathbb{R}}\), \(T\leq T_ c)\) glued along their boundaries L which is the critical isotherm (Hypothesis I). Then they require that magnetic field H restricted to L is infinitesimally cubic in the vicinity of the critical point (Hypothesis II) and that the same form have all curves obtained from L by small smooth perturbations (Hypothesis III). A similar hypothesis is applied to T restricted to the isoenergetic curve \(E=e_ c\) \((e_ c\) is critical energy). These hypotheses and the symmetry of hysteresis curve determine uniquely the general infinitesimal form H and T as functions of (H,E). As result the authors find that \(M\sim (T-t_ c)^{2/3}\), \(\chi \sim (T-t_ c)^{-4/3}\), \(T\nearrow t_ c\) where \(\chi\) (T) is the susceptibility for \(H=0\). Thus the critical exponents \(\beta\) and \(\gamma\) ' have the values \(\beta =2/3\), \(\gamma '=-4/3\).