Hierarchies of Landau-Lifshitz-Bloch equations for nanomagnets: A functional integral framework

Abstract

We propose a functional integral framework for the derivation of hierarchies of Landau-Lifshitz-Bloch (LLB) equations that describe the flow towards equilibrium of the first and second moments of the magnetization. The short scale description is defined by the stochatic Landau-Lifshitz-Gilbert equation, under both Markovian or non-Markovian noise, and takes into account interaction terms that are of practical relevance. Depending on the interactions, different hierarchies on the moments are obtained in the corresponding LLB equations. Two closure Ansätze are discussed and tested by numerical methods that are adapted to the symmetries of the problem. Our formalism provides a rigorous bridge between the atomistic spin dynamics simulations at short scales and micromagnetic descriptions at larger scales.

Emerging magnetic nutation

“…To better understand the space of states of the magnetization, it will be useful to adapt the techniques used in ref. [35] and to to understand the microscopic degrees of freedom that can define the bath in an invariant way it is necessary to implement the program that is sketched in ref. [44].…”

“…Therefore, m(t) becomes a stochastic process, as well; moreover, the noise, that enters additively in the equation for δω, becomes multiplicative for m(t); which implies that its correlation functions acquire a non-trivial dependence on the temperature, defined through the bath. This is, often described as a "breakdown" of the fluctuation-dissipation theorems [34,35]. However, what this, simply, means is that the nonlinearities induce a non-trivial, but quite transparent, dependence of the noise on the dynamics of the magnetization; the two are, just, intertwined in a way that is more subtle than hitherto acknowledged.…”

Multi-fidelity machine-learning with uncertainty quantification and Bayesian optimization for materials design: Application to ternary random alloys

“…Our results demonstrated the efficiency of our a MFGP framework to fuse in the predictions and therefore bridge the gap between two of the most commonly employed atomistic simulation approaches (DFT and molecular dynamics), each of them acting at very different length and time scales. The same methodology remains valid for different scales, and could be leveraged to build materials modelling road-maps 50 by understanding the existing correlations between atomistic methods [51][52][53] and associated higher scale coarse-grained 54 or continuum numerical models 55,56 .…”

Probing magneto-elastic phenomena through an effective spin-bath coupling model

“…The technical details pertaining to the derivation of the expression of δS k (t ′′ ) δη j (t ′ ) can be found, in considerable length, in the supplementary material to ref. [36].…”

“…However the reader can find the framework for writing the corresponding equations for any finite value of τ , in appendix A. For a single spin, in a colored bath, the corresponding equations can be found in reference [36].…”