## 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 toward equilibrium of the first and second moments of the magnetization. The short-scale description is defined by the stochastic 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.

- Authors:

- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- CEA DAM/Le Ripault, Monts (France)
- Univ. d'Orleans (France)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1477321

- Alternate Identifier(s):
- OSTI ID: 1475088

- Report Number(s):
- SAND-2018-10424J

Journal ID: ISSN 2470-0045; PLEEE8; 668215

- Grant/Contract Number:
- AC04-94AL85000; NA0003525

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physical Review E

- Additional Journal Information:
- Journal Volume: 98; Journal Issue: 4; Journal ID: ISSN 2470-0045

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY

### Citation Formats

```
Tranchida, Julien, Thibaudeau, Pascal, and Nicolis, Stam. Hierarchies of Landau-Lifshitz-Bloch equations for nanomagnets: A functional integral framework. United States: N. p., 2018.
Web. doi:10.1103/PhysRevE.98.042101.
```

```
Tranchida, Julien, Thibaudeau, Pascal, & Nicolis, Stam. Hierarchies of Landau-Lifshitz-Bloch equations for nanomagnets: A functional integral framework. United States. doi:10.1103/PhysRevE.98.042101.
```

```
Tranchida, Julien, Thibaudeau, Pascal, and Nicolis, Stam. Mon .
"Hierarchies of Landau-Lifshitz-Bloch equations for nanomagnets: A functional integral framework". United States. doi:10.1103/PhysRevE.98.042101. https://www.osti.gov/servlets/purl/1477321.
```

```
@article{osti_1477321,
```

title = {Hierarchies of Landau-Lifshitz-Bloch equations for nanomagnets: A functional integral framework},

author = {Tranchida, Julien and Thibaudeau, Pascal and Nicolis, Stam},

abstractNote = {We propose a functional integral framework for the derivation of hierarchies of Landau-Lifshitz-Bloch (LLB) equations that describe the flow toward equilibrium of the first and second moments of the magnetization. The short-scale description is defined by the stochastic 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.},

doi = {10.1103/PhysRevE.98.042101},

journal = {Physical Review E},

number = 4,

volume = 98,

place = {United States},

year = {2018},

month = {10}

}

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