Universality of the magnetization ripple

The magnetisation ripple is a microstructure formed by the magnetization in thin ferromagnetic films, with its characteristic texture orthogonal to the average magnetization direction. It is triggered by the random orientation of the grains in a poly-crystalline material, which through the crystalline anisotropy effectively acts like a random field of white noise character on the scales relevant for the description of the ripple. From a physics point of view, the important feature of the ripple is its universal character, that is, the microstructure does not depend on the precise composition and geometry of the grains.

The term 'universality' is linked to a phenomenon where various random systems exhibit the same behaviour on large scales. The simplest example of universality is Donsker's invariance principle, a central limit theorem, which recovers Brownian motion as the limit in law of a suitably rescaled random walk (built from i.i.d. random variables of expectation 0 and variance 1).

The main goal of our work [5] is to show the universality of the magnetisation ripple. Following [4], we describe the ripple as minimizer of a reduced energy functional for the transversal component of the magnetization vector, composed by the sum of three contributions: the exchange energy, the stray field energy, and a noise term that acts like a random transversal field of white-noise character.

To address the universal character of the ripple, we consider approximations of white noise within a wide class of random fields (which are not necessarily Gaussian). This is motivated by the source of randomness in our model: the random grain orientations in the material suggest a modelling through non-Gaussian fields. Our main result shows that the law of the energy functional is stable under approximations of white noise in this class of random fields, in the sense that one can recover (the law of) the energy functional corresponding to white noise as the limit in law of the energy functionals corresponding to the approximations.

At this point, it is worth mentioning that to study the universality of the ripple, it is more natural to work on the level of the energy functional rather than the Euler-Lagrange equation, of which the minimizers are weak solutions. The reason is twofold: On the one hand, we cannot expect uniqueness of minimizers due to the non-convexity of the energy functional. On the other hand, the Euler-Lagrange equation is more amenable to a perturbative approach as developed in [4], allowing only a small-data analysis; however, we want to get rid of the smallness assumption. On a technical level, we use De Giorgi's \(\Gamma\)-convergence to endow the space of functionals with a topology which implies convergence of sequences of minimizers.

One of the main challenges in showing the universality of the ripple arises from the roughness of white noise. This is a common problem in the theory of singular Stochastic Partial Differential Equations (SPDEs), which has become a very active field in recent years. The word singular relates to the fact that the driving noise of these equations is so irregular that their nonlinear terms (which usually involve products of the solution and its derivatives) are not classically defined. We refer the reader to [3] for a more detailed exposition of the theory of singular SPDEs.

In the case of the ripple, the irregularity of the noise has two important consequences:

1. The energy functional is almost surely unbounded from below. Therefore, the energy functional is in need of a renormalization.
2. In a more detailed analysis of the renormalized energy functional (and in analogy with rough-path theory), in order to define certain products of functionals of the noise, one needs to extend the distribution of white noise in a way similar to the lift of a Brownian motion to its first iterated integral, see for example [2, Chapter 3].

A central tool in our probabilistic analysis is the spectral gap assumption on the class of probability measures for which our universality statement holds, that is, the measures satisfy an “infinite dimensional” Poincaré (also called spectral gap) inequality, which provides a good substitute for Gaussian calculus.